CELL THEORY OF NATURE · 1.1 Atoms of atoms. The Atomic Theory of Matter and the Cell Theory of Life are now well established, Leib-niz’s monadology and Whitehead’s philosophy

CELL THEORY OF NATURE

Work in process

David Ritz Finkelstein1

October 17, 2008

1Physics, Georgia Institute of Technology, Atlanta, Georgia. [email protected]

2

Time is the number of motion with regard to before and after.Aristotle, Physics

I would indeed admit these infinitely small spaces and times in geometry, forthe sake of invention, even if they are imaginary. But I am not sure whetherthey can be admitted in nature. G. W. Leibniz [3]

It is, for example, true that the result of two successive acts is unaffected by theorder in which they are performed; and there are at least two other laws whichwill be pointed out in the proper place. These will perhaps to some appear soobvious as to be ranked among necessary truths, and so little important as tobe undeserving of special notice. And probably they are noticed for the firsttime in this Essay. Yet it may with confidence be asserted, that if they wereother than they are, the entire mechanism of reasoning, nay the very laws andconstitution of the human intellect, would be vitally changed. A Logic mightindeed exist, but it would no longer be the Logic we possess. G. Boole [14]

To be sure, it has been pointed out that the introduction of a space-time contin-uum may be considered as contrary to nature in view of the molecular structureof everything which happens on a small scale. It is maintained that perhapsthe success of the Heisenberg method points to a purely algebraical method ofdescription of nature, that is, to the elimination of continuous functions fromphysics. Then, however, we must also give up, by principle, the space-timecontinuum. It is not unimaginable that human ingenuity will some day findmethods which will make it possible to proceed along such a path. At thepresent time, however, such a program looks like an attempt to breathe inempty space. A. Einstein [29]

And so I suggested to myself that electrons cannot act on themselves; they canonly act on other electrons. This means that there is no field at all.

R. P. Feynman [34]

Contents

1 Strata of actuality 51.1 Atoms of atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Simplicity and stability . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.2 Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.3 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.4 Beneath geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.5 The cosmic crystal film . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.6 Praxics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.1.7 Indefinite probability forms . . . . . . . . . . . . . . . . . . . . . . . 171.1.8 Probability vector spaces and algebras . . . . . . . . . . . . . . . . . 171.1.9 The need for full quantization . . . . . . . . . . . . . . . . . . . . . . 181.1.10 A cellular hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.1.11 Quantum time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2 The idea of the queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.2.1 Strata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.2.2 The Clifford algebras of Fermi and Dirac . . . . . . . . . . . . . . . 361.2.3 The origins of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.2.4 The origins of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.2.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.2.6 Q terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.2.7 The origins of g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.2.8 Fields and queues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.2.9 Dynamical law of the queue . . . . . . . . . . . . . . . . . . . . . . . 451.2.10 The vacuum queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.2.11 The cosmic crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.3.1 Canonical quantization . . . . . . . . . . . . . . . . . . . . . . . . . 531.3.2 Full quantization strategy . . . . . . . . . . . . . . . . . . . . . . . . 551.3.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581.3.4 Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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4 CONTENTS

1.3.5 Cellularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591.3.6 The choice of statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 601.3.7 Internal structure of the photon and graviton . . . . . . . . . . . . . 621.3.8 Indefinite forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631.3.9 The representation of symmetry . . . . . . . . . . . . . . . . . . . . 641.3.10 Finiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

1.4 The groups of nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661.4.1 Canonical strata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681.4.2 Metric forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691.4.3 Fully quantum strata . . . . . . . . . . . . . . . . . . . . . . . . . . . 691.4.4 Fully quantum self-organization . . . . . . . . . . . . . . . . . . . . . 701.4.5 Full quantization tactics . . . . . . . . . . . . . . . . . . . . . . . . . 71

1.5 Fully quantum regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 741.5.1 Fully quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 741.5.2 Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741.5.3 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781.5.4 Real quantum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 791.5.5 Root vectors and quanta . . . . . . . . . . . . . . . . . . . . . . . . . 801.5.6 Full Fermi quantization . . . . . . . . . . . . . . . . . . . . . . . . . 811.5.7 Physical Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

1.6 Gravity and other gauge fields . . . . . . . . . . . . . . . . . . . . . . . . . . 831.6.1 History as quantum variable . . . . . . . . . . . . . . . . . . . . . . . 841.6.2 Fully quantum equivalence principle . . . . . . . . . . . . . . . . . . 851.6.3 Weyl gauge strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 881.6.4 Kaluza gauge strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 891.6.5 Queue gauge strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 901.6.6 Fully quantum gauge group . . . . . . . . . . . . . . . . . . . . . . . 911.6.7 The space-time truss . . . . . . . . . . . . . . . . . . . . . . . . . . . 921.6.8 The gravitational and gauge potentials . . . . . . . . . . . . . . . . . 931.6.9 Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

1.7 Unifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 951.7.1 Being and becoming . . . . . . . . . . . . . . . . . . . . . . . . . . . 951.7.2 Gravity and quantum theory . . . . . . . . . . . . . . . . . . . . . . 961.7.3 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961.7.4 Non-commutativity and granularity . . . . . . . . . . . . . . . . . . 971.7.5 System and metasystem . . . . . . . . . . . . . . . . . . . . . . . . . 98

1.8 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

CONTENTS 5

2 Linear praxics 1012.1 Praxics in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.2 Heisenberg and von Neumann praxics . . . . . . . . . . . . . . . . . . . . . 102

2.2.1 The system itself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052.2.2 The equatorial bulge in Hilbert space . . . . . . . . . . . . . . . . . 1062.2.3 Commutative reduction . . . . . . . . . . . . . . . . . . . . . . . . . 107

2.3 Standard semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072.3.1 The orthogonal group . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.3.2 Probability vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.3.3 The probability form . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102.3.4 The linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102.3.5 The projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

2.4 Change is a quantum effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122.5 Simple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122.6 Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1132.7 Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1132.8 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1142.9 States, proper and coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . 115

2.9.0.1 Proper state . . . . . . . . . . . . . . . . . . . . . . . . . . 1162.9.0.2 Coordinate state . . . . . . . . . . . . . . . . . . . . . . . . 116

2.10 Praxiology and its singular limit . . . . . . . . . . . . . . . . . . . . . . . . 1162.10.1 Ritz combination rule. . . . . . . . . . . . . . . . . . . . . . . . . . . 1222.10.2 Probability Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . 1222.10.3 System catenation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232.10.4 Relation to probability . . . . . . . . . . . . . . . . . . . . . . . . . . 1232.10.5 The von Neumann ambiguity . . . . . . . . . . . . . . . . . . . . . . 1242.10.6 Schrodinger’s frozen cat . . . . . . . . . . . . . . . . . . . . . . . . . 1322.10.7 Fundamental law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

3 Polynomial quantum logic 1373.1 Set algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.1.1 The random set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.1.2 The queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383.1.3 Input-output processes . . . . . . . . . . . . . . . . . . . . . . . . . . 139

3.2 Clifford algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413.2.1 Clifford semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1423.2.2 Fermi Clifford algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 1423.2.3 Fermi vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1433.2.4 Grade operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443.2.5 Mean-square form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

3.3 Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6 CONTENTS

3.3.1 Choosing a quantification . . . . . . . . . . . . . . . . . . . . . . . . 1453.3.2 The cumulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.3.3 Algebra unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.3.4 Spinor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

3.4 Fermi algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.4.1 The duplex space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523.4.2 A spin is a queue of semiquanta. . . . . . . . . . . . . . . . . . . . . 1533.4.3 Spin-statistics correlation . . . . . . . . . . . . . . . . . . . . . . . . 155

3.5 Clifford statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583.5.1 Spin-statistics anomaly . . . . . . . . . . . . . . . . . . . . . . . . . 159

3.6 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

4 Exponential quantum logics 1634.1 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4.1.1 Baugh numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644.1.2 Random sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.1.3 Quantum cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.1.4 Bracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664.1.5 Critique of the brace operation . . . . . . . . . . . . . . . . . . . . . 1684.1.6 Queues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4.2 Simplifying quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.2.1 Choose a simple Lie algebra . . . . . . . . . . . . . . . . . . . . . . . 1724.2.2 Choose a vacuum organization . . . . . . . . . . . . . . . . . . . . . 1724.2.3 Choose a faithful irreducible representation . . . . . . . . . . . . . . 172

4.3 Fermi full quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1744.4 Fully quantum vacuum organization . . . . . . . . . . . . . . . . . . . . . . 175

4.4.1 Stratum assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5 Quantum space-times 1795.1 Problems of classical space-time . . . . . . . . . . . . . . . . . . . . . . . . . 182

5.1.1 Structural instability of time . . . . . . . . . . . . . . . . . . . . . . 1825.1.2 Dynamical instability of time . . . . . . . . . . . . . . . . . . . . . . 182

5.2 Earlier quantum space-times . . . . . . . . . . . . . . . . . . . . . . . . . . . 1845.2.1 Event energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1845.2.2 Indefinite probability form . . . . . . . . . . . . . . . . . . . . . . . . 1865.2.3 Feynman space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.2.4 Snyder space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1895.2.5 Segal space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1895.2.6 Penrose space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.2.7 Palev statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.2.8 Vilela-Mendes space . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

CONTENTS 7

5.2.9 Baugh, Shiri-Garakani spaces . . . . . . . . . . . . . . . . . . . . . . 1945.3 Fully quantum event spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

5.3.1 Fully quantum spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1965.3.2 A fully quantum space-time . . . . . . . . . . . . . . . . . . . . . . . 197

6 Fully quantum kinematics 2016.1 History space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

6.1.1 Canonical quantum histories . . . . . . . . . . . . . . . . . . . . . . 2026.1.2 Fully quantum histories . . . . . . . . . . . . . . . . . . . . . . . . . 2036.1.3 Physics without functions . . . . . . . . . . . . . . . . . . . . . . . . 2036.1.4 Field variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2046.1.5 The cosmic crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2066.1.6 The organization of the imaginary unit . . . . . . . . . . . . . . . . 2086.1.7 Statistical form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

6.2 Fully quantum scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2106.2.1 Experiment time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2126.2.2 Momentum vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2126.2.3 Quantum topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2166.2.4 Origins of the coordinates . . . . . . . . . . . . . . . . . . . . . . . . 2176.2.5 Fully quantum fermions . . . . . . . . . . . . . . . . . . . . . . . . . 217

6.3 Fully quantum gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2196.3.1 Fully quantum events . . . . . . . . . . . . . . . . . . . . . . . . . . 2196.3.2 Time form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2206.3.3 Fully quantum gravitational potentials . . . . . . . . . . . . . . . . . 221

6.4 Construction of the vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . 2236.4.1 Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2276.4.2 Antiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2286.4.3 Flavor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

6.5 Fully quantum gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . 2286.5.1 Fully quantum gauging . . . . . . . . . . . . . . . . . . . . . . . . . 229

6.6 Fully quantum metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2316.6.1 Fully quantum probability forms . . . . . . . . . . . . . . . . . . . . 2326.6.2 Fully quantum causality form . . . . . . . . . . . . . . . . . . . . . . 233

6.7 Covariant differentiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2376.8 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2386.9 Fully quantum covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

7 Fully quantum dynamics 2437.1 The history vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2437.2 Higher-order time derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 2457.3 Spinorial dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

8 CONTENTS

7.4 Gravity action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2467.4.1 Cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . 246

8 Output 249

9 ACKNOWLEDGMENT 251

BIBLIOGRAPHY 251

INDEX 258

CONTENTS 9

THINQThink quantum.

10 CONTENTS

Chapter 1

Strata of actuality

1.1 Atoms of atoms.

The Atomic Theory of Matter and the Cell Theory of Life are now well established, Leib-niz’s monadology and Whitehead’s philosophy of organism are cellular, and Von Neumannformulated a cellular automaton model of reproduction, but an atomic or cellular the-ory of nature as a whole is still nascent. Here I [hopefully] develop such a theory to thepoint where it can accommodate the current gauge groups and predict particles and theirinteractions.

Two deep but widespread philosophical reifications seem to have unduly blocked progressin physical understanding since the discoveries of general relativity and the quantum theory,preventing their synthesis within one more comprehensive theory. One is the postulationof absolute space-time, shared by Einstein and Heisenberg, which can only be corrected bya grand extension of relativity. The other is the inconsistent but common formulation ofquantum theory expressed in the term “state vector”, which blocks the required relativiza-tion. Here the common formulation is replaced by one closer to Heisenberg’s and Bohr’s,which it is then natural to apply to space-time.

The formulation is purely algebraic, as Einstein suggested, and so in principle it issimpler than the usual mix of classical differential geometry and quantum matrix algebra,but the algebra has an unfamiliar structural feature: Its elements are not only graded butranked. The algebra is stratified into nested subalgebras. An initial chapter presents themain ideas in familiar language, in preparation for a more formal development in laterchapters. So everything is said at least three times.

An important defect in the current quantum theories is the apparently seamless con-tinuity of time, which is representative of several continuities of special relativity generalrelativity, and the standard model. This leads to unphysical infinities in the predictions ofquantum field theory and to an unphysical singularity at the core of black holes. Infinityin, infinity out.

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12 CHAPTER 1. STRATA OF ACTUALITY

It is understood that time is what clocks meter out. This understanding then permitsthe question, What is time? What do clocks meter out? The ostensive definition givesmeaning to the structure question.

Naturally the continuity of time has no direct experimental support. The absence ofevidence for time jumps is not evidence of their non-existence.

The opposite possibility, discrete time, implies a breakdown of energy conservationfor which there is also no evidence. This possibility is not further contemplated here. Inthe past, discarding continuous symmetries has not led to improved theories, while smallvariations in the structure of the symmetry groups has. Therefore we follow that avenuehere.

We synthesize the continuous and discrete concepts of time in the sense that quantummechanics synthesizes wave and particle. We quantize time, rather than merely discretizeit. This is one way to a finite physics of the domain presently described by quantum fieldtheory and general relativity. A molecular theory of water resolves the singularity predictedat the core of vortices by continuum hydrodynamics of an ideal non-viscous fluid. A cellulartheory of the event space does the same for the gravitational singularities.

Dynamics operates on a higher logical stratum than time in present physical theory.Dynamical variables are functions of the time variable and not conversely. This madeit possible to quantize dynamical variables without quantizing the time variable, whichremained singular. To remedy this omission we make the stratification of physics moreexplicit, formulating a stratified algebra to express it.

Canonical quantization removed an infinity that resulted from the continuity of energyin classical physics, while respecting the continuous symmetries of the Lorentz and Poincaregroups. Canonical quantization repaired only the dynamical stratum, however, and thatonly partially, quantizing some quantities and not others, and it preserved the singularitiesof the lower strata intact.

Many concluded that a fuller quantization should eliminate the universal continuity oftime as canonical quantization eliminated the universal continuity of energy. One impetusfor the present effort is to eliminate infinities resulting from continuity while respecting allempirical continuous symmetries. But in fact the project took root when an encounter withthe ideas of von Neumann made it clear that the logic that Euclid used for points was buta singular limiting approximation to the quantum mode of reasoning that Heisenberg andBohr used for atoms. It seemed clear that in the long run the quantum mode of reasoningmust be used for all strata of physics.

For this purpose a stratified algebra and quantum theory are constructed. Gradedalgebras are familiar tools of quantum mechanics. A stratified algebra has both grade andrank.

Von Neumann proposed to regard quantum theory as a reform of Boolean logics thatreplaced distributive by non-distributive lattices. Boole himself favored an algebraic ratherthan a lattice formulation, and Heisenberg reformed it by replacing a commutative algebraby a non-commutatice one. The algebraic road is the only one that has led to new physical

1.1. ATOMS OF ATOMS. 13

theories, and it is followed here.A cellular logical engine that operates according to a stratified finite-dimensional quan-

tum logic would be finite on every stratum and still have continuous invariance groups likethose seen in nature. This motivated the earliest phase of this project, though neither vonNeumann’s logics nor his automaton theory are prominent in the product.

1.1.1 Simplicity and stability

The quantum theory is often described in negative terms: It is non-deterministic and non-commutative. Here we turn to the positive side of the ledger. What did physics gain bythis “painful renunciation”? (Bohr’s term.) Granted that physics is headed away fromclassical completeness and determinism, but towards what?

Simplicity, it is supposed here. The history of physics in the 20th century has suggestedto some that physics is on a slow climb from searching for a bcompletec and final theory tosearching for a bsimplec and provisory one.

This refers especially to Lie algebraic structure of our theories. A bsimplec (or birreduciblec)structure in general is one with no proper invariant substructures; that is, none but thetrivial two, itself and the empty structure. A simple Lie algebra, however, is required tobe non-abelian as well as having no proper invariant Lie subalgebras; this excludes the1-dimensional case.

A bsemisimplec, or decomposable, structure is a a union of simple structures. A struc-ture that is not semisimple is bcompoundc. A bsimplification strategyc is at least implied bybSegalc [66]. His principle is stretched here to accommodate the Pauli exclusion principle:

Physical systems have simple graded Lie algebras.This suggests that a compound graded Lie algebra that works is only an approximation

to a simple one that works better. The strategy that follows is to make all the graded Liealgebras that enter in the formulation of a physical theory simple by a variation of theirconstants. An arbitrarily small variation will always suffice, provided that constraints,attributable to organization, freeze out some variables.

We adopt this in the following but it does not go far enough; the problem it poses it isstill seriously undetermined. It is sharpened in two stages. First §1.4.5 gives a fully quan-tum (= Q; q = canonical quantization, in all its variant forms) kinematics. To determinedynamics, Q correspondents of the familiar principles of bminimal couplingc and bminimaldifferential orderc — which can be understood as maximal locality — are formulated andapplied to the existing theories of gravity and the standard model.

Group simplicity has immediate consequences for structural stability of the theory [66].This is defined as follows in the present context.

The various Lie algebras that can be defined on a fixed vector space have structuretensors that form a certain curved algebraic bstructure manifoldc in an associated tensorspace. An algebra with a neighborhood in the structure manifold composed of isomorphicalgebras is said to be bstructurally stablec, bregularc, bgenericc, brobustc, or brigidc. If on


the contrary every neighborhood of a given algebra contains a non-isomorphic algebra, thegiven algebra is bstructurally unstablec, bspecialc, bsingularc, or bfragilec. Semisimple Liealgebras are structurally stable and many bcompoundc Lie algebras are not.

Any unstable Lie algebra has structurally stable ones in every neighborhood [66]. Herewe suppose that one of these is more physical than the unstable one.

1.1.2 Locality

Locality has become an indispensable guide for theoretical physics since Newton used itto criticize his own law of gravity. The standard model freezes the gravitational field atthe zero value it has in special relativity, but draws nevertheless on the insights of New-ton, Faraday, and Einstein concerning dynamical locality of gravity and electromagnetism.extending them to hypercharge, electroweak, and strong interactions:

Assumption 1 (Locality) All action is by contact.

In the cellular quantum theory developed here, two cells are defined to be in contactwhen they share vertices, elements of a lower stratum. Then blocalityc restricts them tointeraction by bcontactc. Locality is imposed here by requiring the action operator to be apolynomial of low degree in swap operations.

Einstein’s law of gravity is the simplest localization of Newton’s. To take localityas seriously as quantum theory permits, one should attribute all global symmetries toorganizations of local structures by local interactions.

It is possible, to be sure, to have maximal information about a quantum particle andnot know where it is localized. Canonical quantum theory does not modify dynamicallocality when it introduces this kinematic non-locality.

Faraday incorporated this locality in his pre-quantum concept of field, Maxwell andEinstein developed the field concept into their pre-quantum theories, and Dirac developed itinto his quantum theories. The notion of local gauge invariance, the basis of the theories ofgravity and the standard model, is the peak of locality. Here a quantum reform of canonicalgauge theories is carried out while preserving the locality principle. The gauge group ariseshere from the invariance group of a cell of a stratum beneath that of space-time.

Quantum field theory was derived from a classical theory by bcanonical quantizationc.This was a reasonable first step, but there are signs that the journey has just begun.

Canonical quantization is a partial quantization.

It is lacking in both horizontal and vertical dimensions, in the following senses.In the first place, as was pointed out, canonical quantization leaves deeper strata

unquantized. Classical physics is stratified, in that it can be built up in strata of increasingcomplexity, also called levels, stages, generations, phases, types, and orders. The mediumusually used to represent such stratification is set theory. Sets of each stratum are made


up of elements of lower strata and are the elements of sets of higher strata, forming a greatladder of beings.

The number of rungs below a set on this ladder is called its brankc here. Stratum L,or S[L], consists of the sets of rank ≤ L.

Classical field theories, for example, represent a field as a continuous bfunctionc onspace-time to field values, and represent space-time as a set of events. Therefore a fieldbelongs to a higher bstratumc than the space-time event and the field value, which areelements of its elements, or its second elements. It is assumed here that nature too isstratified in this way. The constructs of bstratumc and brankc are therefore basic for thisdiscussion. They are developed further in §1.2.1.

A set of classical elements is classical. Therefore one cannot assemble a quantum systemout of classical elements. Since canonical quantization partially quantizes the dynamicalstratum, but leaves the deeper space-time stratum unquantized, it breaks the ladder ofstrata. To maintain this ladder the system must be quantum, quantum, quantum all theway down. In the present work, each stratum is a quantum structure assembled from thoseof lower strata by a regular quantification process like Fermi statistics. Near the bottom ofthe ladder is the empty set, which is trivially both classical and quantum. What separatesthe classical and quantum ladders is not their bottom but the stratum-raising operationby which they are assembled. Quantization converts atomic beings into atomic processesof creation and annihilation, while preserving the stratification.

In the second place, a canonical quantization is but a first step even on its own stratum.The canonical relations

qp− pq = i, iq − qi = ip− pi = 0 (1.1)

say that p and q are not quantized. (They have continuous unbounded spectra.) Thiscanonical non-quantization is the main source of infinities in present-day quantum theo-ries. Canonical quantization quantizes some quantities, like the harmonic oscillator energy,eliminating some infinities, but its quantization is unfinished, and infinities remain.

The problem with canonical quantization is that its Lie algebra (1.1) is singular andhas only infinite-dimensional representations, aside from a trivial one. The Lie algebra ofspace-time coordinates and derivations has the same singular nature.

Quantization can be total. The prototype of complete quantization is the so(3) Liealgebra of commutation relations qp − pq = i, iq − qi = εp, pi − ip = εq. No matter howsmall ε > 0 is, these relations allow p, q, and i to have discrete spectra. In an irreduciblerepresentation of these reformed relations, all physical quantities can have bounded discretespectra with a finite number of values.

By a bfull quantizationc is meant one that replaces every singular Lie algebra of everystratum by a nearby simple Lie algebra. In what follows, q means quantum, including thecanonical quantum theory, and bQc (or bqueuec) means fully quantum, simple on everystratum. A full quantization replaces the infinitesimal translations of a canonical quantumtheory by infinitesimal generators of a simple group.


1.1.3 Swaps

In the theory most fully developed here, all commutation relations arise from those whichdefine statistics. The elementary quantum process is a creation or annihilation, and thesepair into bswapcs, or pair exchanges, when they enter dynamics.

We lose nothing of interest by building all physical processes out of swaps. Indeed,the earliest double-valued representations to be studied were of swaps, not rotations, thealgebras being isomorphic.

Full quantization as thus defined is an excessively underdetermined problem. Oneplausibly reduces the possibilities by limiting the Lie algebras to those that result fromiterating one quantification process. These then belong to one of the classical Killing-Cartan sequences of Lie algebras, determined by the quantification process. Here Fermiquantification is adopted and the so(n+, n−) Lie algebras result.

Several questions confront the explorer at once: The standard model uses a globalcausality form g, probability (amplitude) form H, and imaginary unit i. As global constantsthey violate Einstein locality. What is their origin? Common assumptions of batomismc,blocalityc, and bsimplicityc suggest that they arise from local elements of a lower stratumby organization. Whether this is correct remains for experiment to decide.

Atomism analyzes apparent continua into more elemental individuals, atoms, or mon-ads. The atomism presented here resolves space-time processes into atomic (first-grade)processes of a lower stratum.

Notation: NR is the vector space

NR =

N︷︸︸︷R⊕ . . .⊕ R .

and similarly for NC.V1 V2 is the direct sum V1 ⊕ V2 furnished with the difference norm

‖ . . . ‖ = ‖ . . . ‖1 − ‖ . . . ‖2.

SO(V ) is the special orthogonal group of the quadratic space V ; so(V ) is its Liealgebra.

A homogeneous Grassmann polynomial of Grassmann grade g is called a g-adic, andthe system it represents is called a g-ad (bcenadc, bmonadc, bdyadc, . . ., bpolyadc).

The theory most studied here, bϑoc, is based on the line of orthogonal Lie algebrasso(n+R n−R). A theory bϑlcbased on the linear Lie algebras sl(n) is also considered.They replace the global Minkowski space-causality form g, Hilbert-space metric H, andglobal imaginary number i of the canonical quantum theory by local elements of structure.

Simplicity is meant here in the sense of Lie algebra theory. Einstein introduced twoclassical non-commutativities, that of boosts in special relativity, and that of translations


in general relativity. Heisenberg later introduced the deeper quantum non-commutativityof coordinates like position and momentum. The theories ϑo,l deduce both Einstein’s andHeisenberg’s non-commutativities as singular limits of simple quantum non-commutatvities.

The superlative of “non-commutative” is “simple”. The bsimplicity principlec requiresthe groups of all strata to be simple groups. This eliminates commutative subgroups andmaximizes non-commutativity.

Infinitesimal generators of translation groups are called momenta in quantum theory.Those of rotation groups are called spins or angular momenta. These Lie algebras arise herefrom the graded Lie algebras of quantum statistics. All physical operations are reduced topermutations, which in turn reduce to dyadic permutations, swaps. The prototype of allswaps is the move of a man in a game of checkers, which swaps adjacent squares. Spinsand unitary charges are other forms of swaps. All momenta are regarded here as singularapproximations to high-stratum representations of underlying swaps.

A continuum of pre-quantum physical theory is usually assembled by iterating set-exponentiation S → 2S and is represented by a bclassical setc or bc setc, briefly a bseac.Analogously, a fully quantum system may be assembled by iterated Fermi quantification.It is then called a bquantum setc or bq setc; or briefly a bqueuec when it is supposed tobe not only a mathematical system but a physical one. The constituents of a queue arequeues of a lower stratum. The queue is fully quantum.

The hypothesis is that a queue hides beneath every sea, and the project is then tofind it. In the theory ϑo the bvectorc spaces of all strata are Clifford algebras of realorthogonal groups SO(N ;σN ) with signatures σN =

√N . Fully quantum theories based

on the sequences SU(NC) and SL(NR) are also considered. It is most likely, apriori, thatthe quantification process varies from stratum to stratum, but a fixed process is exploredfor the present.

An infinitesimal generator of an orthogonal space-time group or its covering group canbe called a bspinc for short. An infinitesimal generator of one of the unitary groups ofthe standard model is called a bchargec. These two concepts meld here into one, the morecombinatorial construct of a bswapc. The prototype swap, the move of a checkerpiece,annihilates a piece on one square and creates one on a touching square. Queue theorypermits the swap of swaps, constructed by bracing.

It is accepted here that continuous space-time coordinates and field variables are notbasic but emerge from multitudes of organized swaps. This builds on quantum spaceproposals of Segal [66], Weizsacker [81], Bohm [12], Roger Penrose [60], and Vilela Mendes[75] among others.

1.1.4 Beneath geometry

Events of the sea are supposedly completely defined by their space-time coordinates. SomeLorentz-invariant proposals for quantum space-time, however, adjoin momentum-energycoordinates to the space-time coordinates [69]. Some adjoin internal coordinates related


to charges as well. Events of the queue may have more than twice as many coordinates asthose of the sea.

Galileo’s conceptual reformation adjoined a time variable to the space variables ofEuclid. This can be regarded as refining rather than enlarging the pre-Galilean space, byresolving its points into events, atoms of eternal being into atoms of transient process. Itdeepens the description of the same domain. Quantizing space-time further analyzes theevent into parts. It deepens the Galilean reformation.

When we look about us we see objects with color, temperature, weight, and manyother properties besides space-time coordinates. Indeed, nothing with only space-timecoordinates has ever been seen. From a contemporary empirical viewpoint the concept ofpure space-time seems like an archaic myth with a remarkable longevity.

How did this myth arise and persist? The question is asked not for its historical interestbut to help us recognize and reform similar myths that control us today.

For Euclid, a point is that which has no parts, an atomos, and all in his universewas made up of these atoms. Euclid was therefore an atomist. His atom was likely anabstraction from a speck of the medium underlying the diagrams of his geometry, such asa writing slate or a box of sand, and ultimately the flooded fields along the Nile. Visiblespecks are composed of millions of chemical atoms. The timelessness of Euclid’s point seemsto result from its alleged immobility, which ultimately expresses the geocentric cosmologyof Plato and older sources, in which the Earth, being the center of the universe, is atabsolute rest. Evidently velocity and momentum-energy were abstracted away at the sametime as time (§5.2.1). This omission is equally unphysical. Again, we never see such animpoverished object.

Therefore geometry will not be regarded as the substratum of physics but as a high-stratum statistical approximation to a solid. Field theory restores some of the omittedevent variables by attaching them to imaginary space-time events, still supposed to haveindependent existence on a lower stratum. It is a further patch on the myth of Euclid, nota reformation. These underlying events are still unphysical, unobservable.

Theories of the Kaluza-Klein variety enlarge the concept of event to include furtherphysical variables, but they still lack the dynamical dimensions of momentum(-energy),including the canonical conjugates of the extra dimensions.

Kaluza introduced an extra electromagnetic coordinate axis so that he could identifyelectric charge with a momentum along that axis. In this post-quantum era, however, thesecontinua are no longer necessary for this end. We have two kinds of angular momentato build with, spin and orbital, one essentially quantum, the other originally classical.Generalizing these constructs to other groups, we may speak of quantum and classicalgenerators respectively, based on finite quantum systems and infinite classical systemsrespectively. In the time of Kaluza the bquantum generatorsc — spin and charge — werenot widely accepted as basic. Spins were still being modeled as rotating objects as late as[37], for example.

Here atomicity and simplicity are taken seriously, however. Spins and charges need


not be modeled as objects moving in a classical continuum. On the contrary, a classicalcontinuum may today be modeled as an assembly of quantum spins. Roger bPenrosecassembled a 2-sphere in Euclidean space from su(2C) spins 1/2 [60], a construction laterextended to Minkowski space-time [38]. Likewise, Connes attached the Lie algebra ofquantum charges to a classical space-time, instead of attaching a Lie group [21].

What we require of the Kaluza-Klein dimension is an electric charge 0,±1 in units of thequark charge. To represent this quantum variable no classical continuum is required. Onequantum swap in its defining representation is sufficient. In a quantum theory there is noreason to attach an entire manifold with infinite multiplicity to space-time to account for thegauge group generators, as Kaluza did. It suffices to attach the finite-dimensional quantumsystem of the observed gauge charges. Then the gauge Lie algebra enters at a quantumstratum below the classical stratum, as a theory of quantum swaps. And the underlyingevent has only the charge values 0,±1, higher charges resulting from catenation of severalevents. Kaluza-Klein theory created the compactification problem: What energies curvethe gauge dimensions into circles? This problem never arises for spins or swaps.

Then the physicist’s predeliction for parsimony of concepts suggests that the classicalspace-time coordinates also be resolved into quantum swaps.

The quantum physical event has many more coordinates the the usual space and timevariables of special or general relativity, including dimensions usually considered dynamical.Calling its quantum space a “space-time” might then mislead. It will be called “eventspace” here. This in turn is just stratum E of a stratified queue, designated by Q. Thedaughter stratum of E is that of queues composed of events. In honor of Faraday this willstill be called the field stratum and designated by F.

The adjunction of dynamical variables to positional is also suggested by the Trautmanladder, though in a classical context.

1.1.5 The cosmic crystal film

To save the phenomena, the ambient queue field of stratum F must be somewhat like acrystal film, a graphene, a fullerene ball, a truss dome, a cortex, or a bReggec skeleton withfour long dimensions and small extra dimensions [?]. Now the fundamental unit of the fieldis a quantum bsimplexc or bcellc, a fermionic queue of fermionic events. Its variables areswaps with only three values, those of the defining representation, which may always benormalized to 0,±1 by choosing units.

A theoretical surface of zero thickness is sometimes called a membrane or brane. Abfilmc, on the contrary, is allowed here to be several cells thick, like a diamond film.

The cosmic film does not occupy some space but rather constitutes one. The space-timemanifold is [hopefully] a singular limit of the queue film. The usual external and internalvariables of the quanta are now the longitudinal and transverse coordinates relatove tothe film.h The longitudinal directions in the film run along the crystal hypersurface forcosmologically many cells. The transverse directions, normal to the cellular hypersurface,


are only one or several cells in extent. They correspond to the stiffeners of the dome. Thedifference between spins and charges is the difference between swaps along longitudinaland transverse axes of a quantum cell.

The stable particles are then [hopefully] excitations of the crystal film to which thefilm is transparent along its external dimensions. Less stable excitations are less stableparticles.

When the canonical commutation relations are regularized, as in the above illustration,the resulting relations establish a symmetry between q, p, and i. In the ϑo,l theories,therefore, two extra quantum edges are assumed for each cell, to provide a swap thatorganizes over many cells into i and a central algebra of complex numbers. It would beconfusing to call these additional two dimensions the “complex” dimensions, since theyare real; they will be called bcomplexualc. If each cell is a hypercube (for example) withone vertex as origin, then adjacent to the origin are four external vertices corresponding toDirac spin operators γ1,2,3,4 that organize across cells into global space-time in a singularlimit; two complexual vertices γ5,6 contributing to the global complex plane; and 10 internalvertices providing standard model charge operators γ7,...,16.

Classical space-time is to be the singular limit of the external extension of the crystalfilm. The fields and sources in the standard model involve the internal extension as well.The imaginary constant i of quantum probability amplitudes corresponds to the generatori of an so(2) that rotates each of the two complexual dimensions γ5,6 into the other. Theswap i becomes central only in the canonical limit.

The canonical quantum field variables are singular canonical limits of collective prop-erties of the individual cells of the crystal film involving their internal dimensions.

In the film phase, cells of the queue can be addressed approximately by four numericalspace-time coordinates, and the field construct emerges.

The queue has strata corresponding to the field stratum F of the canonical quantumtheory, the deeper space-time event stratum E, and a still deeper differential stratum D.Charges arise from a still deeper charge stratum C. The relation between these strata isno longer functional but is still a relation of set theory, expressed by the bbracec operationI described briefly in §3.1.2 and more fully in Chapter 4.

It is widely supposed today, and in this study too, that physical events have more thanthe classical four space-time coordinates, that the extra coordinates provide field variables,and that they are small compared to the four classical coordinates. This smallness heresuggests that field variables arise in a canonical limit from queue variables of a lower stratumthan the space-time coordinate variables of the events in the queue. In the quantumspace-time of ϑo the basic Fermi operators generate events; the usual scalar coordinatesxµ of Minkowski and Einstein are singular limits of swaps in an orthogonal Lie algebraso(N+RN−R) of the underlying quadratic space; the fermionic quanta of the standardmodel are quantum event themselves; and the gauge bosons of the standard model andgravity are bound catenations of such events.

Replacing the field by the queue as basic variable solves a major conceptual problem


that has beset such quantum studies: how to formulate the general concept of a single-valued field function when the space-time argument is a quantum variable rather than arandom variable (§6.1.3). The field passes for a single-valued function of the event in thecanonical quantum theory because in Q theory, the ambient queue has an empirical struc-ture analogous to that of a long, thin bcrystal filmc. The cells are (say) 16-dimensional, butfour coordinates suffice to locate a small patch of cells; these are the space-time coordinates.Field coordinates are surrogates for the other coordinates of the cell patch.

Space-time coordinates run along the long film-dimensions and field variables alongits short ones. Each basic vector of one stratum is composed of vectors from the nextlower stratum somewhat as a simplicial bcellc is composed of vertices. The long space-timecoordinates commute in the canonical limit, and their values specify a unique patch of cellsin the canonical field, whose short quantum dimensions define the field variables.

Such a crystalline structure does not fit well into a field theory over a manifold, likeKaluza-Klein or string theories, which require one to go far beyond experimental data. Afully quantum theory accommodates quantum experimental data with less strain.

1.1.6 Praxics

The physical distinction between the classical and the quantum is basic to this study.Both classical and quantum physics can be expressed in the terms of hit-and-miss relationsbetween ideal metered sharp input processes and output processes usually represented bythe directions or rays of vectors and dual vectors.

Heisenberg called the components of such vectors “probability functions”, so the vec-tors themselves may be called bprobability vectorsc, or more fully, bprobability amplitudevectorsc. If an input process p is associated with a probability vector 〈p] then an outputprocess p′ is associated with a dual probability vector [p′〈 so that the transition probabilityis

P (p′←p) =[p′〈p][p < p′][p′〈p′][p〈p]

. (1.2)

[p′〈p] is the evaluation of the dual vector on the vector. 〈p][p′〈 = 〉p′][p〈 is the linearoperator or dyadic that maps 〈w] 7→ 〈p] [p′〈w].

Superpositions of input and output processes for a system are called bterminal processesc.Terminal processes correspond to the atoms or points of a Boolean logic. The duality be-tween input and output does not occur in Boolean logics, which purport to deal withatemporal being rather than temporal doing.

The ancient insistence on absoluteness and timelessness still pervades the study oflogic as it did natural philosophy through the Renaissance. Some natural philosophersescaped the dead hand of Plato by changing their job title to physicist and starting newdepartments. Perhaps logic must go a similar route. The temporal, relativistic, stratifiedphysical theory of quantum epistemic processes that has logic for its classical limit is called


bpraxicc here. It does not claim the absolute authority that logic did, but denies thatauthority and declares itself to be empirical and revisable.

One stratum of praxic can be formulated in terms of input and output spaces V, V D,an involutory mapping H : V ↔ V D and a relation ø on V D × V .

When 〉v′] = 〉H〈v], the transition v → v′ is assured, occurs on every trial; the input“hits” the output channel. The symbol H honors Hermite and Hilbert. In the usual vectorrepresentation, it becomes a symmetric tensor of the form H = Hβαe

βeα in the basis eα.The form H and any a real multiple rH (r 6= 0) have the same physical meaning.

When the relation v′ ø v holds, the transition v → v′ never occurs; input v “misses”the outp channel v′. In the usual projective representation, v′ ø v becomes the dualityrelation v′αv

α = 0.To distinguish the two metrical forms that are central in this work, we call H the

bprobability formc (more properly, the probability-amplitude form) and g the bcausalityformc. In the theory ϑo they coincide on a stratum C and differ on higher strata.

In both quantum and pre-quantum versions of process praxics, the probability form Hdefines an involutory symmetry between input and output processes: H2 = 1.

Symmetries in the usual sense are permutations of input processes that respect somestructure of the system, such as H and ø ; or the dynamical development as well. Incontrast:

The probability form is a symmetry of the metasystem.

Namely, it interchanges inputs and outputs. H is not a time reversal in an ordinary sense,since both input and output processes are generally thermodynamically irreversible andfeasible, while the time reversal of an irreversible process is unfeasible.

A bHermitianc operator h is one with the symmetry property

[w 〉HhH−1 〉v] = [v 〉h〈w] =: [w 〉hT〈v]. (1.3)

The quantum principle is usually stated in a vector representation: The relation p′ ø pholds if and only if an associated vector 〈p] and dual vector [p′〈 obey [p′〈p] = 0. Moregenerally put,

Assumption 2 (Quantum Principle) Every isolable physical system has a 1-1 corre-spondence from its input and output processes p, p′ to Hermitian probability operators〈p〈 , 〈p′〈 on an associated vector space such that the transition probability for p → p′

is Tr 〈p′〈p〈 .

In most current practice H is positive definite and physical probability operators are positivedefinite and of unit trace.

A process p is sharp or irreducible if the associated probability operator 〈p〈 is aprojector on some vector 〈p]: 〈P 〈 = 〈p][p〈 .


Vectors used in this way are called probability vectors (more properly, probability-amplitude vectors). They belong to the stratum of beams or collections of systems, not tothe stratum of the individual system they guide. One does not measure a probability vectoron one system, as one does a force or momentum vector; a probability vector represents ameasurement itself. This use of a vector to specify an input or output process for a systemis different from all uses of vectors in pre-quantum physics. The change is at the level ofthe laws of probability (§2.9).

The form H is more descriptively written as 〉H〈 , showing that it accepts two vectors.With respect to a basis 〈α], H is defined by the coefficient matrix [β〈H〈α] =: Hβα.

1.1.7 Indefinite probability forms

The relations ø and H are usually assumed to be contradictory; then H is bdefinitec.To be Lorentz-invariant, however, a fully quantum theory, having a finite-dimensional

probability vector space, must use an indefinite probability form (§5.2.2).There are two well-known ways to give physical meaning to indefinite probability forms,

one associated with bGuptac and bBleulerc, the other with bDiracc.If H is indefinite, then there are probability vectors 〈p] for which the probability form

‖p‖ = [p〈p] > 0,= 0, < 0. They are said to be positive, negative or null vectors. [z〈z] = 0says that on every trial, 〈z] misses the output that it hits on every trial. The inferenceseems unavoidable: There are no trials of 〈z]; null probability vectors represent unfeasibleprocesses. They are still necessary in order to express the basis of one frame in terms ofthose of another.

Moreover if Hp p > 0 > Hn n then the transition probability P (p←n) is negative:of N > 0 trials, transitions occur in PN < 0 trials. Given that the number of trials ofany kind is non-negative, this implies N = 0; inputs of negative signature are unfeasible.In the Gupta-Bleuler interpretations the null cone formed by null vectors is an impassablefrontier for transitions.

Dirac, however, proposes that Nature sometimes extends us credit, allows us to with-draw systems even before we input any. This results in a negative number of systems. Suchan output against future inputs can be regarded as the input of a dual system. This seemsespecially natural for fermions, where there is symmetry between the full and the emptyprobability vector of the many-fermion system, and there is an algebraic isomorphism be-tween increasing the fermion number from 0 and decreasing the fermion number from itsmaximum value.

1.1.8 Probability vector spaces and algebras

For a discrete bclassical systemc the relations ø and H define a discrete state space andtheir symmetry group is the discrete permutation group of the state space.


For classical mechanics the state space is a symplectic manifold, and its group is thecontact transformation group. This group is singular.

For a bcanonical quantum theoryc the group of the ø -relation is commonly an infinite-dimensional unitary group. That is the trouble.

In a fully quantum theory the groups of ø and H are classical groups and the proba-bility vector space is finite-dimensional but indefinite.

Automorphisms of ø are projectively represented by elements of SL(V ), the regularlinear mappings V → V of unit determinant. Endomorphisms of the quantum system areprojectively represented by linear operators V → V , which make up the algebra Alg(V ).Infinitesimal automorphisms of the quantum system are bijectively represented by opera-tors in Alg(V ).

Canonical probability vector spaces are provided with a symmetric bprobability formcH, usually definite. Then the bMalus-Bornc relation holds:

The transition probability is the squared value of a unit dual vector on a unitvector.

Infinitesimal transformations V → V respecting H form the bisometryc Lie algebraso(V ) of the probability vector space V .

The Lie algebras so(3R R, sl(NR), and sl(NC) have no finite-dimensional unitaryrepresentations with definite probability form H.

1.1.9 The need for full quantization

For Faraday and Einstein, both field values and events were classical and their pairing wasmanaged using informal classical logical intuition. For Dirac, the electromagnetic field val-ues were operators and handled by formal algebraic methods, but space-time was classical,as though field-meters and clocks were classical objects. In canonical quantum theoriesin general, the metasystem is ordinarily treated classically and the system quantally. Thespace-time variables of field theory are metasystem variables.

The split of nature into classical and quantum strata is unphysical and creates prob-lems:

It seems to be impossible to represent the action of a quantum system on a classicalone without quantizing the classical system.

Approximating a sufficiently small classical field-meter or target with a quantum sys-tem likely creates a black hole, and neither of the two major principle theories of the day,general relativity and canonical quantum theory, covers this phenomenon.

These considerations suggest that quantum field theories need further quantization, ifonly to work near the Planck energy.

Any classical system is but a crudely observed quantum system.


Canonical quantization modifies and partially regularizes only one stratum of a canon-ical classical theory and leaves deeper strata such as that of space-time still classical andsingular.

1.1.10 A cellular hypothesis

The present bfull quantizationc quantizes and regularizes every stratum by means of aradical

Assumption 3 (Cellular Hypothesis) Physical processes on every stratum are quan-tum cells composed of a finite number of finite quantum processes of lower strata withFermi statistics.

Cellularity is intended here to express stratification. The cell overlies a deeper strata oforganelles and underlies a higher strata of organs. Classically this stratification is expressedby iterated quantification, usually within set theory. Full quantization carries this classicalstratification over into the quantum realm.

The bqueuec, a quantum structure assembled by iterating Fermi(-Dirac) statistics(§3.1.2), is finite and its groups on every stratum are simple Lie groups. The bfull quantizationcstrategy is group-guided:

Reduce a canonical theory to a singular limit of a queue theory keeping exactlythe same groups where possible, and approximately the same groups where not.

Dirac’s Lagrangian quantum dynamics approximates a propagator u for infinitesimaltime as [q(t + dt)〈u〈q(t)] ∼ exp i[q(t + dt〈L〈q(t)], the exponential of a matrix elementof the action. This has no unitary-invariant meaning. Operator exponentials are frame-independent, but exponentiating matrix elements singles out a small family of frames.In this case, the frames singled out are those in which each basis element is associatedwith a definite space-time event at which the Lagrangian coordinates are evaluated. Thisprinciple refers on the dynamical stratum F to an underlying event stratum E. Q respectsthat stratification in its way.

Since there are no classical systems, a sum over classical histories is not a part of aquantum theory but only a heuristic precursor. It is also too singular to be well-defined.In Q theory it survives as a finite trace over a Q history taken to be a queue of events, witha Q dynamics taken to be a dual probability amplitude vector [D〈 assigning probabilityamplitudes to every experimental history vector 〈E]. It is easier to take the q limit if qtheory too is expressed in terms of histories ??.

Fermi quantification maps vectors v ∈ V, u ∈ V D to operators v, u on the Grassmannalgebra of V , subject to

u, v = u(v); (1.4)


This is a full quantum theory as far as it goes, with discrete bounded spectra only, butit covers only one stratum. One may cover all strata by iterating it. This results in thequeue.

Eliminating the field in favor of the queue eliminates the classical puzzle of infinite self-interaction [34]. Present physical theories of gravity and of the standard model [hopefully!]become singular large-scale limits of one fully quantum queue theory that is also valid onand near the Planck scale.

Fully quantum events have coordinates complementary to their positional coordinates,resembling momentum coordinates. Such events are no longer atomic elements of space-time position but of a quantum dynamical process. Classical space-time emerges whenthese event momenta can be neglected because they are small, perhaps 0.

In a fully quantum theory, variables on every stratum have bounded discrete spectra.In particular in the theories ϑo,l the spectrum of the time coordinate is bounded anduniformly spaced in multiples of a btime quantumc bXc, which with c also defines a space-quantum. Due to the non-canonical commutation relations, exact Lorentz invariance isnevertheless preserved. Spectra on lower strata are much sparser than those on higherstrata. The Heisenberg uncertainty principle between space-time and momentum-energyvariables breaks down badly on lower strata and reappears on a higher stratum as a singularlimit. This is large effect, and may be able to explain the large discrepancy between theempirical cosmological constant action density and that expected from field theory, whichis on the order of one Planck unit.

General relativity and quantum theory are dominated by two forms:H]/, a hermitian form on vectors with associated imaginary i, such that Hi + iH = 0,

defining transition probabilities, called the bprobability formc.g, a symmetric bilinear form on space-time differentials dx defining bcausal structurec

of the event stratum, called the bcausality formc. It is measured with chronometers in theclassical limit, and by graviton emission and absorption in the canonical quantum theory.The form g = (gνµ) defines the proper time dτ for a differential displacement dxµ near xµ

so thatdτ2 = −g dx dx. (1.5)

It also defines the action S = −∫m0c

2dτ for a projectile of rest mass m0.Fermi quantification is usually carried out within a unitary quantum theory. It has,

however, a larger invariance group than the unitary. It makes use of neither a probabilityform H nor a central imaginary i, nor does it produce any. If the quantum kinematics hasa group SU(NC), the invariance group of the Fermi algebra is at least GL(NC).

This raises the question of how the canonical g, H and i manifest themselves in a queue.Since they are global in dominion, the strong blocalityc principle of Einstein suggests thatthey are not intrinsic but bemergentc, arising from more local constructs by organizationand spontaneous symmetry breaking. The bfully quantumc theory ϑo assumes that a lowerstratum C has intrinsic constant g,H, i and that these organize into variables of higher


strata like F.The operators of a unitary representation of the Poincare group respect a definite

probability form H on one stratum, while the Poincare transformations being representedrespect a time form g on a deeper stratum. Both the unitary group and the Poincaregroup can be imbedded within SO(N+, N−) groups appropriate to their strata: In ϑo,the unitary representations of the Poincarre group used in physics are singular limits andorganizations of higher-stratum SO(N+, N−) representations of lower-stratum SO(N+, N−)groups, which arise naturally for a queue.

The Riemannian curvature of space-time is then expressible as a near-classical vestige ofthe quantum non-commutativity of the momentum-energy variables of the quantum event.Even in the classical theory of gravity the probability form and the causality form arenot independent, making it plausible that the have a common origin at a deeper stratum.General covariance and all other singular gauge invariances are approximations to a fullyquantum gauge invariance whose gauge group is regular, and the approximation breaksdown at high energy. Space-time [hopefully!] becomes an organized quantum system witha melting point analogous to a Curie temperature. There are no events in the classicalsense:

What has space-time position has momentum-energy, angular momentum, spin, and the standard charges too.

In the canonical limit, the momentum-energy of quantum events may still contribute tothe cosmological constant action density.

To bfully quantizec a physical theory it suffices to assemble every stratum of the theoryfrom quantum elements of a lower stratum with Fermi statistics. All quantum commutationrelations for a fully quantum system derive from the quantum statistics of its quantumelements. The usual canonical quantization is expressible as a first approximation — buta singular one — to such a full quantization, which necessarily quantizes time.

It is easy to see the basic ingredient of queue models. The basic generating operatorsall represent operations of creation and annihilation of quanta of Fermi statistics. Theseact before the organization of space-time as well as after, so it would be misleading to saythat these building bricks are spins, although they have Clifford commutation relations. InFermi statistics, annihilation of a quantum is isomorphic to creation of a dual quantum,so we may say briefly that queue models are made of creation operations, or bcreationsc,represented by creation operators, or bcreatorsc.

Full quantization works on the Lie algebras that occur on every stratum of the canonicaltheory in various roles:

1. Abrelativityc Lie algebra defines the frame transformations for the system.

2. An bexperimentalc Lie algebra defines feasible reversible experimental operations onthe adiabatically isolated system.


3. A bsymmetryc Lie algebra defines the transformations that fix some specified featureof the system such as the dynamics.

Some of these Lie algebras are made up of operators that act on probability vectors. In thecanonical theory those of lower strata act on classical objects, for example, on space-timepoints, or tangent vectors to space-time. In principle, full quantization preserves roles asit regularizes Lie algebras, and its Lie algebras of every stratum act on vectors.

A bfully quantumc system is quantum on all strata.

In the fully quantum theories ϑo,l the probability vector spaces of all strata are realGrassmann algebras. In the theories of the orthogonal group line like ϑo they are Cliffordalgebras as well. The canonical Lie algebras and unitary Lie algebras that occur at variousstrata of the standard model and gravity theory are singular limits or reductions of finite-dimensional invariance Lie algebras so(n) of these Clifford algebras.

Fluctuations about the classical predictions of the electromagnetic field theory helpedto convince Einstein of the existence of photons. Analogous fluctuations about the canoni-cal quantum predictions are expected to signal the existence of quantum space-time events.

A discrete space-time theory must have difficulty accounting for the continuous groupsthat have been the main navigational aids in the quantum exploration. A quantum space-time theory does not; its probability vector spaces are born with these groups or closeapproximations.

Queue theories like ϑo,l replace the classical logical construct of set by the praxicconstruct of a queue. Einstein pointed out that his general relativity did not geometrizephysics, it physicalized geometry, presumably the geometry already used in physics. Ina similar sense, queue theories do not set-theorize physics but physicalize the set theoryalready used in physics, while following in the footprints of the existing canonical quantumtheory as closely as possible. This goes deeper than geometry, which can after all beregarded as an exercise in set theory. The result is no longer a theory of sets as objectsbut a theory of queues as reflexive processes of creation (and annihilation), an example ofa bpraxiologyc.

In a bqueuec theory a Grassmann product of g single-quantum probability vectors canbe regarded as the probability vector of a cell with g quanta as vertices or elements. TheCellular Hypothesis of Assumption 3 can be sharpened:

Every physical system is a queue.

A (many-system) balgebrac, unless otherwise specified, is a probability vector spacethat is also a Grassmann algebra whose product represents the composition or catenationof probability vector processes. The most familiar example is the Clifford algebra thatserves as vector space of a collection of fermions.


1.1.11 Quantum time

The quantization of time leads to several conceptual changes in quantum theory taken upin this section..

Recall that a system is bquantumc, to put it in terms of laboratory experience, whenthe filtration operations for it do not all commute with one another. Only the simple orirreducible case is considered here, with no proper central filtrations, 0 and 1 being con-sidered improper. This is the case of unlimited quantum superposition. The axioms for aprojective geometry provided with a polarity characterize such systems in a coordinate-freeway. Filtrations are then isomorphically represented by the projectors of the probabilityvector space.

The canonical relations originate in the differential calculus in the form

∂x − x∂x = 1, ∂tt− t∂t = 1. (1.6)

and migrate to the Lie algebra of Poisson Brackets of classical canonical mechanics, fromwhich they enter canonical quantum theory.

To be sure, t is not a canonical observable of the system stratum, and ∂t is not a canon-ical observable of any classical stratum, but the instability of their Lie algebra propagatesto that of space coordinates x in a relativistic theory, and integrals over events enter intoobservables and produce infinities.

On the other hand, finite dimensional Fermi algebras, with their graded-canonicalcommutation relations, are structurally stable, represented in finite-dimensional probabilityvector spaces, and require no regularization. They are preserved intact in the present fullquantizations.

Gauge-fixing fields and ghosts are introduced into canonical theories to make themrenormalizable. They are not needed in a Q gauge theory that is finite from the start.The gauge is fixed in actuality by the experimenter, who breaks all symmetries [84]. Thegauge-fixer can be imagined, for example, as a Mandelstam loop system [55], each loopincluding a quantum interferometer, or a generalized voltmeter in the classical limit. Agauge transformation replaces one such loop system by another. A gauge-fixing field is thena surrogate for a loop system in the metasystem. Quantizing it therefore violates Bohr’searly principle of the classical metasystem, but it works. This provides some precedent forthe deeper violations perpetrated here.

In order to eliminate the unphysical ultraviolet and infrared infinities of canonical quan-tum theories, many considered slightly varying the space-time-momentum-energy commu-tation relations to quantize space, time, or both, including bAmbarzumianc and bIvanenkoc[1], bEinsteinc, Heisenberg, bSnyderc, bSegalc, bPenrosec, bConnesc, and bVilela-Mendezc(§5.2). This strategy is not at all iconoclastic, merely biconoelasticc. It is extended to allstrata of assembly here.

Admittedly, Einstein mentions the approach only to declare it daunting [29].


It takes a fine mill to grind time. Time passes untouched through the canonical quan-tization machine. One rationalized the persistence of classical space-time when all else isquantum by allocating space-time to a different stratum of physical theory. Time t is notan observable variable of the system stratum in canonical quantum theory; not because itis unobservable, since it is observed with clocks, but because it is an observable of anotherstratum. The clock on the laboratory wall is not part of the system under study. Thisconcept of stratum merits further discussion.

Physical theories are usually structured, at least tacitly, into mathematical bstratacrecursively generated by bracing and catenation. Making one variable a variable function ofanother, for example making position a function of time, allocates the dependent variable toa higher mathematical stratum than the independent. Canonical quantization accepts thestratum structure of the classical theory, modifies the algebra on the stratum of dynamicalvariables, and leaves the algebras of the deeper strata unmodified, including the stratum oftime in mechanics and space-time in field theory. This agrees well-enough with low-energyexperiments except that it builds in infinities from the start.

But we never measure an infinity on any stratum, having neither divided any physicalline infinitely often nor extended any line to infinity. Euclid built these infinities into ourtheory of space with no experimental evidence for either, only an absence of clear evidencefor finiteness. And since it takes a theory to interpret the evidence, this may simply be anabsence of adequate theory. The ill-defined formulations from which present-day quantumfield theory sets out, with its non-existent integrals over infinite-dimensional spaces andits unbounded operators that cannot be multiplied, reflect limits to our imagination andmathematical powers as much as inferences from experiment.

All these unphysical infinities trace back to our choices of groups. Euclid incorporatedinfinities into plane geometry in order to have translational and rotational invariance underthe Euclidean group. A Lie algebra is called bsingularc or bregularc according to its bKillingformc; a Lie group, according to its Lie algebra. The Euclidean and canonical groupsare singular, the classical groups are regular. Singular groups generally require infinite-dimensional unitary representations, but classical groups have enough finite-dimensionalmatrix representations to make a finite quantum theory possible in which the groups act onfinite-dimensional probability vectors, and expectation values are finite traces. And thereare classical groups near most singular groups, including the canonical groups, though theapproximation is not uniform, and may require freezing some degrees of freedom of theLie algebra. Full quantization is not a well-defined algorithm but a heuristic method, evenmore so than canonical quantization.

Astronomical evidence for the Big Bang already suggests that one might not be ableto extend the time translation operator beyond about 10∼21 s into the past, so that it isnot completely absurd to renounce the time translation group as is done here.

A canonical quantization reduces the infinities of classical physics by replacing somecommutative groups by canonical ones. A full quantization, for example ϑo, eliminates theremaining infinities by replacing every bsingular Lie algebrac — which includes all canonical


Lie algebras — with a nearby regular Lie algebra — for example, an so(n+, n−) —- onevery stratum, with organizations as needed.

The theory ϑo assumes that the constituents of each stratum have Fermi statistics. Thereasoning behind this choice is given in §1.3.6. It unifies spins and charges with a quantumspace-time somewhat as Mankoc Borstnik unifies spins and charges with a classical space-time [58]. Its simple Lie algebras groups are unitary ones su(n+, n−) of various signatures,in the A series of Cartan. A quantum space-time theory based on the D series has alsobeen studied [68].

Segal proposed to quantize the space-time stratum with a simple Lie group instead ofa canonical group on the grounds of structural stability [66]. Since there are always errorsin data, physical Lie algebras should be insensitive to small errors in the commutationrelations; should, in other words, be structurally stable.

A Lie algebra is bsingularc or bregularc as its Killing form is. Thus the canonicalLie algebras h(n) are singular, while the linear Lie algebras sl(n) is regular, as are theorthogonal Lie algebras so(n) and the unitary Lie algebras su(n) of whatever signatures.

Singular Lie algebras are not structurally stable, and regular Lie algebras are, whichincludes the simple ones (§1.1.1).

But finiteness is an even stronger motivation than structural stability for regarding allthe singular groups of physics as approximations to nearby classical groups.

The deepest conceptual problem in quantizing space-time arises from the fact that thetime variable of mechanics and the space-time variables of field theory are not observablesof the system but of the metasystem. Canonical quantization is limited to observables ofthe system, and this limitation has been elevated to a decree: Do not quantize the meta-system. Two observables do not commute if they are complementary; that is, if measuringeither invalidates a prior measurement of the other. It would be hard to understand howtwo quantities can fail to commute if they are not supposed to be measured. Time is usuallydefined today as what clocks read, and as such is measured; but within the metasystem,not the system.

But if the time that enters into the dynamics of the system were to be identifiedwith the time reading of the laboratory clock, it would create a mystery of how the clockon the wall is so rigidly connected to the particle in the experimental chamber. Herethe two times are kept conceptually separate, further relativizing time. It is no longerposited that there are ideal clocks that read the ideal time. Each clock actually readsits own time. The space-time structure connects clocks, keeping them in approximatesynchrony. The limit of ideal time is a singular one, and here underlying regular theoriesare discussed. Special and general relativity introduced new acceptable time variables. Afully quantum relativity introduces many more. It allows different times on different strata,and introduces a complex organization of many quantum variables below the perceptionsof canonical physics to account for the correspondence between these times, a form ofself-organization that degenerates to a monolithic classical time in the singular canonicallimit.


As a Lorentz-invariant model of quantum space-time point that can approach a point ofMinkowski space-time in a singular limit, Segal suggested an so(6R;σ) angular momentumof large quantum number in a 6-dimensional quadratic space of unspecified signature σ.Thus Segal chose the defining representation of an extension of the Lorentz group as theseed of space-time-energy-momentum space. This has a substrate of Minkowski space-timevectors, with Lorentz spinors playing a secondary role. Segal does not consider a stratifiedquantum theory. The conformal Lie algebra so(4, 2) and the Lie algebra so(3, 3) ∼= sl(4R)are also possibilities.

In an independent approach, Penrose combined many so(3) spins 1/2 to make anSO(3)-invariant quantum space that can approximate a Euclidean 2-sphere in a classicallimit. A Segal model can be constructed by similarly combining many so(6;σ) elementaryspins.

The queue theory ϑo is constructed by iterating a Fermi quantification. Fermi statisticsis the natural choice, on the grounds of its regularity.

This assigns odd exchange parity to even rotational parity, but can be reconciled withthe empirical spin-statistics correlation [hopefully] if the vacuum moves a half unit of spinto remote regions S:SPINSTATISTICS.

Dirac spinors transform as probability amplitude vectors of Fermi aggregates of lower-stratum entities [16]. Therefore the substrate of queue theory too will be spinorial ratherthan vectorial. Since a Fermi algebra is also a Clifford algebra, its substrate is appropriatelyspinorial.

Such a full quantization expands every canonical Lie algebra, whatever its stratum, toa linear Lie algebra sl(NR) on the same stratum, decentralizing and conditioning its i.

1.2 The idea of the queue

A bqueuec is a hypothetical Fermi-statistical quantum system whose basic probability vec-tors are generated from 0 by (1) the operations of Grassmann polynomial formation, withaddition representing quantum superposition and Grassmann multiplication representingphysical combination, and by (2) bracing, I(ψ) = ψ, representing physical unitization.The stratum of a queue is the maximal number of its nested braces. This leads to thestratum scheme defined by (1.7). To define a quantum kinematics one must further specifythe brace I, a probability form H, and their physical interpretations (Chapter 4).

The Lie-algebraic commutation relations defining any classical group can be interpretedas a statistics [59] called bPalev statisticsc. Bose statistics is a singular limit of Palevstatistics. The dyads or pairs in a queue obey a Palev statistics. They can be consideredas generalized bosons, or bpalevonsc. Canonical quantum theory uses an infinity of fermionsto make a boson. In fully quantum theory a fermion pair is a palevon, as close to a bosonas desired. This makes the Cellular Hypothesis of Assumption 3 more tenable.

A probability form H on a probability vector space V is not needed to define the Fermi

1.2. THE IDEA OF THE QUEUE 33

algebra over V , but it is needed to represent physical probabilities.Its presence gives the Grassmann algebra over V a second structure, that of a Clifford

algebra over V . The two products are written as v ∨ v′ and v t v′.The generations or strata have probability vector spaces S[L], L = −2,−1, 0, 1, 2, . . .

defined recursively:

S[−1] = ∅S[L+ 1] = Poly ID’ S[L],

where ID : S[L] → Grade1 S[L+ 1]. (1.7)

The dimension of S[L] is dL. The Whitehead-Russell bapostrophec after an operation meansthat the operation is to be executed on the following set element by element. f ’X can beread as (the set of) “the f ’s of the X’s”.

S[−1] is the empty set regarded as probability vector space. The sum of no vectors isthe vector 0 that belongs to every grade and rank but has no operational meaning. Thisis the sole element of the vector space 2S[0]. The product of no vectors is the vector 1,representing the empty input. This is a basis vector for S[1].

The resulting probability vectors support grade and rank operators, designated bybGradec and bRankc. The grade is the Grassmann polynomial grade; the rank is thenumber of nested bracings. Stratum L consists of all probability vectors with rank r ≤ L.

The Fermi system algebra S has a fractal, self-similar structure: A tree isomorphic tothe entire structure grows from each element in the structure. S = 2S is its own Fermiprobability vector algebra. As a result, groups of any stratum are represented naturally onhigher strata. In physical application the recursion is cut off at some stratum; stratum 7suffices for the dynamics of ϑo.

The concept of probability assumes an unlimited supply of systems and an unlimitednumber of trials. In a fully quantum Fermi praxic, the probability vector space of a queueis a finite-dimensonal subspace of the self-Grassmann algebra S generated by the sequenceof generators begun in Table 1.1.

One probability amplitude vector, the bdynamics vectorc of stratum F, is supposed toimprove on the standard model and canonical quantum gravity, at least in the small.

Probability vectors are statistical, and properly belong to a higher stratum than thequantum system they pertain to. This technicality is ignored here to simplify discussions.A system and its variables are assigned to the same stratum as its probability vectors. Ifthis practice leads to confusion it should be corrected.

The exploding sequence of stratum multiplicities of S corresponds to that of classicalset theory. It describes the materiel made available for theory construction by this algebraiclanguage. There is no compulsion for the theorist to use it all, but then it is incumbenton the theorist to delimit the subalgebra that is actually physical. The lowest stratumL able to describe a system of given multiplicity D must have multiplicity dL ≥ D. Apractical upper bound is set on L by the economy principle, of not using more materiel


than necessary. The theory ϑo uses the 4-dimensional stratum 2 for its Lorentz group,the 16-dimensional stratum 3 for the Lorentz group and the other groups of the standardmodel, and stratum 5 for the generic event of a quantum history.

Full quantization thus converts infinite fine structure on one stratum into finite struc-ture on several strata. Structure that has been attributed to the very small is now allocatedto a deeper stratum. Extra classical space-time dimensions of the Kaluza kind are repre-sented by probability vector space dimensions several strata below space-time, and so priorin order of construction. Space-time itself is replaced by a huge-dimensional probabilityvector space, with one dimension per independent event possibility. This elimination of allclassical continua also reduces the number of probability vector dimensions, both ordinaryand extra, from the infinity of field theory to a finite number for a queue.

A fully quantum theory must cover the canonical theory; that is, agree with it whereit works. One therefore constructs one by slightly revising the canonical (or classical)theory on each stratum, not by starting from scratch; by replacing canonical commutationrelations with classical Lie algebra commutation relations. For cross-section computationsone requires queue correspondents of input and output momentum eigenvectors, althoughqueue momentum components do not commute with each other.

To start the inquiry, radical simplifying assumptions seem practically necessary inorder wring experimental statements out of the theory. In the theory ϑo one provisionallyassumes that that the various canonical strata correspond to fully quantum strata whoseprobability vectors lie in corresponding subspaces of one bgrandc probability vector spaceS; and that every stratum is assembled from the previous with quantum statistics in thesame Cartan class. This is less restrictive than it may seem, since it turns out to be easierto make bosons out of fermions with regular statistics than singular. The model ϑo thenquantizes even time.

The main work is then to fit the complexities of canonical quantum physics, stratumby stratum, into the simpler fully quantum framework. Recall that the process of canonicalquantization can be divided into three parts:

1. Construct a family of canonical quantum theories, defined by canonical Lie algebraswith an adjustable parameter ~ in their commutation relations.

2. Express the classical theory as a singular limit of these canonical quantum theoriesas ~→ 0.

3. Choose one of the family to succeed the classical theory, by fixing the optimal valueof ~ experimentally

Correspondingly, one fully quantizes (all strata of ) (say) the standard model in threestages:

1. Define the general fully quantum theory, using one sequence of classical Lie algebrasfor its commutation relations.


2. Express the standard model and gravity as a singular limit X → 0 of such fullyquantum theories.

3. Fix the optimum physical value of X experimentally.

Since time is to be quantized, it is convenient that the quantum constant X be afundamental natural unit of time or bchronc forming a complete set of units with thenatural units of speed c (the rømer?) and action ~ (the planck?). Graded Lie algebras areused to define the statistics of the queue as they are used to define the statistics of fermionsin canonical quantum theories.

1.2.1 Strata

Quantum theory inherits a concept of assembly by stages from classical set theory. For bothclassical and quantum set theory the strata form a hereditary family in which any collectionof individuals of one stratum defines an individual of the next by a stratum-raising processrepresented by a bbracec operator

ID : w 7→ w ≡ IDw ≡ w (1.8)

The brankc of a set is the number of nested braces required to produce it from the emptyset, or from whatever proper elements serve as foundation for the set theory.

To build up a quantum correspondent, one may first translate the pre-quantum pro-cedure into a matrix language resembling that of quantum theory, but kept pre-quantumby commutativity restrictions. To arrive at the quantum theory one then drops the com-mutativity conditions.

Classical sets can be described by rays in a preferred orthonormal basis for a realEuclidean Grassmann algebra Sc, with R, +, ID, ∨, H, and · (§3.2). Superpositions of thebasis vectors are given no physical meaning, or forbidden, in the classical theory. Randomsets are described by (positive) statistical operators on Sc that are diagonal in a naturalbasis. This leads to nested strata S[L] of random sets, S[L] ⊂ S[L+1], related by bracing:

ID : S[L] → SL+1. (1.9)

This sets the stage for quantum set theory. One need only turn on superposition.Canonical Fermi quantification goes from a one-quantum Hilbert space to a Grassmann

algebra over a Hilbert space. The Fermi (operator) algebra over that Hilbert space is theClifford algebra over the bduplex spacec with the bduplex formc. The queue follows thepattern of the random classical set and the canonical Fermi assembly in using a Grassmannalgebra S for its vectors, now assumed to be real. Its vector catenation ∨ is that of Fermistatistics. To iterate Fermi statistics requires converting a high-grade polyadic x of onestratum into a first-grade monadic x of the next by bracing. In this way one constructs


the hereditary family (1.7) of finite-dimensional vector spaces S[L], one for each stratumL = −1, 0, 1, 2, . . . of bracing.

The set theory presently used in classical and canonical quantum physics is the mostfamiliar hereditary family. It is generated by bexponentiationc S → 2S , which catenatesmonads of one stratum into polyadics of that stratum, and bracing, which forms polyadicsof one stratum into monadics of the next stratum.

The exponential set 2S is often called the bpower setc of S. This can lead to confusionhere; if S is a numerical variable, its powers are Sn, not nS . Here 2S is called the (binary)exponential of S.

Correspondingly, a space V of queue probability vectors is mapped into a superspaceby an analogous functor V → 2V , where 2V is the Clifford algebra over V . This processtoo is called (vector space) bexponentiationc.

A vector of brankc r is one with exactly r nested bracings. Its symbol in the barnotation of Table 1.1 is r bars high. A vector of stratum L is one of rank ≤ L. Its symbolin the bar notation of Table 1.1 is ≤ L bars high. Some trivial cases:

The probability vector space for stratum L = −1 is the empty set, of dimension −1.The probability vector space for stratum L = 0 is the one-point set 0, of dimension

0.The probability vector space for stratum L = 1 is R, of dimension 1. This is a

customary representation of a vacuum vector in Fock space.Starting from stratum 0, L catenated bracings still remains within the vector space

S[L] of stratum L. Its dimension is 2L.In canonical theories, space-time events and their coordinates, like time, are allocated

to an bevent stratumc, or stratum E. The probability vector is assigned to a higher bfieldstratumc F.

These stratum designations are retained for the queue. Presumably the present con-struction of a quantum bracing operator I (§4.1.4) is not the last word on the subject.This operator completely ignores the grade of its operand, embracing a million monads asreadily as two, like a mathematical black hole. But tinkering with the brace would takeus far from current physical theories and open too many possibilities. While it seems ab-surdly optimistic to assume that the stratified structure of quantum nature is close to thatof classical sets, accepting classical set theory uncritically, as the standard model does, iseven more optimistic, and yet the standard model already works rather well. Assumingthat classical logic works exactly on all strata but one is betting on a case of measure 0 ina large family. Assuming a small change that leads to regularity is following the successfulprecedent of canonical quantization, with no new concepts, and no new degrees of freedom.Rather, full quantization further prunes vector dimensions already pruned by canonicalquantization.

The main hazard of this exploration is getting lost in the jungle of possibilities. Thestrategy of uniform statistics greatly reduces that risk; but it is another speculative unifi-cation.


A field of coefficients, R or C, is regarded as given from the start. The exponentialspace of a vector space is defined using familiar operations of +, ID = . . ., ∨, and † onvectors. These have the following physical interpretation.

The operational characterization of quantum addition or superposition is suggested byprojective geometry. If the transition probability between an input process and an outputprocess is 0, they are said to bmissc or boccludec each other. One input process w is absuperpositionc of others u and v if and only if every output process that occludes u and valso occludes w. In classical theories the vectors are restricted to a bsis and become statesand a superposition of u and v is either u or v.

The bbracec operator ID = . . . of set theory is indispensable in physics today, un-derlying both quantization and gauging, including general relativization. In physics bIDcrepresents physical unitization or association. When we say in classical mechanics (forexample) that the position of a particle is a function x(t) of time, we express that functionas a set of bracings x, t := IDx ∨ IDt of position values and time values.

To respect quantum superposition, the quantum brace is defined as a linear operator.It converts a probability vector of any grade and rank into a first-grade vector of the nextrank.

Catenation, the multiplication ∨, represents performing input operations in sequence,and composes a sequence of g vectors into a tensor of bgradec g.

Every element can be decomposed into homogeneous parts by bgradec; Gradeg V des-ignates the g-grade subspace of a Grassmann algebra V .

The bexponentialc 2V of a vector space V is defined here as the vector space of formalpolynomials in the braced vectors of V , reduced modulo the commutation relations of thestatistics, usually Fermi. 2V is the vector space of a collection of a variable number ofV -systems, and the vector space of one 2V system.

The brace is convenient when V itself is a space of polynomials, permitting one to usethe same multiplication sign, usually none, for both algebras V and 2V without confusion.The queues of all strata taken together form the minimal family of quantum systems closedunder exponentiation.

To define the exponential space, it remains to specify formally the bracing operator Iand a probability form H. Iterated Fermi statistics without the probability form leads toa nested family of linear groups. The resulting vectors are polynomials in the bmonadicsc(elements of grade 1) of Table 1.1, which are anticommuting vectors of norms ‖ψ‖ =Hψ ψ = ±1.

All dynamical variables of a queue are polynomials in its generators and all theircommutation relations derive from statistics. The generators in turn are expressed interms of +, ID, and ∨.

If any things are fundamental in this theory, at least for the present, they are themeta-processes +, ID,∨.

Quantum theory today has a mixed salad of groups: unitary groups of quantum ori-gin, and canonical, orthogonal, and permutation groups of classical origin. Some canonical


......

......

F 6 22216˜ ˜ ˜ ˜ ˜ ˜ ˜

. . .˜

. . . . . .˜

. . .

E 5 2216 ˜ ˜ ˜ ˜ ˜ ˜. . .

˜ ˜ ˜. . .

˜. . .

D 4 216 ˜ ˜ ˜ ˜ ˜

C 3 16 ˜

B 2 4A 1 2

0 1−1 0

L L dL MONADICS OF STRATUM L

Table 1.1: Monadics resulting from ≤ L iterations of the classical exponential-set functorP, the Grassmann functor P0, or the Clifford functor P, for L ≤ 4, with small samples ofstrata L = 5, 6. The known universe is too small to print the monadics of stratumL = 6in a Planck-size font. Those of stratum5 would fit onto 64 pages of 12-point roman.KEY: L is the tentative stratum assignment of (4.33). L is the absolute stratum number.S[L] is the space of all vectors of rank ≤ L. dL := Dim S[L] = 2L. A bar represents bracingID. A stroke represents 1, the empty. A tilde labels a monadic of negative square.


groups come classical ones by way of canonical quantization, others come from the differ-ential calculus of space-time coordinates. The main orthogonal group is the Lorentz group.The main unitary groups are the U(1),SU(2),SU(3) of the standard model and the SU(∞)of quantum field theories.

Full quantization as practiced here extracts all these groups from the invariance groupsof quantum statistical assemblies or queues.

The Fermi operator algebra alone is not an adequate grammar for quantum physics.It expresses the misses ( ø ) but not the hits (H). It must be augmented with a probabilityform H for the many-quantum assembly. This is generally induced by a one-quantum formalso designated by H. The theory ϑo uses an indefinite mean-square form for this purpose,defined in §1.3.8. The resulting H-algebra has su(N+C N−C) symmetry for complexcoefficients C, so(N+R N−R) for real coefficients R. The Fermi operator algebra istrivially stable against small changes in H because H does not enter into its structure. TheH-algebra is stable against small changes in H provided that H is regular; which is thereforeassumed.

The algebraic grammar used in this work is not dramatically different from someconsidered earlier [39, 40]. Every bsharpc — that is, maximally determinative — inputprocess is still represented by a ray in a recursively generated Grassmann-algebraic vectorspace.

What is added is the strategy of full quantization and its application to the standardmodel and general relativity. The identification of the organization of i, the quantized i,as the organization of the metrical structure of space-time and gravity is a corollary.

Houses are built from the bottom up. Some try to construct the next theory from thebottom up – that is, from intuitively appealing axioms; I admit that I have. This is a trap.Our intuition develops in our least critical, least experienced years. It is rarely quantumor relativistic enough.

One might consider constructing a theory from the top down, from raw experimentaldata, instead. This too is impractical. Only theories tell us what the experiment is about.

The next theory develops from the most advanced form of the last, according toSchwinger; from the bottom downward, according to Laughlin. Trees grow upward, down-ward, and sideways all at once, outward from their center. Here we imitate trees, notbuilders.

In the theory ϑo, some basic constructs +,∨, ID,R,g,H, i are supplied at or near thebottom. The causality form g at the field stratum F is variable because the cellular contentof the cosmos is variable. It is likely that the same holds for H and i. The bdynamicscis specified by a bdynamics vectorc D〈 , usually exponential in an action operator, thatassigns a probability amplitude E〈D to the experiment specified by an experiment vectorE〈 . The theory ϑo draws its dynamics vector from the existing theories of gravity andquanta.

It is likely that the global quantum i is emergent like g, because of how it composeswhen systems are composed. Infinitesimal symmetry generators of separate systems add;


and finite symmetry transformations multiply; but i’s equate. This signals that i comesfrom outside the systems being composed, from a place that is not changed by composition,as from stratum G, the metasystem. The centrality of i presumably results from the lawof large numbers, so it must emerge on a high stratum; stratum G will do.

A similar argument applies to g, which is shared by all systems at a common event inthe canonical theory. Einstein’s general relativization can be seen as a singular limit of aquantum gauging of g (§1.2.7).

The emergence of H is discussed in §1.2.4.The model ϑo is expressed within a general-purpose self-Grassmann algebra S = 2S

large enough to represent a queue of any finite stratum number.By a bbilinear spacec is meant a vector space provided with a bilinear form. Ev-

ery probability vector space is provided with a bilinear form H that defines its bassuredtransitionsc or hits ψ → Hψ and the relative transition-probability amplitudes Hφψ. A bi-linear form with this physical interpretation is called here a bprobability formc. It vanishesfor forbidden transitions.

If V is the vector space of a queue of stratum L then a bClifford algebrac over V desig-nated by 2V is the vector space of the queue of stratum L+ 1. 2V consists of polynomialsin the vectors v ∈ V identified modulo the Clifford Clause v2 = ‖v‖ = H v v.

This raises the question of how nature determines the probability form H. The initialspace S[0] = R has a natural norm ‖r‖ = r2, the numerical square. If a norm is given onV it propagates to 2V = Poly(V ) in a well-known natural way. One may then propagatethe norm from one stratum to the next. Unfortunately this convenient form is positivedefinite, and therefore unsuitable for the probability form of a fully quantum theory.

Queues of any finite stratum are free of the infinities mentioned above. All theirvariables have discrete bounded spectra. Every canonical quantum theory is as close asdesired in any given domain of experiment to a queue theory.

A canonical quantum theory is an approximation to a better queue theory.

In this scheme, a queue of stratum 0 has only one vector, the number 0. The queue ofstratum 1 has the 1-dimensional vector space R. The first non-trivial queue is the quantumbit, or bq bitc, of stratum 1, with two-dimensional vector space.

As several quanta combine into one, polynomials in monadics form polyadics, whichform an associative algebra, graded by algebraic grade and generated by individual vectors.This many-system algebra of a system is not to be confused with the operator algebra ofthe system. In the case of odd statistics, adducing the multiplication operation enlargesthe vector space from V to a Grassmann algebra 2V , an exponential growth in dimension.

In the exponentiation process every polyadic of the current stratum serves as a monadicof the next stratum. This process is represented here by the bracing operator ID, a lin-earized version of the bracing operation . . . of set theory. To construct an algebra thatis its own exponential, S = 2S, it suffices to close S under finite superposition and multi-


plication, and bracing.Bracing is used in theoretical physics to express the idea of “at”; for example, to link

the variables of classical mechanics to the time at which they are measured, and fieldoperators with the point of space-time at which they act. But it is usually left implicit andtherefore classical, part of the infrastructure of the theory, without operational physicalmeaning. To complete the formulation of the physical theory one should define bracingitself by laboratory operations, as one does superposition (in (2.14)) and multiplication.That cannot yet be done, Instead, such operational meaning is given to many structuresformed by bracing. It is supposed, however, that under sufficiently drastic condition a setcan separate into its factor event elements.

Bracing cannot be identified with physical binding, as of a proton and an electronbound into a hydrogen atom. But physical examples suggest that bracing underlies physicalbinding. Electric binding of atomic electrons to the nucleus rests on the bracing of electricbfieldc variables to their space-time points.

A passage from a theory of yes-or-no questions — predicates — about an individualto a theory of how-many questions — occupation numbers — about a collection can beencalled bquantificationc; William Hamilton used the word in 1846. In quantum physics itwas somewhat inappropriately named “bsecond quantizationc”. Here the older word isretained and extended, for the sake of interdisciplinary consistency. In classical theorya quantification defines a functor from state spaces of individuals to state semigroups ofcollections generated by the individual state spaces, with the semigroup product operationrepresenting the symmetric union or xor of collections or classes. In both classical andquantum usage, the product operation that is introduced by quantification melds twosystems into one.

In the full quantification of ϑo a many-quantum vector space is a Grassmann algebraover the one-quantum vector space. Its Grassmann product ∨ is analogous to the classicalpartial operation por, for which

v por v = v ∨ v = 0, (1.10)

defining no ray. In such quantum theories, the classical semigroup becomes a quantumGrassmann algebra, thanks to quantum superposition.

Iterated Fermi quantification provides a nested sequence of linear groups SL(N), oneeach for each iteration. The variables of any stratum are finite matrices, becoming arbitrar-ily large as the stratum climbs. The spectra of these observables are finite and bounded.Space-time and quantum can then be fitted together into one space-time-quantum constructof several nested strata; six non-trivial strata suffice here. The lower-stratum algebras areused here for the internal Lie algebras of the standard model; the higher-stratum ones forspace-time and Fermi-Dirac quantification (graded) Lie algebras.

The classical analogue of a queue is a brandom setc, a classical finite pure set asa statistical object; the bpure setsc being those generated from nothing by unions and


braces. (§3.1.1). This analogy is useful as a guide to setting up the algebraic language forqueues. The classical structure has multiple strata of fan-out and fan-in, and is random ina classical sense. The classical finite pure sets are elements of a group S, the state spaceof the random set, with the xor operation as group product. It is convenient to representrandom sets in the real Grassmann algebra S generated by finite addition, bracing, andcatenation, subject to the Grassmann Relation that the square of a unit set is 0. The onlyelements in S interpreted as descriptions of random sets, however, are the monomials inthe basic unit sets, which represent sets of every cardinality, without their superpositions.Their rays form a ray basis for S isomorphic to S, whose elements correspond to theclassical sets. Statistical descriptions of a random set are probability operators ρ : S→ Shaving the rays of S as eigenrays and probabilities as eigenvalues. S is a bself-Grassmannalgebrac in that

S = 2S. (1.11)

Stratum −2 has the empty vector space. Stratum −1 has the one-point vector space 0.Stratum 1 is the bit, with two-dimensional vector space. These are seeds for the whole treeof classical recursive, hereditarily finite random sets.

The canonical quantum algebra in use today replaced classical probabilities by quan-tum probability amplitudes, and modified the algebra of addition and multiplication. Ithas only one quantum stratum, and no quantum unitizer ID. This poverty blocks theformulation of stratified quantum theories, leaving quantum theory in what seems to be astage of arrested development.

The quantum analogue of the classical random set is a hypothetical quantum element,the quantum set, or queue, whose vectors are the elements of S, now with unlimitedsuperposition, without the restriction to S. S is used here to replace, regularize, and extendthe canonical algebra, replacing both Fermi-Dirac and Bose-Einstein canonical algebras.

The bcanonical quantizationc strategy in a nutshell is:Express the classical canonical system and its algebras at the dynamical stratum as

singular limit of a canonical quantum system and its associated algebra.This tacitly freezes the deeper-stratum algebras in their classical forms and perpetuates

a singularity at the dynamical stratum.The bfull quantizationc strategy is analogous but extends to all strata:Express the canonical quantum theory and its algebras at all strata as singular limits

of a fully quantum theory and its stratified algebra.For example the theory ϑo replaces all canonical Lie algebra relations [p, q] = i of the

singular theory by fully quantum Lie algebra relations

[p, q] = r, [r, p] = q, [q, r] = p (1.12)

in an orthogonal Lie algebra, on all strata.


1.2.2 The Clifford algebras of Fermi and Dirac

Two major Clifford algebras of the standard model must be reconstructed within the queuealgebra S⊗SD and so can be used to guide its design: the Dirac Clifford algebra of electronspin with four anti-commuting generators γµ; and the bFermi operator algebrac of (say)an electron assembly with generators ψξ(x) distributed over a classical space-time variablex and a spin-charge variable ξ.

These are not strictly comparable. The Fermi Clifford algebra is constructed over thedirect sum of a vector space and its dual. Namely, it is a Clifford algebra over a neutralduplex space NR NR, reviewed in §3.4.1. As the operator algebra over a Grassmannalgebra, it fits naturally into S⊗ SD.

The Dirac Clifford algebra is constructed over a Minkowski space-time tangent space.It is also presented as a Fermi algebra over semivectors by Brauer and Weyl [§3.3.4]. Ineither case it has anti-commuting tensors γµ of integer spin. This violates the usual bspin-statisticsc correlation for particles, but that applies to quantum particles, which belong tostratum E, while spin γµ seems to reside on stratum C.

The Fermi algebras used in physics impose anti-commutation relations on spin 1/2vectors in keeping with the bspin-statisticsc correlation.

The Dirac algebra used in physics imposes anti-commutation relations on operators γµ

that transform as Minkowski vectors and as operators on spinors. This is no violation ofthe usual spin-statistics correlation, which is asserted only for the particle stratum E. Theγµ, being so few in number, can be assigned to the operator algebra of a lower stratum,here designated as stratum C while its absolute stratum is being decided. Evidently 4 =C < E = 6 as conservative trial estimates.

1.2.3 The origins of i

In a real-number formulation of this fully quantum theory, the central i of the quantumequation of motion, the canonical Lie algebra, and quantum superposition, is replaced byone real antihermitian dyadic r not intrinsically different from p and q. The operator rthat becomes i in the singular limit is supposedly fixed in magnitude at its maximumpossible absolute value by a physical organization process like magnetization, and can beconsidered to be a quantized i except for its magnitude. Suitably renormalized r becomesan antisymmetric i = −iH such that the eigenvalues of −i2 are ≤ 1. In the ambient queueand in the canonical limit, i→ i, a constant.

Such an i-organization is the opposite of complexification, a mathematical processwhich adjoins an i from outside the algebra. Complexification enlarges a real algebra toa complex over-algebra of fourfold greater real dimension; while i organization reduces areal algebra to a subalgebra of half the dimension, isomorphic to a complex algebra. Thei of complexification is absolute and central. The i of ϑo is contingent on organizationand non-central except in a singular limit, and is expected to be a frozen form of a gauge


variable.Organizations of i and g on stratum F out of elements on stratum C can be compared.

Indeed i and g have similar basic tasks, to convert generators of infinitesimal transforma-tions to real variables, the conversions being bilinear and linear respectively:

g : vµ 7→ gνµvνvµ, i : q 7→ q = iq. (1.13)

Einstein replaced the constant chronometric form of special relativity by the variable oneof general relativity on the basis of the Gravitational Equivalence Principle.

The existence in present quantum theory of a constant global gauge generator i floutsthe Einstein locality principle underlying general relativity, by establishing a remote com-parison of vector phase angles. The basic relations of a local theory relate variables ata point to those in its infinitesimal neighborhood according to Einstein. The fully quan-tum version of this principle is that the basic relations relate a bcellc to adjacent cells only.Then the fully quantum theory must be chosen to be real not only for the sake of structuralstability but also for the sake of upholding the strong locality requirements underlying thetheory of gravity, localizing i and g at the same time. The imaginary i should be thelimit value of a dyadic gauge variable i associated with a gauge connection and a physicalinteraction. With the dyadic 8i, the familiar exponent of quantum gravity, ix

∫(dx)R, for-

merly bilinear in the dyadics of events, is now trilinear i in the dyadics rather than dyadic.This trilinearity [hopefully] unifies the gauge fields with their sources and approximatesthe usual Y-shaped Feynman diagrams of the standard model. (§7.4).

It is not difficult to surmise which interaction this might be, if any. In an early study iwas proposed for a Higgs variable connected with the electroweak interactions. Nowadaysthis is obviously wrong. The coupling to i is as universal as quantum theory. It cannotbe limited to the electroweak interactions. The only known interaction as universal as iis gravitation, but this is a tensor force. It seems that the gauge theory of i is the gaugetheory of some totally new and equally universal scalar interaction.

1.2.4 The origins of H

A Fermi statistics has the commutation relations

∀v ∈ V, u ∈ V D : uv + vu = u v (1.14)

where the hats are worn by creation-annihilation operators and the circ operation evaluatesthe dual vector u on the vector v. The Fermi commutation relation (1.14) neither assumesnor provides a probability form, nor an imaginary unit, not a causality form for its inputvectors v. They must have another origin. They could be provided from the bottom up,from the top down, or from both top and bottom at once.

An operation is local if it can be carried out point by space-time point. The probabilityform H defines the transition amplitude Hv u between two input vectors. Is H local? If notit is likely to be emergent., provided by a higher stratum.


The probability form seems to be local in the canonical theory; no integral kernel isneeded to form the H-adjoint of an operator. Therefore H is induced from the bottom up.

To maintain Lorentz invariance of the fully quantum theory, the induced H must beindefinite, on every stratum where it exists. And to reconstruct the standard model Hmust become definite in the singular canonical limit.

The fully quantum theory ϑo, which constructs its variables within the line of Liealgebras so(NR) appropriate to Fermi statistics, sets out to fully quantize a loop theoryof gravity, for the reasons indicated in §1.2.3, with i emergent from global organization ona higher stratum and H induced by one on a lower stratum.

Quantum variants of the brace and de-brace operations are linear operators on Sassumed to obey the ladder conditions

IDI− IID = 1, ID1 = 0, ID = H−1ITH = (HIH−1)T, (1.15)

in which T indicates the transpose. These commutation relations between I and its adjointID define a separate Lie algebra with three generators I, ID, 1 that is structurally unstableand therefore suspect ([66, 32]); but this instability seems harmless, since only a finitenumber of I strata are used, stopping here at stratum G=8.

Classical physics carries finite groups, Lie groups, and some functional groups witharbitrary functions as parameters; including permutation groups, the Poincare group, thecanonical group of classical mechanics, the diffeomorphism group of gravity, and the gaugegroups of the standard model. Many of these groups are singular. Nevertheless canonicalquantization seeks unitary representations of them all.

In a full quantization they are first approximated by simple Lie groups, here finite-dimensional linear groups acting on strata §[L] in one space S, by assuming ad hoc organi-zations as necessary.

The so(16R) of stratum-4 vectors is large enough to include the local bgauge groupscof gravity and the standard model. It is used in this way in the theory ϑo.

Each stratum of a physical theory has its own Lie algebra to be simplified. A canonicalquantum field theory has a quantum field stratum F and a classical event stratum E; in afully quantum theory all strata are quantum.

The Kaluza-Einstein-Mayer theory [30, 31] has extra dimensions on the stratum ofspace-time differentials (stratum D) and tangent vectors but not in the space-time eventstratum (stratum E). Somewhat similarly, ϑo has extra dimensions on a lower quantumstratum C. Its stratum E has the four observed macroscopic dimensions, and inherits themicroscopic dimensions of stratum C and D as well.

Quite varied catenation processes connect these strata in a canonical quantum theory.One passes from stratum D to E by integration, and from E to F by quantification withan appropriate statistics. In classical set theory, nevertheless, one mode of catenation,set formation, generates the self-exponential set S = 2S of all finite sets, which forms acommon syntactical framework for all other modes of catenation.


The strata of ϑo are treated more uniformly. They all fit into a fully quantum theorybased on one statistics with simple Lie algebras of one Cartan class. Stratum L has areal Grassmann algebra S[L] ⊂ S within the embracing self-Grassmann algebra S. This isanalogous to classical random set theory, which is its projectively commutative restrictionof S to rays of one basis.

Classically, one integration x =∫dx carries us from infinitesimal differentials to cos-

mological distances; from stratum D to stratum E. The quantum correspondent of suchan integration is exponential space formation P. There seems to be little sign of a distinctphysical space-time unit in the range of sizes between subnuclear and cosmic.

One possible indication of such a unit, to be sure, might be the cosmological constant,a certain zero-point energy per unit space volume, or zero-point action per unit space-timehypervolume. The space-time cell can be chosen in the form of a light diamond or doublecone. A hypercube about 0.1 mm across contains one unit ~ of bzero-point actionc. Thereseems to be no effective cell structure of that scale.

Therefore a single application of the exponential functor P must make the bcosmologicalleapc from subnuclear to cosmological strata. Roughly speaking, each monadic of a queuestratum corresponds to a quantum vertex of that stratum, made up of vertices of the parentstratum beneath. The cosmological leap must go from a merely large dL to a cosmologicallylarge dL+1:

The electron and the cosmos are one stratum apart.

A glance at the values of dL in Table 1.1 shows that the first candidate for the cosmologicalleap is that from stratum 5 to stratum 6. Moreover, stratum 6 is already so large thatthere seems to be no physical motive at present to place the bcosmological leapc at a higherstratum. ϑo therefore pegs the charge, differential, event, field, and meta-strata at thevalues shown in (4.33).

1.2.5 Notation

Rank and stratum are related by r ≤ L; neither determines the other.A bracketed argument as in ω[L] and S[L] generally specifies a stratum.The bhyperexponentialc 2L is defined by

20 = 1, 2L+1 = 2(2L). (1.16)

Then2L = 1, 2, 4, 16, 65536, 265536, 26, . . . , forL = 0, 1, 2, 3, 4, 5, 6, . . . .

A basis for the vectors of the strata A, B, . . ., G will be indexed by lower-case formsa, b, . . . , g of the stratum designations. For example se, se′ , . . . designate basis elements forS[E].


A basis for the Lie algebra of each stratum is indexed by corresponding upper-caseLatin letters as in ωA, ωB, . . . , ωG. For example the ωE are operators on S[E], and spanthe Lie algebra so[E]. The index A (say) can be regarded as a skewsymmetric pair aa′ witha traceless condition.

By a bcanonicalc quantum theory is meant one with at least some operators in thecanonical relations

[qα, pβ] = iδαβ , α, β = 1, . . . (1.17)

defining the canonical Lie algebra h(n). An banti-canonicalc quantum theory is one withanti-commutation relations [qα, pβ]− = iδαβ among its basic variables.

A bcanonical quantizationc respects canonical Lie algebras on every stratum where theyoccur, representing them in a Hilbert space for the dynamical stratum and and preservingthem intact for the lower strata. A bfull quantizationc replaces all canonical Lie algebrason all strata by Lie algebras of classical groups, which are simple groups of isometries ofvector spaces.

Quantum systems are considered with real vector spaces as well as complex. If V isa real vector space then C ⊗ V is a complex vector space; if U is a complex vector spacethen RU designates — improperly — any real vector space V such that U ∼= C⊗ V . Theconcepts of the real and complex theories are structured so that an operation R extractsa real theory from the complex.

For either a real or complex vector space, an borthonormalc frame of reference Fconsists of vectors of norm Hψ.ψ = ±1 Hψ.ψ = ±1 that are mutually orthogonal: Hψ.ψ′ =0 for F 3 ψ 6= ψ′ ∈ F . A vector of positive norm represents a feasible input-outputoperation. The interpretation of a vector of negative norm is discusssed in §1.3.8.

1.2.6 Q terminology

A queue has a many-system algebra with the usual stratum-preserving Grassmann oper-ations of vector addition + and composition b∨c, with the customary physical interpre-tations. It also has a brace operator I on its vectors that raises stratum by 1. This is aquantum version of the brace operator of set theory, which presumably developed from con-sidering collections of entities like sheep, stones, or taxpayers, but I applies to collectionsof quanta. It can also be written with braces or a bar:

Is = s = s. (1.18)

I metaphorically boxes its argument, converting many into one, and a polyadic into amonadic.

A many-quantum vector is a tensor with a string of indices like Tzy...ba, one for eachquantum. The operator I converts this tensor to one with a single compound index, writtenas Tzy...ba. The compound index zy . . . ba may then enter as a single index in the stringof indices on a tensor of the next stratum.


Since classical set theory has worked reasonably well in quantum theory, only changesin it necessary to conform to the quantum principle are made here.

Since the vector space of the standard model has a probability form, one seeks aprobability form for the queue. A probability form H on a vector space V induces severaldifferent forms H on the vector space 2V of a Fermi assembly. A H is assumed to be ananti-automorphism of the Grassmann algebra:

H(u ∨ v) = Hv ∨ Hu. (1.19)

Stratum L = 1 has vector space R, and it is natural to assume that the hermitianadjoint H of a real number is itself. This H propagates up the strata if one makes a suitableassumption about the effect of bracing and multiplication on the norm. The theory ϑo usesthe bmean-square normc (E:MEANSQUARENORM) for H.

In addition organization provides a quantum probability form H and imaginary i thatprovide the canonical H and i in a singular limit.

The composition of no systems is the trivial queue whose vector space is the emptyset; this starts the recursive construction at stratum −2 The quantum bbracec operator Isis a linearized version of the usual brace operation s of classical set theory.

Left-multiplication by a vector in a many-system algebra is a linear operator interpretedphysically as a quantum bgenerationc — creation — operation, and usually called a creator.

A set is usually a mathematical object, not a physical one. A queue is supposed to bea physical process, not to be confused with any of its vectors. Unless otherwise declared,queues have Fermi statistics.

The theory bϑoc [hopefully!] is a fully quantum variant of the standard model andgravity and their space-time. It has operators that generate events themselves, and theevents have Fermi statistics.

If vectors in the space V represent input processes, those in the dual space V D representoutput processes, so that w(v) =: w.v, the value of w on v, is the relative transitionprobability amplitude.

These vector processes are system–metasystem interactions. Operators represent pro-cesses that go on entirely within the system, when no one is watching. Nevertheless, sincea dyadic product of vectors is a linear operator, vectors and generators are assigned to thesame stratum here.

1.2.7 The origins of g

In the canonical theory a field g(x) of causality forms guides classical point projectiles likeplanets along their trajectories. Quanta with spin like electrons require more guidance,provided by a spin form γµ(x). In a quantum theory of gravity these forms become oper-ators on the graviton field as well as on its material sources. In the weak field limit theycreate and annihilate gravitons.


If a full quantization enlarges the event group from SO(3, 1) to (say) SO(10, 6), pre-sumably these forms grow correspondingly. The spinors of SO(10, 6)

SPINORBriefly put, Dirac defined how spins move in special relativity by mapping momentum

vectors p = eµpµ in the cotangent space (dM)D

of Minkowski space-time into Clifford-algebra elements γp = γµpµ so that Clifford’s law holds in the form

(γp)2 = g p p, (1.20)

and imposing the equation of motion

[γp+m, q] = 0 (1.21)

on dynamical variables q not explicitly depending on time, in which m is the quantum restmass. Here γ is called the (Dirac) bspin vectorc.

In classical general relativity g and γ become functions g(x) and γµ(x). In a canonicalquantum gravity g(x) and γ(x) generate gravitons.

The Dirac spin algebra is both a Clifford algebra over MInkowski vectors and anoperator algebra over spinors. Its operators are observables in a generalized sense, of aquantum system that may be called a bspinc, whose probability vectors are spinors.

For a quantum theory with probability vector space V one also requires a probabilityform H : V → V D and (especially to form an action principle) an imaginary unit i : V → V .

The 16 basic polyadics sc ∈ S[C] for C = 3 give rise to an operator algebra Alg S[C]that accommodates the spins and charges of the standard model and gravity. The fourmonadics among them are used for the spin vector γ of special relativity. In turn the fourbasic monadics arise from a lower stratum B.

This transition B → C is represented by the transition from stratum 2 to stratum 3of Table 1.1. The sb are already quantum, not classical, and require no regularization. Inqueue theory all monads are creation (and annihilation) processes, and all monadics arecreators.

Stratum D ≈ 4 is too small for the canonical Lie algebra L(xµ, ∂µ, 1) of coordinatesxµ and differentiations ∂µ of classical special relativity; but the canonical Lie algebrah(xµ, ∂µ, 1) is a singular limit of the orthogonal Lie algebra so(6) ⊂ so[C].

General relativization too is an analysis of a system into atomic elements, thoughclassical: namely, the analysis of space-time (stratum E) into tangent spaces (stratum D),one at every event. This classical analysis in turn is regarded here as a singular classicallimit of a quantum analysis of a field queue of a dynamical stratum F into cells of stratumE which are also events, each endowed with a fixed causality form g. It can therefore beexpressed using the brace operation ID on S. The variability of the gravitational space-timemanifold then derives from the variability of this collection of cells.

When the variables xµ, ∂µ of general relativity are fully quantized they become oper-ators representing generators of so[C] within so[E]. Provisionally stratum E is two levels


higher than stratumC. The fully quantized coordinates and momenta and i now generatea regular Lie algebra so[C] instead of a singular canonical limit h(4).

In the classical theory a change in frame in a space-time manifold M is representedby an element of Diff[M]. Since all is quantum, physical changes in frame might beautomorphisms in SO[E] of an event vector space S[E] relative to a probability form Hbut they cannot actually be point mappings of some classical state space. Diff[M] doesnot represent the physical group of quantum gravity any more than the canonical group ofclassical dynamics represents the unitary group of the quantum theory. This conclusion isentered into line E of the tabulation in (4.33).

Canonical quantization is nevertheless a useful guide to a full quantization, which maybe required to have a canonical quantum theory as singular limit. Canonical quantizationof gravity in the Einstein form introduces the bprobability formc H of the quantum theoryand converts the classical causality form g to a canonical quantum hermitian operator g(x)that generates gravitons.

Canonical quantization unifies both the commutator and the Dirac-Poisson BracketLie algebra into one commutator Lie algebra in which the elements of gνµ(x) of g(x) andthe canonically conjugate variable πνµ(x) no longer commute but obey canonical relationssubject to gauge constraints. The integral

∫dτ , where dτ2 = −dxgνµdxνdxµ, is now a

graviton generator as well as a proper time operator. The canonical quantization leavesstratum D and E classical, and quantizes only stratum F.

A fully quantum theory like the theory ϑo still has quantum strata D, E, and F corre-sponding to the canonical D, E, and F. The sea variables dxµ, xµ, gνµ correspond to queueoperators in associated Lie algebras so[D], so[E], so[F ], respectively. These Lie algebrasare represented within Clifford algebras of successively higher strata D, E, and F, in a waythat respects the stratum structure of present canonical quantum field theories.

1.2.8 Fields and queues

Once coordinates fail to commute, the usual classical bfieldc concept is not available. Itrequires the construct of functional relation; a fully quantum theory has the concept ofrelations between quanta but not that of functional relations. But canonical quantumfield theory already suggests an alternative pathway: A canonical quantum field is alsoa catenation of quanta; a field operator represents the input or output of one quantum.Catenation can take over many duties of functional relation. A field operator ψ(x) inputsor outputs one quantum at x. In bFermi statisticsc, ψ(x) is an operator on the vectors ofa quantum catenation; the field as a q system can be reconstructed from a queue whoseelements are quantum events. Here a vector of each stratum is a catenation of associationsof elements of a lower stratum obeying a certain statistics, in the way that a simplicial cellis a catenation of its vertices. Physics without functions is discussed further in §6.1.3.

A queue is not a field in the familiar sense of a functional relation between a field-variable and a space-time. Einstein and others advocated a bunificationc of bfieldc and


space-time into one construct; Feynman proposed to eliminate the field. The queue doesboth.

The quantum spaces of ϑo have vector spaces in one stratified bself-Grassmann algebracS. Some dimensions in one stratum C of the algebra of such a quantum space will beassigned to space-time, other dimensions of the same bstratumc will be allocated to internaldegrees of freedom of a bGrand Unified Theoryc the bstandard modelc, or to the complexplane. Dimensions of stratum F in the same algebra are to be used for dynamics andbgaugec transformations, including general covariance as a singular limit. All these vectorspaces and their isometry groups and Lie algebras are represented in the stratified bself-Grassmann algebrac S.

1.2.9 Dynamical law of the queue

In canonical theories, classical or quantum, a kinematical algebra gives some relationsamong dynamical variables at one time, and a dynamical law gives additional relations be-tween dynamical variables at different times, reducing the kinematical algebra to a quotientalgebra called the dynamical algebra. Here it is necessary to fully quantize the classicalconstruct of natural law.

The hypothesis of a fixed absolute fundamental law of nature seems to have no placein a fully quantum theory. Such a law controls the system only between input and out-put processes. Quantum measurement always violates the dynamical law of the systemmeasured.

The space-time event at which a field is measured or a quantum is created is notdetermined by the system but by the experimenter in the metasystem, so canonical fieldtheory couples system field variables to metasystem space-time variables. Its most seriousinfinities therefore arise from its theory of the metasystem, not of the system. To regularizethe theory by quantization requires quantizing at least the spatiotemporal variables of themetasystem.

This contradicts early strictures of Bohr requiring the metasystem to be describedclassically; but it accords with Bohr’s later view on this matter [13].

To avoid possible conceptual inconsistency, it must be remembered that the experi-menter and most instruments cannot function or even survive as such if they are observedwith maximal quantum resolution, as a vector for the metasystem implies. For example,measuring the positions of the electrons in a structure like an organic molecule or a crys-tal probably breaks the bonds of the structure. A maximally observed experimenter canno longer experiment. Bohr avoided this inconsistency when he restricted the descriptionof the experimenter to a classical stratum of resolution, which can be non-invasive. Asufficiently low-resolution or b gentle determinationc of the metasystem is described by abprobability operatorc ρ, not a vector. Generally an experiment is described by two proba-bility operators, an input one ρ0 and an output one ρ2, for the two ends of the experiment[45, 40].


A probability operator ρ representing a gentle determination of the metasystem canbe expressed in terms of maximal determinations by a spectral resolution

ρ =∑n

pnPn (1.22)

with probabilities pn and singlet projectors Pn. But this does not mean that performingthe gentle measurement described by ρ entails performing one of the lethal measurementsdescribed by the Pn.

A fully quantum theory modifies the canonical construct of dynamical law. Classicallythe law is treated as an extra-cosmic principle that governs the cosmos; an irresistibleaction of the unknown on the known. Quantum physics, however, recognizes that mostof the cosmos by far is necessarily unknown, and that every experiment begins and endswith violations of the system dynamics. People and instruments cannot function as suchwhen they are sharply determined at the quantum stratum of resolution. Therefore thebmetasystemc — the part of the world that studies the system — must remain almostunknown during an experiment.

Furthermore the metasystem is necessarily enormous compared to the system. Thenthe unknown source of dynamical law need no longer be extra-cosmic, and the violationof dynamical law is not mysterious. Quantum theory permits one to replace the absoluteconcepts of the known cosmos and the unknown extra-cosmos by the relative concepts ofa queue and its metasystem, neither of which admits complete symbolic representation.The dynamical law now appears as a surrogate for the metasystem. It is no longer extra-cosmic, merely extra-systemic. The law of the system might be a partial description of themetasystem.

It is therefore useful to provide a standard representation for the metasystem. Themain function of the metasystem during an experiment is to emit and absorb systemsselectively. It is common in thermodynamics to represent a heat bath for a system, whateverits actual composition, as an ensemble of many replicas of the system. There is a reasonableextension of this practice to quantum theory. One may represent a general source for asystem as an ensemble of many replicas. An input vector then represents a mode of selectionfrom this assembly, and dually for an output vector. If the system has Fermi statistics, themetasystem algebra can be represented by the Fermi algebra over the probability vectorspace:

The metasystem is one stratum higher than the system.

The queue system being on stratum F, the queue metasystem is assigned to stratum G =F + 1.

A special or general relativistic theory admits no invariant decomposition of historyinto instants. This makes history vectors more convenient than instant vectors, in the wayintroduced by Dirac and developed by Feynman, Schwinger and others. History vectors


support the representation of actions on the system between the first input and the lastoutput process, rather than actions at one reference instant in that history, propagatedto other instants by a given dynamics, Each term in a trace over histories representsan amplitude for transmission of the system through a dense network of filters betweeninput and output. These filters represent drastic but adiabatic interventions in the systemdevelopment. They change the dynamics without performing determinations. The traceover such histories expresses the adiabatically isolated dynamical development in terms ofmany different adiabatic ones.

A dynamical law is mathematically associated with a history vector that assigns prob-ability amplitudes to every other history vector. In the singular canonical quantum theorytime is a continuous real variable and in a suitable frame the quantum phase of the historyvector is the classical action divided by ~. Quantum phase, however, is relative to a choiceof basis and has no absolute meaning. The Lagrangian function is regarded as the matrixelement x′〈L〈x of a Lagrangian operator between coordinate values at two infinitesimallyseparated instants. Up to divergent normalization factors, Dirac showed, the exponential ofthe Lagrangian matrix element for an infinitesimal time can be replaced in the integral bythe matrix element of the exponential of a Hamiltonian operator, a unitary transformationfor an infinitesimal time:

ei x′〈L〈x dt/~ ∼ x′〈e−iH dt/~〈x (1.23)

It is this unitary transformation, not the exponent Lagrangian, that has invariant physicalmeaning in the quantum theory. The usual concepts of Hamiltonian and Lagrangian areuseful in the classical limit but even in the canonical quantum theory the invariant way todescribe quantum dynamics has been by a history vector, a Dirac-Feynman path amplitude.

A history vector used to express a dynamics in this way is called a bdynamics vectorc.It represents what can go on in the system between input and output operations. Suchglobal constructs are often used in gravity theory and the standard model.

Using a history vector to describe a dynamics does not mean that the dynamics is aquantum entity like the system. Every vector has one foot in the metasystem and one in thesystem. The dynamics vector puts its weight on the foot in the metasystem. One might usea vector in 2R to describe the orientation of a macroscopic linear polarizing filter, althoughthe orientation is regarded as a classical variable in the metasystem. The dynamics vectoris supposed to be determined as precisely as one likes by many experiments under the sameconditions, and therefore to be a classical object. Classical objects described by quantumvectors are also familiar from superconductivity, as descriptions of quantum organizationof many electron pairs. It seems likely that the dynamics vector for stratum F is alwaysan order parameter of a quantum organization of the metasystem on stratum G.

In one extremely singular classical limit, the system is supposed to approach the entirecosmos, the metasystem disappears to infinity, and the contingent quantum law becomesthe absolute classical law. Quantum theory relativizes absolute law rather in accord withcentury-old expressions of C. S. bPeircec, Ernst bMachc, and other antinomians.


In traditional terms, the algebra of the system is a necessary feature of the theoryand each element of the algebra represents a contingent feature. One assumes that thesystem has a fixed kinematical algebra of operators, giving the actions that can be carriedout on the system and relating them. Here this algebra is a representation algebra for aLie algebra defined by a structure tensor C, in a quadratic vector space. This assumesgood cosmic weather and large human resources, permitting experiments to be repeated asoften as desired. Within that idealized framework we assume a free choice of experimentalactions, represented by elements in that algebra. Features of experience that we take tobe necessary elements of physical theory we incorporate in the structure of the algebrarepresenting what can happen; those that we recognize as contingent we express in thestructure of an element of the algebra representing what actually happens.

The event variables that enter into the fully quantum dynamics of stratum F areusually operators of the previous stratum E. Although (as has been pointed out) space-time variables that fix the location of an input/output action are clearly metasystemic, ofstratum G, those that enter into the field operators of stratum F are presumably of theevent stratum E and so have smaller spectra. In the theory ϑo, E = 5 and G = 7.

If t[E] is a time operator on stratum E and has N values in its spectrum, with a spectralinterval of X, then one expects t[G], the time variable of the experimenter on stratum G, tohave ∼ 22N values in its spectrum. From the fact that t[G] is quasi-continuous one cannotinfer that tE is.

It was originally supposed that X is of the order of magnitude of the Planck time juston dimensional grounds, involving just the gravitational constant G, ~, and c. However westill know gravity only at the classical stratum, and our quantum theories have only muchlonger times, like the Compton periods of the particles. For current quantum experiments,gravity can be relegated to the metasystem.

The ratio of the Compton time of (say) the electron to the Planck time is a bLargeNumberc of physics. This Large Number is currently accounted for by quark confine-ment and asymptotic freedom [?]. In a fully quantum theory, low strata have a naturalmicroscopically small limit to their space-time dimensions that makes a certain kind ofconfinement unavoidable. Naturally one hopes that this confinement is the origin of quarkconfinement. It is tentatively conjectured here that the time quantum X of the queuehas an order of magnitude suggested by the particle masses and coupling constants of thestandard model, like a Compton time for the most massive quantum, not the much smallerPlanck time, which emerges in higher strata. The Large Number can then be related to thelarge multiplicities of strata ≥ 6. In the most satisfactory outcome, this quantum accountof the bLarge Numberc would converge to the Wilczek theory of the Large Number in acanonical limit.


1.2.10 The vacuum queue

Probability amplitudes for experiments have the generalized Malus-Born vector form A =D〈E, evaluating an experiment vector with a dynamics (vector). To use the theory requiresa lexicon of experiment vectors expressing basic experimental input-output operations.

For many experiments the experimental process is best described relative to an am-biant bvacuumc vector, which represents no input or output of quanta, and serves as thebackground for more interesting experiments. The condition of minimum energy seemsindispensable for a useful concept of the vacuum, so we restrict the concept to the singularlimit of a time-independent action. We divide the time into early, middle and late eras, inwhich the radius of the system is growing, near maximum, and shrinking. Then In practicethere is is a good approximation to a vacuum in the middle era, which can be arbitrarilylong in a singular limit.

Of course, no experiment is done in a vacuum; there is usually an experimenter around.What is called the vacuum is a singular limit of the metasystem. In actuality there arealways perturbations of the system by the metasystem not taken into account in the dy-namical theory.

In canonical field theories a vacuum vector is an instant vector of minimum energy,and therefore of minimum Hamiltonian. The dynamics vector determines a Hamiltonian,and the Hamiltonian determines a vacuum projector. Usually the minimum energy is setto zero, and then the vacuum instant vector 〈Ω0 is independent of time. A canonicalvacuum history vector might have a discretized form that begins and ends in the vacuum,connected by a chain of do-nothing identities.

This raises the general problem of relating fully quantum history vectors to instantones (§6.1.1).

Instant vectors are presumably eigenvectors of a time operator; a typical time operatoris given in (5.17).

In a fully quantum theory, time on stratum E has a uniform discrete bounded spectrumbetween extreme times ±T . The eigenspaces of time have dimensions that depend on time,and they are not unitarily equivalent. Here it is supposed that we live in middle times,and the vicinity of the eigenvalue t = 0 is taken as typical for present physics.

In practice one draws on experimental data as well to guess at the vacuum. These pro-cedures of the canonical quantum theory must have fully quantum correspondents (§6.4),involving the experimenter and metasystem more explicitly than the canonical limit does.

1.2.11 The cosmic crystal

On a macroscopic scale and at low energies, the cosmos near each event today resemblesMinkowski space-time. It seems to have Poincare group symmetry and to support but breaka canoncal group. The translation subgroup and the canonical group, not the Lorentzgroup, force the continuity and homogeneity of event space. From a more microscopic


viewpoint, they may be illusions of low resolution. From the time of Kaluza to todayit has become increasingly plausible that we are astronomically long in only four of ourdimensions, but submicroscopically short in most of them, like a membrane. Sorkin andSamuel developed this membraneous perception of the field into a qualitative theory ofthe cosmological constant, by an analogy between quantum fluctuations of space-time andthermal fluctuations in fluid membranes [71, 65]. The cosmos is even more anisotropicthan a DNA molecule or a soap film, with a larger aspect ratio between long and shortdimensions.

The model that is most appropriate in a quantum framework is not a membrane buta crystalline film. Quantum models are not constrained by the classical assumption thatthe extra dimensions are those of classical continua, as in Kaluza-Klein theories and stringtheories. All that is needed of the extra dimensions is the gauge Lie algebra they provideat each event. Classically, to be sure, it takes an infinite continuum of events to supporteven a finite-dimensional Lie algebra; but in the quantum theory, N independent quantumevents suffice to support an so(N) or su(N) Lie algebra. These events may constitute justone queue cell of a cosmic crystal film.

The problem is no longer what makes the short dimensions of the cosmic crystal soshort. In a fully quantum theory the cell is a given architectonic unit and a crystal filmthickness is about 1 atomic unit, requiring no explanation. Now the problem is how thefour long dimensions of the cosmic film grew so large.

But this is at least a familiar kind of phenomenon, calling for no unusual sources ofcurvature. Presumably the long space-time dimensions grew like the long dimensions of asnowflake or a graphene molecule: by organizing small queue cells into a long thin queue.

Any mathematical formulation of this structure should be tested by comparing itsBrillouin zones with the observed particle spectrum. A fit would acknowledge the insightof Newton and Fresnel, who argued for an adamantine ether in the face of much skepticism.The evidence for their crystal has not diminished with the years, if one replaces theirassumption of broken Galilean invariance by one of near-Poincare invariance. This crystalhas no rest frame and its vector has continuous symmetry as well as discrete structure.

The extra dimensions mean that each quantum cell of a fully quantum system of theworld can have a high-dimensional kinematic Lie algebra, for example, the 255-dimensionalsl(16), while residing in a multicellular system with a lower-dimensional local Lorentzsymmetry so(3, 1). Iron atoms in a magnet have a reduced symmetry because of themagnetic field they themselves create, and exhibit a Zeeman splitting in consequence.Presumably the quantum cells of the cosmic crystal film too are distorted by the anistropyof the ambient film.

The crystal film vector can be regarded as composed of struts connected by pins. Eachstrut in the truss symbolizes a spin of a classical group; and may itself be a sub-trussassembled from spins of lower-dimensional groups, by addition, bracing, and compositionTwo struts are pinned by the elements they share. If the cells of the cosmic film areN dimensional, they may organize into four cosmological dimensions if they abut their

1.3. QUANTIZATION 57

neighbors on faces of N − 4 dimensions.As a result of how displacements of the structure are distributed among its cells, a truss

dome can be much stiffer against local longitudinal displacements, which couple its longdimensions to each other, than against transverse displacements, which couple longitudinalto transverse dimensions, by a factor on the order of the aspect ratio. In gauge theory,gravitational gauge transformations couple longitudinal dimensions, and electroweak andcolor gauge transformations couple longitudinal with transverse, with couplings in the ratioof the Large Number. This suggests that the Large Number is essentially the aspect ratioof the cosmic dome.

In the fully quantum theory the possibility arises of a self-organization like a spin-alignment that forms a macroscopic coordinate, the total spin, but also freezes it at thepeak of its spectrum, so that in the canonical limit it does not appear as a coordinate atall. The theory ϑo uses this possibility to model i. It sets aside 4+2 = 6 vector dimensionsof the cell for macroscopic organization: 4 for space-time and 2 for the complex plane.Since i is a scalar, the possibility must be examined that its organization is responsible fora cosmic inflation.

In the theory ϑo, the quantum variables proper of the standard model, those withno classical correspondents, like isospin and color, are creators for the short or transversedimensions of the cosmic crystal dome. These are conveniently represented by operatorson stratum 3 of the queue, which has an sl(16) Lie algebra.

A truss that is a quasi-continuum can be assembled by stratum 6, within a kinematicLie algebra sl(264K) acting on fields of events. This suggests that the Large Number canconveniently be represented by the multiplicity of stratum 6 compared with stratum 5 orlower.

By stratum 6, exponentiation approximates Fermi-Dirac quantification. The ingredi-ents of such models are intrinsic in the self-Grassmann algebra S. The non-commutativetransport of classical gravity appears as a classical vestige of the non-commutativity ofmomenta in a quantum space-time.

While the ordinary infinitesimal concept of locality breaks down when the spectra ofspace-time coordinates are discrete, a quantum correspondent can survive as a principle ofbcontiguous actionc. This asserts that dynamical interactions and basic operators couplecells in the cosmic dome that share some elements.

1.3 Quantization

The canonical quantum theory of bBohrc and bHeisenbergc taught physicists to think aboutnature at the atomic stratum in a new way. It represented each adiabatic physical operationon an isolated atom — and ultimately on any isolated system under study — by a matrixof relative transition probability amplitudes. Classical transition probability matrices arecomposed of binary truth values 0 and 1 for deterministic processes, and of positive real


transition probabilities for Markov processes. The matrix for a canonical quantum processis composed of complex numbers instead.

The canonical quantum theory thus revises the theory of probability for quantumprocesses: Catenation of two operations is still represented by the product of their matrices,classical or quantum. This product sums probabilities or amplitudes for all paths in theindex space of the basis. Where the classical physicist adds probabilities for all paths,however, the quantum physicist adds probability amplitudes for all paths and squares theabsolute value to form a probability.

The canonical quantum theory also relativizes the basic construct of state. In theclassical theory all experimenters share the same absolute states, differing only in thecoordinates labeling the states. In the quantum theory each experimenter has a referenceframe of proper vectors, different from all other experimenter frames.

Relativity transformations between reference frames are also complex matrices. Filtra-tions, represented by projection operators P obeying P 2 = P = P †, that do not commutefor quantum systems. When two filtrations do not commute, determining one propertyinvalidates a prior determination of the other.

That class-defining filtrations commute was first explictly postulated by Boole, a lo-gician and psychologist of the 19th century, who spoke of “mental acts of election” [14],quoted on page 2, and recognized that this commutativity was empirical and subject tocorrections that would be important.

Each variable is defined by filtration operations that sort according to the value of thevariable. The non-commutativity of filtrations suspended the commutative laws for themultiplication of variables that had been tacitly assumed for centuries. Classical filtrationswere assumed to have no effect on the system filtered but able to provide complete deter-minations. Quantum filtrations are incomplete but omnipotent: any transition in a systemcan be effected by appropriately chosen successive filtrations.

Bohr described the quantum way of thought as a “painful renunciation” of classicalthought in the atomic realm. Quantum non-commutativity interferes with the Cartesianprocedure of completely representing the physical system under study by a mathematicalobject, and with the Laplacian ideal of a fundamental all-determining law. It reconstructsclassical bontologyc, a metaphysical theory of absolute beings, as a limiting approximationof a quantum bpraxiologyc, a more operational metaphysics, of operations on operandsknown only through those operations, and then not completely but only statistically. Thevectors of quantum theory are not ontological in the usual sense but bpraxiologicalc; theyare verbs describing actions rather than nouns describing absolute objects (§2.10).

Bohr’s “renunciation” is a relativization, and was quickly compensated by deeper phys-ical understandings of atoms, molecules, crystals, and nuclei. It also formed conceptualunifications unimaginable in a classical ontology.

According to the Bohr correspondence principle classical and quantum physics convergefor experiments on such large scales of action that ~ → 0 in comparison. The differencebetween classical and quantum theories is important on the scale of ~ and never entirely


disappears.1900 physicists naively treated a solar system as a set of masses, an atomic electron

cloud as a set of electrons, a line in space or time as a set of points, and a randomvariable as a set of its possible states, a state space. One catenation process, set formation,served on every stratum. They tacitly supposed that the constitutive relations found innature could be faithfully represented by those that hold among sets, and that usefulphysical probabilities could be found by counting sets of mutually exclusive mathematicalpossibilities. It was also taken for granted that an ideal measurement need affect the systemno more than counting a mathematical set affects the set. Briefly, physics was assumed tobe identifiable with a part of mathematics, the physical process with a symbolic one. Theseassumptions are part of an bontologyc that reduces the main task of physics to finding apart of mathematics that corresponds to reality. It uses a bstate strategyc to do this:

Represent a physical system by a mathematical set of states, changes of the ref-erence frame by permutations, our physical actions on the system by mappingsof states, and physical variables by real functions of states.

The bcanonical classical strategyc is the state strategy supplemented by a canonical codicil:

Represent dynamics by a continuous one-parameter group of canonical transfor-mations, with time as the parameter and a Hamiltonian function as generator.

The state strategy works reasonably well for celestial mechanics and much of elec-tromagnetics. Set theory thus serves classical physics as a “philosophical language” [88],although one that is still seriously context-dependent, in that one cannot read the physicalmeaning of a set from its set-theoretic formula, and highly redundant, in that most setshave no physical meaning.

Quantum theory replaced the state strategy by a bvector strategyc:

Represent input-output processes by vectors. Represent physical actions, in-cluding filtrations, by the matrices of transition probability amplitudes thatthey define between vectors. Represent physical variables by weighted sums oforthogonal filtrations.

The matrix product represents sequential action. A bprojectorc P (a matrix withP 2 = P = P †) represents a bfiltrationc. A weighted sum of orthogonal projectors, calledan bobservablec or bvariablec, encodes a multi-channel sortation, each term correspondingto one channel, and each weight being a value accepted into that channel.

1.3.1 Canonical quantization

In addition to defining a new kind of theory, the quantum theories, Bohr and Heisenbergprovided a heuristic strategy, canonical quantization, for reconstructing a quantum originalfrom a classical approximation. The canonical quantum strategy supplements the quantumstrategy with a bcanonical codicilc:


The commutators of certain chosen basic variables are their bPoisson Bracketsctimes −i~, and time is central.

To extend the process of canonical quantization, three stages implicit in the usualformulation are first made explicit in general terms:

1. Subquantization: Choose basic state variables q, p of the classical system whose linearcombinations will serve as subquantum vector space V .

2. Quantification: Generate the many-subquantum vector space W = Poly(V ) of poly-nomials in the subquantum vectors, defining a subquantum statistics by commutationrelations.

3. Correspondence: Express the original classical algebra as singular limit of the many-subquantum operator algebra.

For example, consider the quantum harmonic oscillator in units with mass m = 1, elas-tance l = 1, natural frequency ω = 1, and ~ = 1. If x is its coordinate and p its momentum,generating the operator algebra A(q, p) modulo the commutation relation [x, p] = i. TheHamiltonian is

H =12

(p2 + x2), (1.24)

Stage 1, subquantization: The system has a subquantum with one-dimensional vectorspace spanned by the basic vector a = 2−1/2x+ ip.

Stage 2, quantification: The subquantum has bosonic statistics , and every subquantumhas the same energy E = ~ω = 1. In this case the oscillator energy is the sum of thesubquantum energies:

H = Nω = a∗a. (1.25)

Stage 3, correspondence: The classical commutative algebra A0(q, p) of the oscillatorconsists of the elements of A(q, p) operators of the form : f(q, p) :; here f is an arbi-trary polynomial expression in q and p, and the colons : . . . : indicate that every productwritten between them is the commutative product on the non-commutative operator alge-bra defined by the bWickc bnormal orderingc, in which creators a∗ stand on the left andannihilators a on the right.

Commonly the subquantum vector space of stage 1 is not brought in explicitly duringthe canonical quantization, the canonical relation [q, p] = i~ of step 2 is not interpreted asa statistics from the start, the quantification is tacit, the only quantum algebra mentionedis the many-quantum algebra of step 3, and the subquantum and its statistics are firstdiscovered after quantization has been carried out. Full quantization, on the contrary,begins with a quantification and a statistics.

Canonical quantization cannot be relied on to produce the physical constituents of adynamical system. It can produce modes of collective oscillation instead, like phonons. One


can canonically quantize the electromagnetic field, the elastic vibrations of a carbon crystal,or the transverse vibrations of a steel E string. The resulting subquantum is the photon,the crystal phonon, or the string phonon. It is known that this quantization does notproduce the atomic constituents of crystals or strings, and we are not required to supposethat it does for the electromagnetic field. Here it is supposed that all these subquanta arelike phonons; that photons too are excitations of a collective crystal-like system, namelythe quantum space-time complex itself. Heisenberg suggested that since photons are atleast approximately massless, they are, like phonons, Nambu-Goldstone bosons associatedwith a symmetry-reducing organization of the system, even though they have spin 1.

Space-time is presumably not composed of things that move in space-time, and there-fore its constituents cannot be found by a canonical quantization, but a full quantizationputs forward candidates for the quantum elements of quantum space-time.

1.3.2 Full quantization strategy

Full quantization is full relative to canonical quantization. Canonical quantization quan-tizes one or two strata, and not completely, leaving a radical. A fully quantum strategyquantizes all lower strata, and its quantizations are completed, no radical survives.

Graded canonical Lie algebras, Fermi or Bose, are the standard statistics for step 2.Neither classical space-time nor Bose statistics is completely satisfactory for field theories,due to their singular natures, which call for infinite-dimensional representations. In thelandscape of Lie algebras, however, any canonical Lie algebra is surrounded by orthogonalLie algebras of various signatures, as well as by other algebras, both singular and regular,like a singular peak surrounded by several watersheds separated by divides. Any of thesewatersheds corresponds to a simple Lie algebra having the canonical Lie algebra as asingular limit, but having greater structural stability and fewer infinities. One of these ishopefully more physical as well. By suitably adjusting some parameters in the theory, anyof these neighbors can be made experimentally indistinguishable from the present theory bypresent-day experiments, but inevitably marked differences between the predictions appearunder extreme conditions.

Canonical quantization effects only the dynamical stratum, while full quantizationworks on all strata of physical theory, including that of the time variable. The deeperstratum of space-time, in particular, lacks the canonical classical structure, but still has aLie algebra structure, based on Lie Brackets of space-time vector fields instead of PoissonBrackets, which are Lie Brackets of vector fields over phase space. This Lie algebra toowill be regarded here as a vestige of quantum non-commutativity.

The canonical commutation relations have an essentially unique irreducible represen-tation, a remarkable and convenient property. Segal [66], Palev [59], and Vilela Mendez[75], for varying reasons, replace the singular canonical commutation relations by regularcommutation relations, which, unfortunately, lack this uniqueness of representation, andraising the question of which of the infinity of representations of these relations is physi-


cal. Fermi algebras are regular and yet have unique irreducible representations, so bFermistatisticsc reduces the problem of choosing representations to the problem of choosing theFermi algebra.

Full quantization regularizes the statistics on all strata of theory, using anticommu-tation relations rather than commutation relations, resulting in a finite and structurallystable theory. It specifies a unique representation for each stratum. Fermi statistics hasunitary Lie algebras su(p, n) instead of canonical Lie algebras. The Fermi algebra modelpursued here approximates the the graded Lie algebras of the various strata of the systemunder study by those of various strata within one bFermi algebrac Alg S, a quantum ana-logue of random set theory. It reconstructs the canonical Lie algebras, including that ofBose statistics, as singular limits of simple unitary Lie algebras su(p, n) of Cartan class A,as Palev did with the linear Lie algebras sl(n). It is not yet clear which commutation rela-tions come closest to experiment, but likely it is not the canonical, which are bstructurallyunstablec (5.1.1).

The matrix language of the canonical quantum theory is still the most practical one,and it is used here too, with additional deep structure: the vector space supporting thematrices is also a Fermi algebra and a matrix algebra..

Full quantization sacrifices the concepts of blocalityc, bunitarityc, and bfieldc, but asingular limit of classical space-time resurrects them. The fully quantum construct thatreplaces the canonical quantum field is the bqueuec of stratum F, typically of enormousgrade.

A vector of a queue has the formal structure of a chain in a classical simplicial complex.These simplicial chains are subject both to quantum superposition and orthogonal relativitytransformations. In the classical interpretation, however, a product of three states like abcis associated with the solid triangle of convex linear combinations of its vertices a, b, c,representing statistical mixtures of the three; while in the present interpretation a productabc of three vectors is associated with the space of coherent linear superpositions of a, b, cwith arbitrary probability amplitudes.

The sacrifices mentioned are compensated at once by promising unifications. Somequalitative experimental predictions are clear from the start:

1. For a system of a given extent, a large violation of the Heisenberg uncertaintyrelation is predicted at very high energy, when the organization producing i melts down,as if i~→ 0 with increasing energy.

2. Space-time meltdown is accompanied by a bdegravitizationc’ analogous to demag-netization above the Curie temperature.

A canonical field is a single-valued function of the event coordinates. The extensionof quantum praxics to space-time events makes this field construct impossible, since thequantized space-time coordinates do not commute and cannot all be specified at once, andsingle-valuedness loses invariant meaning Field theory in the usual sense must thereforebe relativized to a basis when the space-time is quantum (§6.1.3). What survives fullquantization is the more general concept of the stratified system, now fully quantum on


many strata. Single-valuedness of the field as a function of the event, a difficult problemwhen field value and event are separate quantum entities, becomes trivial when the fieldvalue and the event are identified, and the field becomes a quantum set of events.

The theories of the standard model and gravity indicate that the gross structure of theambient fully quantum space-time is highly anisotropic in the sense some of its dimensionsare very long, others are very short. The ratio of the extensions of the long and shortdimensions can be called the baspect ratioc of the system. In queue theory this anisotropyis likened to those of a snowflake, truss dome, and DNA molecule: The long dimensionsresult from the binding of many cells, the short dimensions are composed of one or severalcells. The cosmic dome is about one cell thick and cosmologically-many cells long. In thissimile, the two long dimensions of the dome stand for the four space-time dimensions ofastronomy, usually called external. The short dimension symbolizes ∼ 10 real dimensionssupporting an SO(10) usually called internal, of a theory like the Grand Unified Theoryof Georgi and Glashow [43]. The long and short dimensions of the fully quantum are notclassical continua as in the inspiring construction of Kaluza, or discrete as in an abstractsimplicial complex, but are composed of spins, as suggested by Roger Penrose, quantumvariables proper, spinning in various numbers of dimensions, depending on stratum.

In addition, in the theory ϑo, one lower-stratum spin component is aligned amonghigher-stratum cells, as electron spins in a magnet are aligned among atoms, to accountfor the quantum imaginary, and another for the Higgs phenomenon. These spins assem-ble themselves not relative to some space-time infrastructure but intrinsically, by sharedelements.

Queue cells do not occupy space-time, they constitute it.

In a fully quantum theory, coordinate variables are sums of spin variables of someorthogonal group of enormous dimensionality; the central coordinates, the longitudinalones in the truss metaphor, are composed of enough spins so that the law of large numbersmakes their sum approximately central; and the non-central, transverse to the truss, arecomposed of so few spins that their non-commutativity is salient.

Basic coordinates along all these dimensions, being sums of quantum spins, have dis-crete, finite, evenly spaced spectral values of various spectral multiplicities, like the mag-netic quantum number of an atom.

Under ordinary macroscopic observation, the classical limit, the long coordinates blurinto classical continuous unbounded variables, and the short ones, being properly quan-tum, disappear from view, except for several spins in each cell that enter into long-rangecorrelations with those in other cells and manifest themselves as the classical electrical andgravitational fields. The class of functions X → Y , usually designated by an exponentialY X , is well defined when X and Y are classical spaces, defined by commutative coordinatealgebras, or when X is classical and Y is quantum, but not when they are both quantumspaces. The statement that the field variables are single-valued functions of the space-time


variables is not strictly meaningful when the space-time variables do not commute but canbecome meaningful in some conditions in the singular limit of canonical quantum theory,and then may hold or not.

The physical topology of a line (say) in a classical space can be built unconditionallyinto the kinematics, as for a world line, or it can be a contingent consequence of dynamicalbinding, as for an iron wire. The continuum topology of a line R in a fully quantumspace-time has no invariant kinematic expression and so is likely a singular limit of a queuestructure, which is expressible in terms of the kinematics of S.

Gauge theory can be regarded as an enlargement of the gauge group of gravity beyondthe space-time translations. The present study truncates the gauge group of gravity to fitwithin the orthogonal group SO(S). The diffeomorphism group is presented as a singularapproximation to an orthogonal group of high but finite dimensionality. Einstein intro-duced dynamical interactions that account for the approximate local flatness and stiffnessof space-time, but postulated its continuity as part of the kinematics, so one must considerwhether new interactions are needed to account for the apparent continuity of low-energyspace-time within a fully quantum theory. It is assumed here that self-organization andlarge numbers suffices.

The vector space of the fully quantum theory bϑoc is formed by iterating Fermi statis-tics. Its enveloping vector space is a finite-dimensional subspace SF ⊂ S of the bself-Grassmann algebrac S.

1.3.3 Regularization

A quantum system with finite-dimensional matrices, like a spin, still supports continuousgroups, but its quantum variables, being finite-dimensional matrices, have discrete boundedspectra free of infinities. Predicates of a quantum system are represented not by sets ofstates but by flats of a projective geometry, or equivalently by projection operators in avector space affiliated with the system. Points of the geometry represent bsharpc deter-minations, lines, planes, . . ., represent bcrispc determinations, and the generic probabilityoperators represent bdiffusec determinations. bCompletec determinations assign definitetruth-valus 0 or 1 to every possible determination of the system. They are represented bypoints of state space in classical physics, and do not exist in quantum physics.

Since the matrix elements are transition probability amplitudes, the quantum theoryrenounces determinism; since its predicates do not commute, it renounces complete de-scription. In exchange it enjoys purely finite observables and continuous symmetry groups.

The canonical quantum strategy did not fulfill all the historic hopes of the quantumreconstruction. Quantum theories still contain physical continua, such as space-time, andstill give infinite answers to some reasonable questions, presumably because they still con-tain continua. Moreover they contain bquantum variables properc, quantum variables thatdisappear in the classical limit, such as electron spin and the isospin, hypercharge, andchromospin of the standard model. The canonical quantum strategy gave precious little


guidance about how to discover and represent these.

After the fact, the canonical quantum strategy was extended to admit quantum vari-ables proper by allowing graded Lie algebras defined by anticommutation relations, as wellas ungraded Lie algebras defined by commutation relations. Thus extended, the canonicalquantum strategy is less divergent in some respects but still retains continua and theirattendant infinities. Full quantization is a more powerful physical regularization processthan canonical quantization.

1.3.4 Quantification

Quantization began as a formal regularization process for radiation thermodynamics, andevolved into a procedure that produced a powerful theory of atomic spectra and dynamics.From the classical viewpoint it appears as a mysterious formal prescription, often describedas meaningless. Conversely, one who describes it as a formal prescription adopts a classicalviewpoint.

From the quantum viewpoint, however, the process of quantization has a clear physicalmeaning, whose understanding impels significant further developments in the process. Ev-ery quantization is also a dissection, a transformation of one into many. The commutationrelations defining the quantization are the commutation relations defining the statistics ofthe many.

The process of quantization set out from a pre-quantum theory of an unbounded andunresolved continuum, a 2N -dimensional state space S. The canonical quantum theoryregards S as a surrogate for a quantum space S with non-commutative coordinates. Quan-tization replaces the commutative Lie algebra of certain basic continuum coordinates of thephase space S by a non-commutative Lie algebra defining S that is closer to being simple.Here this quantization Lie algebra, whatever its structure, is interpreted as a statistics,describing how quantum units combine into the aggregate quantized system. This univer-salizes an interpretation already used in many special cases.

The quantization Lie algebra, as vector space with Lie product forgotten, becomesthe vector space of a hypothetical quantum unit of which the quantum system is actuallycomposed.

Thus every bquantizationc process can be understood to infer the existence of a quan-tum unit or monad, and to resolve the given continuous system into a polyad of suchmonads.

A quantification is then implicit in every quantization. The commutation relations ofthe quantization are the statistics of the implied quantification. The linear space spannedby the operators that enter into these commutation relations serves as the vector space ofthe quantum monad of the implied quantification.


1.3.5 Cellularization

Full quantization produces theories with regular groups on every stratum, either orthogonal(as assumed here) or unitary.

When the continuum is quantized, a fully quantum dynamics must then account forthe emergence of a connected continuum in a singular limit. This continuum is simplyassumed in the classical theory. The example of general relativity is not much use forthis. There Einstein had to add new gravitational variables to a theory without them.Here instead one must reduce an infinity of local gravitational field variables to a finitenumber of non-local queue variables. Einstein had to account dynamically for the stiffnessand approximate flatness of space-time with a new gravitational action. No new forces areneeded in a fully quantum theory to account for the apparent continuity of space-time,[presumably!] merely a bcanonical limitc and fermionic statistics (§4.4).

In a fully quantum theory classical space-time curvature arises as a classical vestige ofthe quantum non-commutativity of gauge invariant (kinetic) momentum-energy.

The program proposed is to deduce classical gravity as a macroscopic quantum phe-nomenon, a vestige of quantum theory surviving in the near-classical limit as a result ofan off-diagonal long-range cell organization analogous to magnetization and superconduc-tivity: a “bgravitizationc”.

1.3.6 The choice of statistics

Every quantification process, and therefore every quantization process, requires us to choosea graded Lie algebra for its commutation relations, which define its statistics. The choiceof statistics is one of the most vexed questions raised by this project since its inception,because the choice has affects far up the trail, but must be made at the outset. Moreover,groups of any Cartan family A, B, C, or D have groups of other families both as subgroupsand as approximants in singular limits, so the some effects of the choice of Lie algebra canbe subtle.

Ordinarily quantum physics is pragmatic on this issue, using both Bose and Fermistatistics as experiment suggests. This has been practical for the the field stratum F. Ina fully quantum theory, a quantum statistics must be chosen for lower strata B – E aswell. To narrow the daunting choice, it has been optimistically assumed here that all thestatistics in nature are of one kind, leaving corrections to this uniformity assumption untilits predictions are in.

A queue theory requires tensor products and vectors of very many dimensions to rep-resent queues of many elements. The symplectic or exceptional Lie algebras are useless forthis purpose and need not be considered for the line of statistics. They may still enter asreductions of higher stratum classical groups resulting from organization.

Then the main choices to replace the canonical Lie algebras are the Lie algebras of realorthogonal matrices like the Lorentz group, or groups of complex unitary matrices like the


unitary group of a Hilbert space, with whatever signatures work best. Correspondinglyone must choose between (graded) Lie algebras generating real or complex commutationrelations defining a Grassmann algebra or a nearby Clifford algebra.

The real Fermi statistics is explored here before the complex because the absolutecentral imaginary is unstable and gauging the operator i might provide a theory of gravityconvenient for full quantization.

The usual Fermi statistics builds a many-quantum vector space 2V as the Grassmannalgebra over the one-quantum vector space V . This seems at first like an unstable construc-tion. If one made the least generic change in the right-hand side of Grassmann’s Relationv2 = 0 it would become v2 = ‖v‖, a quadratic form in v. The Grassmann algebra overV would become a Clifford algebra over V . However the Fermi algebra makes no use ofany form on V ; it is the Clifford algebra of the duplex form, which is determined by thevector space structure of V . The Fermi operator algebra does not vary, and its GrassmannRelation is an identity.

This then fixes the initial line of groups as the groups of the Grassmann algebras S[L],the linear groups SL(2LR). These Lie algebras provides fully quantized p, q, r satisfyingthe fully quantum commutation relations (1.12).

The vector space V of the dynamical stratum F uses a probability form H in thecanonical theory. This is stable if H is regular, which is therefore assumed. It reducesthe linear group to an orthogonal group on its stratum. If there is a central imaginaryi, it reduces the orthogonal group to the unitary group of the canonical theory; with thedifference that the unitary group has a non-compact form SU(N+CN−C) so that specialrelativistic invariance is possible. Then the canonical SU(∞) has to appear in the canonicallimit, in which N →∞.

Canonical commutation relations are found on deeper strata as well, for example be-tween space-time position xµ and differentiation with respect to position pµ = ∂µ. Thesecan be represented as differential operators on the function space L2(R4). This canonicalLie algebra, to be sure, is not a symmetry algebra of the canonical dynamics, which looksvery different from its Fourier transform. But it may be assumed to be a symmetry of thecanonical kinematics of the event stratum. It is singular and produces infinities, and mustbe fully quantized here. The most economical Lorentz-invariant simple variations of thecanonical Lie algebra h(4) of xµ and ∂µ for µ = 1, 2, 3, 4 would be the Lie algebras so(6;σ)of Segal and Vilela-Mendes. For the reason already indicated, ϑo uses the so(10, 6) theoryof quantum space-time. This has an so(6;σ) subtheory of the Segal-Vilela-Mendes kind.

Fermi algebra is useful for representing skew-symmetric operator-valued tensors orforms but it contains no operator-valued symmetric tensors that can be used for a variablechronometric form g. In particular, the duplex norm ‖ . . . ‖Dup on the duplex space W =V ⊕ V D vanishes identically on both subspace V and V D.

The mean-square form provides a Minkowski metric form on the first-grade vectors ofstratum 3, used in the model ϑo as the seed of the chronometrics of special and generalrelativity.


Special relativity favors a real orthogonal group, the Lorentz group, but this originallyrepresented experience at a macroscopic classical stratum, where quantum superpositionis impossible and the quantum i is therefore unnecessary in any case.

Every quantum system in use today has a unitary group, not an orthogonal one.Smaller internal Lie algebras are both unitary and orthogonal, as su(2) = so(3). The

youngest group in particle physics, color su(3), is not an orthogonal group. The argumentfor the complex Fermi statistics is strong.

If the real Fermi statistics is adopted, then the central imaginary i of the canonicalquantum transition amplitudes has to be reconstructed as a singular limit of a non-centralfully quantum imaginary operator i of a real quantum theory of the Stuckelberg kind [72].

Both real and complex Fermi statistics are finite dimensional. The real is more stable,with smaller center, but the instability associated with a central imaginary i causes no in-finities. The real quantum theory is pursued in ϑo mainly because Einstein locality suggestsit, and because the quantum imaginary i might be the order parameter of gravitization.

1.3.7 Internal structure of the photon and graviton

One crevasse that ϑo must cross is that between the regular fermion and singular bosonfields of the standard model and gravity. The vectors of ϑo form a Grassmann algebraincorporating the exclusion principle. Therefore ϑo faces the old problem of how to assemblebosons — the photon, graviton, and the others of their ilk — from more basic fermions,but brings to bear on the problem a new tower of quantum strata.

Perhaps the way that bosonic alpha particles are assembled from fermionic nucleonscan be used as a guide, but the great rest-mass and binding energy incurred by such asmall assembly must be avoided.

This might be possible because the association between small size and high momentumis a canonical one:

The Heisenberg uncertainly relation fails in a fully quantum theory.

Moreover the Bose-like statistics for a pair of fully quantum fermions is the sl(n) statistics ofPalev. If ω is the column of basic coordinates and momenta of the quantum, the canonicalquantum relations of Bose statistics have the schematic form

Bose : [ω, ω] = ihε (1.26)

where ε is a canonical skew-symmetric symplectic form. Scale factors X, E have been setequal to 1, so h is a dimensionless version of ~. The fully quantum commutation relationsof Palev, which follow from Fermi statistics, have instead the form

Palev : [ω, ω] = Cω (1.27)


with structure tensor C for the relevant Lie algebra sl[L] of stratum L. Bose is inho-mogeneous in ω, Palev is homogeneous. For an extreme example, ω = 0 violates Bosegrossly but satisfies Palev exactly. That is, canonical quantum coordinates and momentacannot both vanish; but fully quantum coordinates and momenta can, at ranks below theorganization of i→ i.

Comparing Bose and Palev makes it plain that h and therefore the effective actionquantum ~ must include a factor l giving the maximum eigenvalue of the spin compo-nent ωY X in the symplectic XY plane for the stratum. The quantum number l varieshyperexponentially as 2L with the rank L.

Assume for example that the organization of i occurs on the field stratum F ≈ 6. Thenaccording to Table 1.1, Planck’s constant is effectively smaller on stratum E ≈ 5 than onstratum F by a factor that can be as much as 26/25 > 1020 000, kin to the Large Numberof the bcosmological leapc. This is in accord with the experience that the event stratum issuccessfully treated classically, as if ~ were 0.

TO DO Complete this estimate 080910

1.3.8 Indefinite forms

The real indefinite Minkowski form g = eν ⊗ eµgνµ on space-time requires a real indefiniteprobability form H on S in order to permit finite-dimensional representations of the Lorentzgroup by isometries. The probability form is not Minkowskian in general but may havearbitrarily many positive and negative elements in its diagonal form, depending on spin.

Each orthonormal frame in S then defines a decomposition S = S+ S− into twosubspaces with definite forms, positive and negative. The elements of S+ = in that framerepresent feasible input processes. Plausible ways to interpret vectors of bnegative normcare provided by Dirac and by Bleuler and Gupta.

In the Dirac interpretation nature has credit and can incur a system debt. A deposit ofa negative number of fermions is merely withdrawal of a positive number of dual fermionsthat happens to occur at the beginning of the experiment, when the withdrawal mustbe from the vacuum. The dual of a particle with positive energy is a dual particle withnegative energy. The antiparticle is a dual particle with energy reversed, so that particleand antiparticle can both have positive energy. Dual particles are usually pictured asholes in a sea of particles, but this does not seem obligatory. A positive net number ofinput quanta and a negative net number of output quanta result in a negative transitionprobability. This may well be forbidden, but it is not meaningless.

The need for such credit can arise when relativity transformations couple a systeminto its dual system while conserving the norm, much as Lorentz transformations couplespace coordinates into the time coordinate while conserving the proper time. This is easierto formalize for finite-dimensional fermionic systems than for infinite-dimensional bosonicones.

This outrageous application by Dirac of quantum concepts to the metasystem again


violates a tenet of canonical quantum theory, but that does not guarantee that this one iscorrect too.

In the Gupta-Bleuler interpretation, a vector with negative norm represents an impos-sible input operation, like the input of a scalar photon. Some transitions cannot even betried, let alone succeed. The number of trials of such an input and the number of transi-tions are then both 0, and so the transition probability is never negative but merely 0/0,undefined.

An indefinite norm is natural on a vector space W that is a Clifford algebra over avector space V . For elements of grade 0 or 1, the norm is the square, up to a constant.This can be extended to any grade by defining the bmean-square normc

‖γ‖ =Tr γ t γ

Tr 1=

Tr γ2

Tr 1= Grade0 γ

2 (1.28)

for a Clifford algebra W of linear operators γ on a spinor space Ψ. Clifford elements ofnegative square have negative mean-square norm, and are inevitable.

This norm is invariant under SL(Ψ). This includes SO(V ) but becomes much largerthan SO(V ) as DimV grows.

1.3.9 The representation of symmetry

Classical set theory has no symmetries; every set is special. Queues are as asymmetric asclassical sets but for a discrete overall sign-reversal symmetry

R : S→ S, I 7→ −I, (1.29)

which induces an isometry of S. All the symmetry groups of present physics must beapproximated by non-symmetry groups of operators on S.

This shortage of exact symmetries does not eliminate the theory, since there are noobservable exact symmetries in nature either, all symmetries being broken by their veryobservation [84]. The problem at hand is to allocate the small and large energy changes ina system to changes in the surface or deep structure of vectors like those in Table 1.1.

1.3.10 Finiteness

More important for the present study, such simplicity also permits finiteness. The simplegroups have a rich spectrum of finite-dimensional representations, with indefinite prob-ability forms for the non-compact versions. If the group of physical transformations ofa quantum system is a simple group of matrices, which are then finite-dimensional, thevariables of the system are also such matrices, with finite discrete bounded spectra. Aquantum angular momentum is a familiar example.

This algebraic simplicity was the compass for the second stage of the present explo-ration.


The superlative of “quantum” is “simple”.Non-commuting variables may make a system quantum but they may still have con-

tinuous spectra.All variables on every stratum of a fully quantum theory have discrete bounded spectra,

finite both in the large and the small.The move from classical theory to canonical quantum theory exchanges an allegedly

complete theory for one that less complete but is closer to being simple; but not closeenough. Canonical quantum theory is infested with infinities because certain of its groupsare compound: the bPoincare groupc, the bcanonical groupc defined by the canonical com-mutation relations, and such functional groups as the bdiffeomorphism groupc of generalrelativity, and the functional bgauge groupcs of the standard model all require infinite-dimensional representations, in which most operators have undefined products.

The semisimple case reduces to the simple. A direct sum of simple matrix algebrasrepresents a statistical mixture of simple possibilities. One measurement suffices to deter-mine which of the simple possibilities is the actuality, and then the others can be discarded.Therefore the semisimple case is usually omitted here.

Non-simplicity of a Lie algebra is indicated by its bradicalc.

Terminology : For any Lie algebra a, the bderivedc Lie algebra Da is that generated by theLie products x × y : x, y ∈ a. A Lie algebra a is bsolvablec if ∃n : Dna = 0. Thebradicalc of a Lie algebra is its maximum solvable . A Lie algebra is bsemisimplec if andonly if its radical is 0.

The existence of a radical implies action of one group element on another withoutreaction. It implies an bidolc in the senses of Francis bBaconc and bNietzschec.

The canonical quantum theory shrank the radical of the kinematical Lie algebra enor-mously but not entirely, and left radicals on other ranks untouched. Here the radicals ofall the Lie algebras in the formulation of gravity and the standard model are reduced to 0.

The first group to simplify is the bkinematical groupc of the system, consisting of allreversible physical operations on the system. For a spin this is already simple; for a particleon a line it is bcompoundc. In classical physics filtrations as infinitesimal generators belongto the radical. Clearly a system with a simple kinematical group is a quantum systemin the contemporary sense that its filtrations do not commute. A system with a simplekinematical group is also quantum in the old sense of being granular, since all its physicalquantities have discrete bounded spectra. The superlative of “quantum” is “simple”.

Often a singular limit of such a simple theory is convenient, for example to bring thepowerful tools of the differential and integral calculus into action, but this limit generallyintroduces unphysical infinities too. The proposal is that the divergent quantum theoriesof current physical interest are approximations to simple finite theories, though usuallysome variables of the simple system have to freeze for the singular approximation to work.

Moreover, the noxious compound case is never required by any experiment. If one


admits the possibility of frozen degrees of freedom, as in ferromagnets and crystals, thecompound groups are rather arbitrarily chosen singular limiting cases of simple groups thatare just as consistent with experiment.

Therefore this arbitrarily introduced non-simplicity, the root of structural instabilityand the infinity syndrome, can be gently corrected by a small variation of Lie algebraicstructure. Every compound system has simple systems in its neighborhood. They canbe reached by adding a few frozen variables and slightly varying the commutation rela-tions. Recovering a simple quantum theory from a singular theory in this way is calledbsimplification by quantizationc. The bdeformation quantizationc of Flato, Fronsdal, andcoworkers [9, 41] simplifies one group on one stratum.

The continuum of the complex matrix elements of quantum theory, seems harmless sofar. These continuous variables are probability amplitudes, and integrals over them neverappear, and so cannot produce infinities. Assuming that the probability amplitudes andthe probabilities that they define can take on a continuum of values, amounts to assumingthat an experiment can be repeated as often as desired, and that the limit as the number ofexperiments approaches infinity has meaning. This assumption of unbounded resources isclearly unphysical but does not seem to have outlived its usefulness for theoretical physics,at least if the allowed volume is also large enough to avoid black holes.

There is no physical difference, actually, between fully quantum theories based onreal, rational, or integer values for the relative transition-probability amplitudes, due tothe finiteness of the number of experiments we can carry out. Measured probabilities arealways rational, the ratios of counts. One could begin with a vector module over theintegers, and develop the vector space and its coefficient field recursively, but this wouldseem to be a presently pointless exercise.

Simplicity is still not a sharp enough guide for this project, however. Each of thesingular groups of the present theory abuts several quite different simple groups, in theway that the Galilean group abuts both SO(3, 1) and SO(4). One simple group with theproper canonical limit must be selected by other criteria. This is done here by the choiceof statistics.

1.4 The groups of nature

Nature seems to require a description with stratum structure; for example, classical fieldtheory has at least a stratum F of field histories f(x), a lower stratum E of space-timeevents (xµ), and a still lower stratum D of differentials dxµ.

Canonical quantum physics tacitly inherited the classical stratum structure. For ex-ample, quantum fields occupy a higher stratum than the space-time events on whichthey depend. But ordinarily only one stratum is described with a non-commutative non-deterministic quantum theory, and other strata are assumed to be classical, hence singular.Canonically quantizing one physical stratum and leaving the lower ones classical necessar-

1.4. THE GROUPS OF NATURE 73

ily broke long-standing constitutive relations between strata. If a theory has explicitlyclassical strata it cannot be simple or structurally stable. Each stratum has its own groupsto be simplified. Finiteness, structural stability, and true simplicity require us to quantizeall the strata used in a theory and the relations between them.

Even if each stratum were simplified separately, the entire structure would still not besimple. Quantum superposition must operate across strata for total simplicity. A strongerstrategy is needed to choose how to simplify each stratum and also the totality of strata.

Classical set algebra provides infinitely many mathematical strata of assocation relatedby a common statistics, the classical analogue of Fermi-Dirac statistics: An element mustbe in a set or not, the occupation number is 0 or 1, multiple occupation is forbidden; andthe order of elements in a set is ignorable.

The stratum counter is set to -1 for the empty stratum. Stratum 0 has the empty set1 = for its sole element. In general, stratum L + 1 is the exponential of stratum L.Peirce, Peano, and von Neumann put the natural number n on stratum n. Here we takeall the real numbers as given on all strata, for use as quantum probability amplitudes. Theoperator I maps each stratum into the next.

In bfull quantizationc, the quantum strata are exponentially constructed by one uni-versal generator of strata I.

About the association symbol I: Braces s and the bar s are more familiar synonymsof I but clumsy to iterate; Peano’s functional notation is more powerful. The quantumvariant of his ι = . . . is designated by I to avoid confusion with i.

Bracing is also called bassociationc, bmonadizationc, and bunitizationc. Bracing dis-rupts associativity: abc differs from abc.

The finite strata of classical set theory support only finite groups. To support acontinuous group, a classical state set must belong to an infinitely high stratum, whereundesired infinities are then endemic. In contrast, strata , . . . , 7 of the queue algebra Scarry the Lie groups used in present-day physics or simple approximations to them. Andfully quantum theory, like classical set theory, is fractal in that an isomorph of the entirefamily tree sprouts from each node of it. The groups of the root recur in the fruit.

Since the state strategy worked as well as it did, the canonical quantum strategy madebut a small change in it, and specifically in the associated algebra of random variables.Since the canonical quantum strategy works even better, the least change in it is madehere that results in a simple algebra of quantum variables for every stratum.

The heuristic process of bfull quantizationc. replaces the classical random-set algebraS by a bqueue algebrac bSc whose Grassmann Relation expresses the exclusion principle ofquantum physics as well as the limitation of classical sets to single occupancy.

Like the classical random set algebra S, the bqueue algebrac S constructed and usedhere is constructed of nested strata tied together by one generator I and its adjoint ID =H−1ITH. Like the Fermi algebra of electron theory, S is a Clifford algebra over S as wellas a Grassmann algebra over S.

The strata of the queue algebra S have simple Lie algebras. Indeed, the strata 2 – 7


support all the continuous groups of current physics or passable approximations to them,with room to spare; fewer strata may suffice.

An individual system with input vector space V is designated briefly by I[V ] and calleda V quantum.

A full quantization replaces the compound classical space-time continuum by a queuewith simple groups, and specifies the kind of simple group to choose at each fork in theroad. At the same time it represents existing experimental symmetry groups as closelyas necessary. It greatly reduces the number of possibilities, but still leaves a great many,mostly connected with the quantum bstratumc structure of physics.

To proceed constructively requires some alignment of physical strata with mathemat-ical. The mathematical strata have absolute stratum numbers L = −1, 0, 1, 2, . . .. Thephysical strata are given floating addresses A, B, . . ., G so that their alignment with math-ematical strata can be adjusted to experiment. The stratum assignments used in this studyare summarized in the next two sections.

Canonical quantum field theory too has several strata of association. Macroscopicvariables belong to the higher strata, while some quantum variables proper are alreadyfound in the lower. Three higher strata D, E, and F have counterparts in canonical quantumtheory: .

1.4.1 Canonical strata

The event stratum E of the canonical quantum theory is conveniently assigned to theevents x of classical space-time, with both long “external” and short “internal” dimensionsin Kaluza-type theories.

The differential stratum D supports space-time differentials dx whose aggregation isx =

∫dx.

It is not always immediately clear to what stratum in this hierarchy a construct belongs.A space-time is used in the metasystem of field theory to locate field meters and targets.To measure the field at a point, one puts a field meter or quantum detector at that point, ametasystemic operation. The space-time event then seems to belong to the metasystem, ahigher stratum than the field system. But on the other hand, in the mathematical theorythe field variable is a set of pairs of events and field values, putting space-time events intoa lower stratum than the field system. The resolution of this puzzle seems to be that anisometry of one stratum can binducec isometries of every higher stratum (§3.3.2).

In particular, if there is a discrete spectrum of time and space coordinates with spacingbXc in the event stratum E, then the spectrum of time in the metasystem stratum G, dueto parallelism, has much smaller spacing ∼ X/N where N is the number of event elementsin a metasystem element, a cosmologically large number. The time read on the wall clockis not the same time that drives an atomic vibration in the system; its spectrum of values ismuch finer and larger. Presumably both testify to an underlying set dynamics that drivesthem both.


Thus the metasystemic time can be bquasicontinuousc even though the system timehas a conspicuously discrete spectrum. This can lead to a serious underestimate of X if itis not taken into account.

1.4.2 Metric forms

Two familiar metric forms pervade the standard model, one for chronometrical structureand one for statistical structure.

The bcausality formc g defines causality and proper time through

dτ2 = −g dx dx = −gνµdxµdxν (1.30)

and is used to construct action principles at least in the canonical limit. In ϑo, the classicalgravitational potential tensor gνµ is a singular limit of a dynamical variable 〉g〈 on stratumF, the bgauge potentialc, that in the canonical limit generates gauge bosons like the gravitonor the standard model gauge vector bosons.

The bprobability formc 〉H〈 defines the hits and the relative transition probabilityamplitude

A = φ〉H〈ψ (1.31)

between two vectors. Its canonical limit is the Hilbert-space metric form of canonicalquantum theories. The probability form transforms any input process 〈 i into an outputprocess i〈 = 〉H〈 i for which the transition o〈 i is assured, called the (total) b reversalc of〈 i. In the real positive definite case 〉H〈 is the real Hilbert-space metric form of the realquantum theory.

Kaluza-Klein theories provide a clue to the origins of the chronometric form. In thosetheories the components of the event coordinates consist of four commutative space-timedifferentials dxµ and N infinitesimal elements dξ of an internal Lie group. The chronometricform therefore consists of three parts, the external part gµ′µ, the gauge vector boson gµξ,and the internal part gξ′ξ. The last of these is a quadratic form on the Lie algebra and canbe identified with the Killing form kξ′ξ of the Lie algebra.

In fully quantum theory all the differentials belong to a simple Lie algebra. The firsttwo parts of the chronometric are absorbed into the third, which is all that remains. Atleast on one cell of the appropriate stratum, one has:

The chronometric form is the Killing form.

1.4.3 Fully quantum strata

Ranks are tentatively assigned as follows in model ϑo.


Notation: bSL[V ]c is the uni-determinantal automorphism group of the linear space V . Itis generated by second-grade elements of the Clifford algebra 2V , acting as inner automor-phisms of 2V , and restricted to act on the first-grade subspace of 2V .

bSO[L]c is the automorphism group of the vector space S[L]. SO[L] is infinitesimallygenerated by second-grade elements Grade2 S[L+ 1], whose commutator Lie algebra isisomorphic to so[L].

The lowest stratum with enough dimensions for the events of cosmic history is stratum5, which allows for 2216

events. Therefore tentatively E ≈ 5. This permits F ≈ 6 and D ≈ 4.Rank 3 has SO [3] = SO(10, 6). Rank 2 has SO(3, 1), the Lorentz group of special

relativity. Therefore in ϑo strata C ≈ 3 and Bapprox2 have been assigned. Stratum Cprovides the group of the unit cell of the cosmic dome.

Therefore the probability form of stratum-B quantum theory is isomorphic to theLorentz causality form of space-time. This seems to be the germ of a unification of quantumtheory and gravity.

With 4 generators set aside for the Lorentz group, the remaining 12 generators ofstratum C are available for internal dimensions and i. This seems just enough.

Ranks −1, 0, and +1 have trivial Lie algebras of dimensions −1, 0, and +1.

1.4.4 Fully quantum self-organization

Self-organization is required to happen on or below stratum F, producing central eventcoordinates xµ, the central constant imaginary i, and the Higgs field. This sets a lowerbound on the number of subquanta composing these quantum entities; there must be atleast enough to organize themselves.

The 216 basic monads of stratum 5 could only form a near-condensation, with fractionalfluctuations as large as ∼ 2−8. Rank E could reduce fractional quantum fluctuations to∼ 2−216

. Fully quantum variants of the Dirac and Maxwell equations may then apply toone E quantum, since they are both real, and so do not require the organization of i.

Tentatively, the event stratum E is chosen to be stratum 5 and the organization of iand the Higgs variable is assigned to a field stratum F = 6.

A large number of elements is necessary but not sufficient for the organizations hy-pothesized. The choice of statistics is also critical.

In the canonical limit, the operator i is to become central and its square is to becomefixed at −1. This suggests that i is the suitably normalized resultant spin vector of apolarized aggregate of spins of an orthogonal group on stratum C beyond the Lorentzgroup, and implies bosonic-type statistics either for the constituent spins of i themselvesor for the quanta of some strong field that polarizes them.

On the other hand, the event cooordinate operators xµ are not only central in thecanonical limit but also exhibit a dense quasi-continuum of values of macroscopic or evencosmological extension.


Once the physical meaning of coordinates is defined, as by radar, the fact that theevent coordinate values in the vacuum form a quasi-continuum without macroscopic voidsis not conventional or tautological but experimental, like the absence of large voids in thedepths of the Earth. For example, radar coordinates stop at event horizons. The manifoldassumption of the canonical theory needs theoretical explanation in the fully quantumtheory. Since i and the xµ are different basis elements for the same Lie algebra, it mustalso be explained why i is effectively constant while xµ is effectively a free variable.

The easiest explanation in both cases is based on Fermi statistics. If events havefermionic statistics, then the absence of voids in the position spectrum in the vacuum may,like the absence of voids in a planet, be a consequence of the self-gravitation of the massdensity of the vacuum, measured by the bcosmological constantc.

The canonical quantum concepts of bself-organizationc, spontaneous symmetry break-ing, degenerate vacuum, or off-diagonal long-range order, have been important for thestandard model, so it is natural to seek their correspondents in a fully quantum theory(§4.4). In quantum mechanics systems ordinarily have singlet ground projectors. A ran-dom Hamiltonian almost always has a singlet ground projector. But the energy of a spins in the vacuum is 0 for all 2s + 1 basic vectors. For spins with appropriate symmetry,ground multiplets are ordinary; and a fully quantum system is composed of spins.

1.4.5 Full quantization tactics

Fermi quantification is defined and iterated here for use in full quantization.In quantifying the fermion an extension of the one-fermion theory is useful, uniting

the fermion and its dual into one bduplexc fermion of double vector dimension.If V is a one-fermion vector space, vectors v ∈ V are usually associated with input

operations and dual vectors u ∈ V D with output operations for the same fermions. Onemay also interpret both v and u as input operations for one duplex fermion, whose vectorsare the linear combinations of the usual fermion vectors and their duals, making up thebduplex vectorc space W = V ⊕V D (§3.4.1). The duplex fermion can be either the fermionor its bdualc.

For any vector space V there is a natural hermitian norm ‖ . . . ‖Dup on the duplexspace W = V ⊕ V D =: DupV , here called the bduplex normc:

∀v ∈ V, u ∈ V D : ‖v + u‖Dup = 2w.u, (1.32)

in which w.v is the value of w on v [62].A form of signature 0, and its bilinear space, are called bneutralc or bKleinianc. The

duplex norm ‖w‖Dup is bneutralc. Its polarization is the bduplex formc HDup.A neutral norm on any space W is isomorphic under GL(W ) to its own negative:

‖w‖ ∼= −‖w‖. (1.33)


The duplex norm ‖w‖Dup requires no metrical form on V for its construction; itsinvariance group goes beyond the usual isometry group of a quantum theory with vectorspace V , and includes the linear group SL(V ).

The original vectors of V and V D are bsemivectorsc of W .The orthogonal group for the duplex norm ‖w‖Dup is taken as the relativity group of

the duplex fermion. This relativizes the customarily absolute distinction between inputand output, and between quantum and dual quantum.

Now a fermionic quantification V → 2V , W → 2W is constructed. When V is thevector space of an individual quantum I V , 2V is the vector space and algebra of a Fermiassembly of such individuals, and 2W is the Fermi operator algebra of that assembly.

2W is still the Grassmann algebra of polynomials

2W := Poly(W : w2 = ‖w‖) (1.34)

identified modulo the Grassmann relation, as indicated.The familiar bFermi operator algebrac over V , FermiV , is the Clifford algebra over

the duplex space W = V ⊕ V D with the duplex norm ‖ . . . ‖Dup. This is

FermiV := 2VD ⊗ 2V = 2V⊕V

D:= Poly(V, V D), (1.35)

the finite-grade polynomials in V and V D modulo the Fermi-Dirac relation

∀v ∈ V,w ∈ V D : (v + w)2 = 2w.u. (1.36)

FermiV is itself the operator algebra of 2V := GrassV , the Grassmann algebra over V ,which may be taken to be a vector space for the Fermi assembly. But any isomorphic spacesuch as GrassV D = 2V

Dis just as good for this purpose.

The step from the vector space of the individual to that of the Fermi assembly isrepresented by the functor Grass. This takes care of the incidence relation ø defining themisses, but provides no probability form H defining the hits.

Iteration of Fermi starting from the empty Clifford algebra leads to the bself-Fermivector algebrac

S = 2S. (1.37)

The Fermi operator algebra of a Fermi assembly of individuals with vector space V isAlg 2V , the algebra of linear operators on a Fermi vector algebra V , or an isomorph.

It is mathematically possible to interpret any Clifford algebra 2V as a quantum setalgebra. V holds the vectors of the quantum elements, and 22V holds the vectors of aquantum set or queue of such elements. The queue may have any number of elementsup to the dimension of V . If one specifies a basis and suspends superposition the queuereduces to a classical set.

The bqueue strategyc is also a simplification strategy, because queues have simplekinematical groups.


A physical system represented by vectors in the power space over a given vector spaceV is thereby endowed with a graded Lie algebra, that generated by vectors of V , thebgenerating vectorsc of the system.

The usual Fermi and Bose statistics are not simple as (graded) Lie algebras, due to the0’s in Fermi law ψ2 = 0 and the Bose law [1, q] = 0. But the generating vectors of thesealgebras provide familiar examples of generating operators.

bCliffordc called his algebra a bgeometric algebrac and restricted it essentially to theclassical differential stratum. bFermi statisticsc (§1.4.5, (1.35)) uses a Clifford algebra not asa geometric algebra but as a q algebra, to represent a quantum power set. The elements ofthis Clifford algebra are vectors, as in Dirac’s Clifford algebraic formulation of Fermi-Diracstatistics [26]. The geometric interpretation is then a classical vacuum of the quantuminterpretation. Assumption 3 implies that physical systems have Fermi statistics. SincebFermi statisticsc covers Palev statistics as well this is not immediately absurd, especiallyin view of the re-analysis of the bspin-statisticsc correlation in §3.4.3.

Dirac constructed Clifford algebras that unite a Grassmann product with a causalityform, as in the Dirac equation, or a probability form, as in Fermi-Dirac statistics [26]. Thebmean-square formc on S[2], stratum 2 of the self-Grassmann algebra S, is Minkowskian,and remains indefinite for all strata ≥ 3. It is easy to see from the matrix representationof the Clifford algebra that the signature σL of the mean-square form of stratum L is, forL > 2,

σL =√

2L. (1.38)

This leads to the chronometrical hypothesis (Assumption 8): that the probability form(a pseudo-Hilbert space metric) 〉H〈 for vectors of stratum F and the causality form 〉g〈for space-time vectors of stratum E have a common origin in the probability form of alower stratum B, where they coincide (§1.4.3). C is the lowest stratum whose isometrygroup SO[C] includes a Lorentz group, an approximation to the Poincare group, and theunitary groups of the standard model. The stratum used for this purpose will be calledthe bcharge stratumc C. In the model ϑo C ≈ 3.

S is used in a heuristic process of bfull quantizationc, which converts non-simple Liealgebras into simple Lie algebras of queues by small variations, inferring frozen variableswhen necessary. This is recapitulated in §4.3 and then tested on bgravityc.

In field theory, the space-time manifold is part of the metasystem and is used to fixwhere and when we make our field determinations. Its continuity produces infinities infield theories and has no experimental basis, one can only say that if space-time has atomicelements then we have not yet wittingly resolved them; though in the present state ofphysical theory we might trip over them without recognizing them. This leaves open thepossibility explored here, that just as quantum spins of so(3) are quanta of bPenrose spacec,quantum spins of a higher so(n) are quanta of space-time-matter.

Here we extend quantization five strata down from the field stratum F with a simpleLie algebra on every stratum.


1.5 Fully quantum regularization

To simplify the groups is to regularize the theory; to fully quantize the theory is to simplifyits groups.

A canonical quantum theory may have a single-quantum stratum and a many-quantumstratum, related by Fermi or Bose statistics. For a gauge theory like the standard model athird stratum is essential, for differentials dxµ and infinitesimal displacements, which thegauge vector field represents. This stratum is supposed to be lower than the single- andmany-quantum strata, but classical nonetheless. In the fully quantum theory the strataare related by the quantum bracing operator I.

1.5.1 Fully quantum dynamics

The fully quantum kinematics is founded on a vector space for quantum histories, which arequeues of quantum events. A fully quantum dynamics is defined by one specific dynamicsvector 〈D in this space. A maximal experiment is then represented by a dual vector E〈such that the transition amplitude for the experiment is E〈D. In canonical theories anexperiment vector 〈(x) can be defined by a sequence of points (x) = x1, x2, . . . , xNapproximating a classical path, and a dynamics vector can be cast into the action form

〈D = eiR

(dx)L/~ (1.39)

with action density L.In a fully quantum theory classical space-time is replaced by a fully quantum space.

The basic form survives: There is a dynamics vector, and its phase has a contribution fromeach cell of the quantum space, now combined not by integration but by addition (3.20).

Without the time continuum, one can no longer assume that vectors must obey a dif-ferential equation that contains time differentiation only linearly, as canonical quantumtheory does. This can at best be a property of the canonical limit of classical time. There-fore one should expect observable corrections to this widespread assumption. Since thetime continuum is an artifact of a singular limit, small corrections involving second orhigher time derivatives, or integral transforms, can probably improve the approximationof continuous classical time but cannot make it exact.

1.5.2 Gauge

A c gauge theory is a dynamical theory whose dynamical variable is a connection on afiber bundle F → B → S whose base space S is space-time, whose fiber F is the space offield values at a point [?], and fiber group G : F → F is the gauge group. The prototype,general relativity, has gauge group Diff = Diff(S), the diffeomorphism group of the space-time manifold. Its Lie algebra is

diff = Polyµ(x)∂µ, (1.40)

1.5. FULLY QUANTUM REGULARIZATION 81

linear combinations of ∂µ with coeficients polynomial in xµ.It has not been possible to say exactly what a q gauge theory is, due to the great

singularity of the c concept, which combines the algebraic singularities of the base spaceand of the space of connections. It is in the q spirit is to preserve the group Diff exactly, onthe grounds that it operates on a differential stratum D < F, deeper than the field stratumF. Diff is part of the concept of a classical history, and q sums over classical historieswithout modifying them.

We may specify the connection by a gauge differentiator Dµ(x). We avoid some singu-larities by taking G to be simple. Some of the infinity of commutation relations responsiblefor the other singularities are, schematically,

[D,x] = 1,[x, x] = 0,[1, x] = 0,

[D,D] = F,[F, F ] = 0,[F, x] = 0,

... (1.41)

In a bcanonical gauge transformationc, an isomorph of a gauge group G acts on a spacefx of field variables at each space-time event x. It is proposed here that gauge groups ariseas the representations within the relativity groups of higher strata of the quantum relativitygroups of lower strata. In vartheta(2008) the field variables and space-time coordinateswork on an event stratum E, two ranks of association above a charge stratum C whoserelativity group G[C] is the gauge group “of the first kind,” here called the C gauge group.

In a canonical gauge field theory there is also a much bigger gauge group “of the secondkind” acting on the field stratum F. If X is the space-time manifold, the gauge group ofthe second kind is GX , the space of (differentiable) maps g : X → G. GX acts on thespace F = fX of differentiable fields, which are maps X → f . Here the gauge group of thesecond kind is the relativity group of stratum F, or the F-gauge group.

In a fully quantum theory there is thus a kind of gauge group for every stratum L,namely the relativity group of that stratum, called the L-gauge group. Some are shown in(4.33).

Gauge theories are particularly attractive to theorists because they enable us to de-duce a dynamical interaction from an assumption of symmetry, under plausible ancillaryassumptions. A gauge theory produces a most-favored action principle for study, deter-mined up to several parameters by the gauge group and general principles like maximallocality and minimal coupling. Such a gauge action can be fitted to experiment withrelatively minor surgery.

Fully quantum gauge theories are as fertile for action principles as canonical ones,but still require ad hoc assumptions, such as self-organization like the Higgs phenomenon.


Some experimental data must be used up in defining these organizations. It remains to beseen whether there will be any data left over to test the theory.

The most obstinate singularities encountered are those of gauge theories, especiallygeneral relativity. Gauge field theory has been so fruitful that it seems worthwhile to fullyquantize it and eliminate its infinities.

The argument by Einstein from gauge invariance to singularity is one of the beautifularguments of theoretical physics:

A gauge transformation could be the identity in the past and present and not in thefuture and still be infinitely differentiable. It would then change some future variablesbut no past variables. Therefore gauge invariance implies that the past variables do notdetermine the future ones. Therefore it must be impossible to cast the variational equationsinto canonical Hamiltonian form, which determine the future by the past. Therefore it mustbe impossible to express the velocities in terms of the momenta. Therefore the Hessiandeterminant of the Lagrangian with respect to the velocities must vanish:

∂2L

∂q∂q= 0. (1.42)

This is clearly a structurally unstable condition, but it is a consequence of generalcovariance. Therefore it must be concluded in any fully quantum theory that generalcovariance – and any other canonical gauge invariance, since they are all singular — is anapproximation to a fully quantum gauge invariance, whose gauge group is regular. Theusual gauge theory breaks down when the canonical limit does, for example at the highestenergy.

The singularity of present gauge theories also shows up in the divergence of integralsover histories of the gauge field. If one integrates over all histories then each history isaccompanied by a wildly infinite set of all its gauge transforms, having the same physicalmeaning. This redundancy causes the integral to diverge badly, by an infinite factor thatcounts the redundant occurrences of a given physical field. It has been possible to eliminatethis infinity by the BRST method ([80], Chapter 15) which selects — or at least favors— those gauge potentials that obey some gauge-fixing condition and excludes — or atleast attenuates — redundant copies of each physical field configuration. This methodintroduces bghost particlesc that are designed never to appear in experiments. The theoryremains structurally unstable but is renormalizable with infinite renormalization factors.This is generally understood as indicating that the renormalizable theory is the singularlimit of some finite theory.

Here a queue theory provides that finite theory. In it the gauge group and its rep-resentation become finite-dimensional, and the divergent integration over gauges becomesa finite trace over histories of the system. A fully quantum gauge group is not a func-tion group but a classical group. In the theory ϑo it is an SO(n+, n−) of some stratum.The usual functional gauge group is a singular approximation to the fully quantum gaugegroup. The redundancy factor is merely the dimension of the representation space of


the fully quantum gauge group. There is no need to fix the gauge and break the gaugeinvariance, or introduce ghosts. One simply does the trace.

The condition that the present determine the future is already weakened in canonicalquantum theory, but a quantum determinacy survives in the canonical quantum theory:The dynamics defines a unitary transformation that maps present vectors into future vec-tors. This quantum determinacy allows us to hit with certainty any one desired targetin the future that we choose to, telling us exactly what input will accomplish the desiredoutput. If after such an input we change our minds and insert some other output target,the quantum determinacy does not predict the outcome with certainty as classical theoriesdo. Since vectors do not determine what happens in all possible experiments, the quantumdeterminacy is a weak statistical kind of determinacy.

In canonical gauge theory this weak determinacy is weakened further. The past gaugepotential does not determine the future gauge potential, because it does not determine theexperimenter’s free choice of gauge frame.

This reduced determinism is still too much for structural stability, and is further re-duced in a fully quantum theory. Any fully quantum time variable is a finite matrix,although usually a large one, Newton’s concept of time derivative does not apply. It isimpossible to increase the time coordinate of one frame above a certain maximum eigen-value. Different time values even have eigenspaces of different dimensionality, smallest forthe extreme time values and greatest for the middle time value. Therefore the passageof time does not define a one-parameter group on the vector space, although there is aregular, fully quantum correspondent for the canonical diffeomorphism group.

General relativity describes how the matter in space-time influences the causalityform in analogy to how electric charges influence the belectromagnetic fieldc. Einstein’sbgravitational fieldc and Maxwell’s electromagnetic field have both matter-determined(Newtonian, Coulombian) and matter-independent (radiative) parts in each frame. Thematter-determined part, however, is determined by generalizations of the Gauss Relation−∇·E = ρ. These are differential bconstraintsc arising from the existence of field variableswhose rates of change do not appear in the dynamical action principle.

These constraints result from the singular nature of the relevant full gauge group. Agauge transformation

Dµ(x) 7→ Γ(x)Dµ(x)Γ†(x), x 7→ x (1.43)

couples space-time coordinates x into the gauge coordinates in D without a reverse couplingof gauge coordinates into space-time coordinates, creating a radical in the commutatoralgebra.

A fully quantum theory, we have seen, can eliminate the fundamental distinction be-tween field and space-time variables; they are both coordinates of the event, possibly withdifferent scales of organization. It thereby regularizes these singular full gauge groups andthus eliminates the gauge constraints, resurrecting them only in a singular organized limit.The photon and graviton presumably have small masses, though possibly cosmologically


small.

1.5.3 Spin

What remains of the classical distinction between matter and field in the canonical quantumtheory is the separation into integer spin bosons, which can organize into classical fields,and half-odd-spin fermions, which can organize into classical matter. In a fully quantumtheory, it seems, half-odd-spin is anomalous, a result of vacuum organization of integer-spinelements as seen in a singular limit.

080530 . . . . . . [Merge above with following, eliminating redundancy]Canonical gauge theories have not only global conservation laws for some kind of charge

but also local conservation laws: not only is total charge constant but its current is con-tinuous, the charge increase in any region is the charge influx through the boundary of theregion. This brings in the topological concepts of the boundary and current-continuity.In canonical quantum theories a finite-dimensional gauge group on the stratum D of dif-ferentials induces in higher strata both a finite-dimensional representation connected withgauge charge conservation and an infinite-dimensional enveloping group connected withgauge current continuity by bNoetherc. These are called gauge groups of the first andsecond kind by Dirac. To leave room for the groups of other strata, they are called D andF (stratum) gauge groups here.

Charge conservation can survive in a fully quantum theory as a commutativity relation

[E, Q] = 0 (1.44)

between skew-symmetric operators of charge Q and energy E. The non-locality of thetheory suggests that there is no exact fully quantum correspondent of current continuity.The first grade of stratum D of ϑo supports an so(7, 5) and the second-grade represents it.The Lie algebra of space-time-momentum-energy is modeled in this Lie algebra.

The stratum D operations ωα5 ∈ so(7, 5) are not local in space-time in the sense ofthe corresponding partial derivations ∂µ, but in a weaker sense. Consider for example thetransformation ω45 ∼ i that represents energy in stratum D. ω45 replaces each vertex s4

of a cell by another, s5. Two cells differing by one vertex contact each other on maximalsubcells or faces, and are contiguous in the first degree. Briefly, ω65 couples cells of thequeue that are contiguous in the first degree.

A gauge system is generally believed to result from microscopic organization, withgauge fields as order parameters. Sommerfeld, for example, noted that the electromagneticfield can be reinterpreted as a vibrating relativistic elastic solid [70]. The bVolterra-Burgersctheory represents bdefectsc in a crystal organization by a gauge field, the Burgers vector.This theory has been extensively developed in a classical framework. Can queue theorydescribe quantum defects in the vacuum organization, in a quantum version of Volterra-Burgers theory?


For any stratum L, the orthogonal group SO [L] is the automorphism group of thequadratic space S[L].

The groups of classical gravity and their proposed fully quantum correspondents are:

1. Lorentz and Poincare Groups←SO[C].

2. Diffeomorphism Group←SO[E].

3. Canonical Group←SO[F].

Any gauge theory has analogs of this triad of groups in its D, E, and F stratum gaugegroups and its kinematical group.

The self-Grassmann algebra S includes the stratum-F vector space for the queue. Thismakes possible a simplifying assumption that accommodates gauge theory.

Assumption 4 (Fully quantum gauge) The C gauge group is the orthogonal groupSO[C] of the first grade of stratum C. The E gauge group is the orthogonal group on theevent stratum E induced (in the sense of §3.3.2) by the C gauge group. Gauge invariance ofthe canonical quantum theory and the classical theory are singular limits of an organizationof the q vacuum on stratum F .

This organization of the set stratum F is described in Chapter 6 and (4.33).

1.5.4 Real quantum theory

The Lorentz Lie algebra is both the real matrix Lie algebra so(3, 1) and the complex matrixLie algebra sl(2C). It is not clear yet whether the fully quantum theory that best replacesthe complex canonical one is better chosen to be real orthogonal or complex unitary.

The use of i in the canonical theory is to provide an absolute connection betweenoperations and observables; that is, between the Lie algebra of infinitesimal isometries,which are skew-hermitian, and the Poisson Bracket Lie algebra of classical observables,which are hermitian. The real theory is harder than the complex theory because it lacksthis connection, and because in it one cannot diagonalize all normal operators. On theother hand, the real theory virtually compels one to account for the i of the canonicalquantum theory as a self-organization, which can have important physical consequences.

Moreover, complex Lie algebras are not structurally stable within the manifold of realLie algebras, due to the instability of the centrality condition [i, a] = 0. The canonicalquantum i is already non-central in the weak sense that it does not commute with timereversal T . Ultimately one should expect a real quantum theory, therefore, in which i andtime t result from self-organizations, perhaps the same one.

Nevertheless, the complex imaginary i seems to be responsible for no infinities, and socould be tolerated. One could postpone the real theory and work within the manifold ofcomplex Lie algebras.


Here instead a real quantum theory is studied from the start. The main purpose forthis choice is to explore new gauge phenomena arising from the bquantized imaginaryc i,which seems to be a candidate for a bHiggs fieldc.

The main goal of bfull quantizationc is still to regularize present field theories andeliminate infinite renormalizations. But hopefully once again a quantization will produce atheory that also fits experiment better than its singular ancestor, and has greater conceptualunity. One can account for the success of complex quantum theories up to now by exhibitinga physically plausible candidate for a bquantized imaginaryc i that has the classical i as asingular limit, and constructing a vacuum in which i is frozen (§6.4).

Since the dynamical i is not central, the quantization relation between classical Pois-son Brackets and quantum commutators must be weakened and generalized. To make asymmetric operator S from an antisymmetric one A using an antisymmetric i, the productiA does not always work. It is replaced by an anticommutator:

Assumption 5 (Full quantization rule) Canonical commutation relations of the forms

[A,B]+ = i~C or [A,B]− = C

are limits of fully quantum commutation relations of the forms

[A, B]+ =~2i, C or [A, B]− = C (1.45)

respectively, in the singular limit X→ 0, A→ A, B → B, C → C.

1.5.5 Root vectors and quanta

Heisenberg’s path from discrete atomic spectra to bquantum non-commutativityc was ele-gantly explained by bConnesc ([21] page 37). In the classical theory of integrable systems,the observables are almost periodic: q(t) =

∑qωe−iωt, and the algebra of observables is

the group algebra (convolution algebra) of the discrete spectral group — the multiplicativegroup of the system frequencies e−iωt). Both the algebra and the group are commutativein classical mechanics. In atomic spectroscopy, however, the frequencies have two indices,as in e−iωnmt labeling the input and output to the transition.

If E1 < E2 < E3 are atomic energy strata, the transition 2→ 1 is found immediatelyafter the transition 3 → 2 but not immediately before. The spectral group is thus empir-ically non-commutative: Therefore its group algebra, presumed to be still the algebra ofobservables, is non-commutative.

A discrete spectrum of transition frequencies practically implies non-commutativity.The essential point for this non-commutativity is not the discreteness, however, but thefact that any observed frequency belongs to a transition of the system, not a state of thesystem. Non-commutativity alone does not guarantee a discrete spectrum. In Heisenberg’soriginal quantization, qp − pq = i~. This guarantees that neither p nor q are quantized.


They are continuous, not discrete. The canonical quantum theory implies no granularityof either p or q, but leads to granularities of energy further along the road depending onthe choice of a Hamiltonian operator. From the viewpoint adopted here, the canonicalquantum strategy is only a first step in the direction of a fully quantum strategy.

A deeper kind of bnon-commutativityc that necessarily leads to granularity in a quan-tum theory was formulated by bKillingc and bCartanc before the quantum theory. This isthe bnon-commutativityc of a bsimplec (or bsemi-simplec) Lie algebra. If a is a simple Liealgebra and H ⊂ a is a maximal commuting independent (Cartan) Lie subalgebra of a,then there is a basis of simultaneous eigenelements Ψn ∈ a of all h ∈ H:

h×Ψn = ΨnEn(h). (1.46)

This says that Ψn increments each h by the quantum En(h), a broot vectorc of a. The h aretherefore granular properties and En(h) is the size of the granule of h in the Ψn dimension.Quanta correspond to root vectors of a Lie algebra interpreted as a quantum statistics.

The canonical Lie algebra lacks eigenelements Ψ with non-zero granule sizes, sinceamong its h ∈ H is the scalar 1, and 1 × Ψ = 0 for all Ψ ∈ a. When the Lie algebra a isbsimplec (or semisimple), there is a complete set of eigenelements Ψ, defining granule sizesof all the Hα.

The strata of fully quantum theory are all quantum but are analogous to those ofclassical set theory, which are all classical. One or possibly two strata of canonical quantumbfieldc theory are quantum, and all the lower strata are classical. Here all of these strataare quantized:

1.5.6 Full Fermi quantization

bFull Fermi quantizationc is a heuristic process that presents a singular physical theory assingular limit of a regular fully quantum theory whose vector space is a self-Grassmannalgebra within S. The present version approximates the Lie algebras of the singular theorywithin linear-group Lie algebras sl(n) . The resulting fully quantum theory is simple andfinite, and hopefully works better than the singular theory. A bfull quantizationc is setup for the bstandard modelc and bgravityc, where all observables have bounded discretespectra.

The correspondence principles of fully quantum theories state that the fully quantumtheory agrees with the canonical one in a certain limit X→ 0. This provides simple regularvariants of present-day singular gauge theories but is not strong enough to define the fullyquantum theory completely. Further specification is needed to define the various charges,groups, and masses that are built into the fine structure of the fully quantum set theory.Analogously, the correspondence principle did not lead to definite predictions in the domainwhere ~ is not small. The possibility of arbitrary terms of order X must be excluded, likePauli magnetic moments in the Dirac equation. Pauli moments were excluded by the


bminimal couplingc principle and by specifying a formal expression of the classical theoryon which gauging, the replacement ∂ → ∂ +A, could then be carried out.

The gauging process that led to general relativity is similar, and is referred to asbgeneral relativizationc. In general relativity, the relativistic dynamics was defined up toone cosmological constant by the assumptions that the dynamics was gauge invariant andlocal of bminimal differential orderc. The condition of minimal differential order can beregarded as a condition of maximal locality, permitting the action to couple only events inthe smallest possible cluster of nearest neighbors. Analogous assumptions are made in fullyquantum gauge theory and define an action function up to a coupling constant. To makepredictions of experiments in the current ambient vacuum, however, additional parametersdescribing the organization of the vacuum may be required.

1.5.7 Physical Lie algebras

We ask here which physical Lie algebras require structural stabilization, as in full quantiza-tion. In the following listing, Lie algebras are allowed to be graded unless otherwise stated,and so the list includes Grassmann and Clifford algebras, defined aby anticommutationrelations rather than commutation relations.

1.5.7.0.1 Statistical Lie algebras A bstatistical Lie algebrac is a Lie algebra gen-erated by input-output operators of elementary systems and their graded commutators,representing the statistics of the systems. It is used to generate associative operator alge-bras whose elements represent physical actions and transformations.

Example 1: The three-dimensional Fermi Lie algebra generated by one 0-grade element1 and two first-grade elements a, a∗ , with [a, a∗]− = 1, [a, 1]+ = 0 = [a∗, 1]+.

Example 2: The three-dimensional Bose Lie algebra generated by one 0-grade element1 and two first-grade elements a, a∗ , with [a, a∗]+ = 1, [a, 1]+ = 0 = [a∗, 1]+.

Example 3: The three-dimensional Clifford Lie algebra generated by one 0-gradeelement 1 and two first-grade elements a, b , with [a, a]− = 1 = [b, b]−, [a, b]− = 0,[a, 1]+ = 0 = [a∗, 1]+.

In the present work all the other algebras, Lie or associative, that occur in physics arebuilt out of statistical Lie algebras.

1.5.7.0.2 Kinematical Lie algebras By a bkinematical Lie algebrac is meant a Liealgebra of the infinitesimal isometries of a vector space with the commutator as Lie product,used to represent infinitesimal operations on the system under study.

1.5.7.0.3 Generating Lie algebras By a bgenerating Lie algebrac is meant one thatgenerates a kinematical Lie algebra. Examples: The three-dimensional canonical Lie alge-bra h(1) with [p, q] = i~ generates the kinematical Lie algebra of operators on the Hilbertspace L2(R) with the commutator as Lie product.

1.6. GRAVITY AND OTHER GAUGE FIELDS 89

1.5.7.0.4 Invariance Lie algebras are subalgebras of kinematical Lie algebras leavinginvariant some structure of the system such as the Hamiltonian. Example: The PoincareLie algebra of special relativity.

Full quantization derives all these Lie algebras from one source, statistics. It usesstatistical Lie algebras as generating Lie algebras, uses these to form kinematic Lie algebras,and finds the symmetry Lie subalgebras within the kinematic Lie algebras from experiment.This replaces the old assumption that “everything mathematical object is a set” with aquantum version: “Every physical system is a queue.”

To define a queue it remains to choose the statistics.The Clifford statistics is selected for study here for the following reasons.A set contains each element with occupation number 0 or 1, and is unchanged by

interchanging two of its elements. bFermi statisticsc has these properties, but is structurallyunstable, Grassmann’s Relation a2 = 0 being the singular limit λ→ 0 of Clifford’s Relationa2 = λ. Clifford statistics also includes a structurally stable Palev version of Bose statisticsas well. Clifford algebra is used in classical physics to describe the chronometrical structureat one space-time point, and in quantum physics as the spin algebra of leptoquarks; it isassumed here that these are vestiges of the deeper Clifford statistics of the elements ofquantum space-time. So it is Clifford statistics that is iterated here to form the fullyquantum theory.

In the best outcome, the operations of a fully quantum theory would have a uniformphysical interpretation so that every polyadic would have a physical meaning that couldbe read from its structure, with no redundancy. This would make queue theory a universalphysical language in a strong sense, but one without self-checking. This is considereddesirable here, but is not assumed.

Dynamical symmetry algebras like the Poincare Lie algebra of space-time, or the uni-tary Lie algebras of the bstandard modelc, were traditionally treated as absolute, givenonce for all. Set theory has no intrinsic symmetry, however; all its sets are different. Sets(finite!) are generated from the empty set 1 by Clifford multiplication (x2 = 1) and theoperation of unit-set formation or bbracingc ID : x 7→ x (§4.1.4). Since there can be no“fundamental symmetries” in a queue theory, where all vector rays are intrinsically differ-ent, it is assumed here that all interesting symmetries are contingent symmetries of somefully quantum structure like the vacuum that can be represented as a queue.

For example the Poincare Lie algebra is presented as a singular limit of a simple Liealgebra fixing a quantum organization that has Minkowski space-time as a singular limit.

1.6 Gravity and other gauge fields

If physical space-time is indeed a fully quantum space, then bgravityc is likely a vestigialquantum effect, in that the non-commutativity of the infinitesimal translations of Riemanncan be vestiges of the non-commutativity of swaps of quantum events in the near-classical


limit. The causality form introduced by Einstein is well-known to be the Clifford formof the Dirac Clifford algebra with generators γµ(x), and these can be a higher-stratumorganization of the Clifford vectors sα of elements below the event stratum E.

These ideas lead into the following familiar conceptual developments.

1.6.1 History as quantum variable

Heisenberg, emulating Einstein, set out to work solely with observables, and ultimatelyencoded operations of observation in single-time operators Q of his quantum theory. Buthis dynamical equations

[d

dt− i~−1H(t)− ∂

∂t,Q(t)] = 0 (1.47)

concern not his alleged observables at one instant but observable-valued functions of timeQ(t).

In a relativistic theory the construct of instant is relative to the space-time frameand it is convenient to formulate dynamics for histories, extending over time, rather thanfor instants. In a q theory of gravity this is all the more convenient, since quantummeasurement at one instant demands infinite energy resources of the experimenter, andtherefore should not be taken seriously, except as a singular limit ~→ 0.

In a Q theory most experimenter frames do not even diagonalize time.Therefore the present full quantization is based on history vectors, not instantaneous

ones. It is diachronic, not synchronic.A bhistoryc of a variable q in the usual synchronic quantum theory might be understood

as an operator-valued function of time q(t). This Heisenberg history is not the historythat is meant here. If variables at different events are to be assigned values, they mustcommute, they must be independent variables, and the dynamical “laws” that relate themin the synchronic theory must be overridden. Indeed, processes of control of a system, byan experimenter, like observation, override the system dynamics. A Heisenberg history q(t)obeys the dynamics. The quantum history studied here overrides the dynamics. It is a pre-dynamical history, represented in each frame by a Dirac-Schwinger-Feynman probabilityamplitude vector 〈E] for histories. In Q theory these form a finite-dimensional historyvector space, designated below by S[F]. One then express dynamics not by a relationamong operators but by a subsidiary condition, restricting the vectors 〈E] to a subspaceof S[F]. A one-dimensional subspace will usually serve. Then a typical unit vector of thisray is selected and called the dynamics vector 〈D].

Fourier transformation figures prominently in the q theory. For example, generatingfunctions are history probability amplitudes Fourier-transformed from field-variables tothe dual variables called sources. The Fourier transform is as structurally unstable asthe canonical commutation relations on which it is based. It makes no sense in a finite-dimensional algebra of variables. Fortunately, the Grassmann Fourier transform is basedon canonical anti-commutation relations of a Clifford algebra, and is structurally stable. Q


theory uses the Grassmann Fourier transform wherever q theory uses the classical Fouriertransform.

Operators of a Q theory operate on history probability vectors 〈F ] describing thecourse of an experiment. Vectors 〈F ] need more than vector-space structure to describehistories. Naturally we use a stratification structure for this purpose and assume that a Qhistory is a catenation of Q events as a c history is a catenation of c events. The space S[F]of Q history vectors 〈F ] is assumed to be one stratum higher than an underlying vectorspace S[E] of Q events described by event vectors 〈E]:

S[F] = 2S[E]. (1.48)

In this Q is more faithful to the c structure than canonical quantization, which suspendsthis exponential relation.

The singular sum over q histories is now a singular limit of a finite-dimensional traceover Q histories as the dimension grows.

Finite quantum instants of the canonical quantum theory arise when a time operatort is chosen, a singular limit t⇒ t is taken, and an eigenvalue of t is selected.

In the fully quantum strategy of Assumption 3, all systems are catenations of simplesystems with Fermi statistics. This permits the theory to be structurally stable.

To permit replication of an experiment its metasystem must be protected. Thereforehigh-resolution quantum measurements are restricted here to small regions well removedfrom ourselves and our instruments. We will therefore assume that the usual macroscopiccontinuous space-time construct still works well enough in the metasystem, though not inthe system.

A dynamical law is represented as usual by a history probability dual vector [D〈 ∈DS[F], assigning a probability amplitude [D〈E] to any experimental history vector 〈E] ∈S[F]. One vector in a vector space does not give enough information to describe an arbitrarydynamical development, which calls for a one-parameter family of transformations. Herethe stratification of S[F] supplies the missing information, in both q and Q theories, byproviding the event construct. The dynamical vector [D〈 entangles adjacent events in thedevelopment it defines.

1.6.2 Fully quantum equivalence principle

1.6.2.0.5 Classical equivalence principle The Galileo-Einstein classical bequivalenceprinciplec states that the effects of gravity are locally equivalent to the effects of anbaccelerationc. This means that to lowest order in the size of the experimental neigh-borhood one acceleration simulates gravity for all bodies and fields in that neighborhood.It is implied that the entities of the theory all have laws of transformation under acceler-ation, and that the theory is invariant under these transformations. Taken literally, theclassical equivalence principle implies vanishing btorsionc at every point, since torsion isabsent from special relativity and is not introduced by acceleration, but clearly a small


torsion cannot be excluded experimentally, and must be expected on grounds of structuralstability.

Acceleration is a time-dependent translation x → x − 12at

2, a special case of a gaugetransformation of coordinates. Invariance under acceleration is intended to imply invari-ance under the group generated by accelerations. Accelerations have local Jacobian de-terminant unity at every event, in a preferred coordinate system, called bunimodularc.Such transformations are also called bunimodularc and form a group, the bunimodulardiffeomorphismc group UDiff[M] of the manifoldM. The unimodular condition is strongerthan the bspecialc condition, which refers to a global determinant that does not exist fordiffeomorphisms. The group that accelerations generate is surely not the diffeomorphismgroup Diff(R4), but it can be the unimodular diffeomorphism group UDiff(R4) of differen-tiable transformations of local Jacobian determinant 1 in a unimodular coordinate systemon stratum E. Einstein understood that UDiff invariance and Diff invariance have the samephysical consequences; invariance under Diff he called “general” covariance; invariance un-der UDiff is called bunimodulat covariancec. UDiff transformations of events on stratum Einduce transformations of the field on stratum F that form an isomorphic group designatedby UDiff[F].

Here invariance under at least UDiff is assumed in the classical limit in order to satisfythe classical bEinstein Equivalence Principlec:

Gravity is a fictitious force that is locally equivalent to a gauge transformation in thegroup UDiff[F].

UDiff, however, still requires transformed coordinates xµ′ to depend only on the co-ordinates xµ, though possibly nonlinearly. A transformation to bharmonic coordinatesc(de Donder coordinates) xµ′, each of which obey the covariant wave equation x = 0, isan example of a more general coordinate transformation, in which the xµ′ depend on thegravitational potentials as well as xµ.

1.6.2.0.6 Gauge equivalence principle Modern gauge theory generalizes the Ein-stein equivalence principle:

Gauge equivalence principle All forces are fictitious forces derived from gaugepotentials that are locally equivalent to gauge transformations.

This reduces to the equivalence principle in the case of gravity, so it will be called thebgauge equivalence principlec.

1.6.2.0.7 Fully quantum equivalence principle The full quantization leading to ϑo

replaces the unimodular diffeomorphism group UDiff by a finite-dimensional Lie subgroupof SO[E]. To avoid notational clutter the entire SO[E] will be used for now, subject to laterreduction:

UDiff ⇐ SO[E] (1.49)


Invariance under this fully quantum correspondent of Diff is more general than generalcovariance even though it is finite-dimensional, in that it mingles space-time coordinateswith momentum-energy coordinates of the event.

Einstein described gravity with a symmetric tensor. Clifford algebra does not includetensors symmetric in two vector indices; the symmetric part of sνsµ is a mere scalar.

The Minkowski bilinear form gνµ(x) is enough to guide point planets but not electrons,which carry significant spins. Quanta of spin 1/2 represent their spin in a Dirac Cliffordalgebra at each event x, with generators γµ(x), whose Clifford product defines and is definedby the chronometric g(x) at that event. This is a natural way to represent gravity in aClifford algebraic fully quantum theory. One arrives at the construct γµ(x) by gaugingthe flat-space-time Clifford algebra, whose generators γµ do not depend on x. Now weformulate a fully quantum correspondent of this gauging process.

The γµ originate on a deep stratum C, the event x on a higher stratum E. The structureγµ(x) transforms under the groups of both strata.

[To be continued.]XXXIt is supposed that a quantum bilinear form 〉g〈 on vectors of a lower stratum C un-

derlies the classical Minkowski form of special relativity. g(x) designates the Minkowskianform of unimodular relativity, a symmetric tensor field of determinant everywhere unity inspecial coordinate systems, reducible in any one reference frame to a fixed Minkowski formg0 by a local frame change Λ(x):

g(x) = Λ(x) g0 ΛT(x). (1.50)

LetDµ = ∂µ + Γµ (1.51)

be the covariant differentiator for the differential manifold M of space-time according togeneral relativity. The term ∂µ is a translation generator. For a quantum moving in aclassical gravitational field, ∂µ is the quantity that is conserved if translation invarianceholds. Therefore it represents the total momentum(-energy). For a spinless quantum, theterm Γ is missing. Therefore Γµ can be regarded as the spin momentum. What is left, Dµ,is then the orbital momentum.

For vector fields Γµ is of type Γλκµ(x). More generally, it is of type ΓCµ(x) where C isa Lie algebra index, specifying the transformation that acts on the entity being transportedin the direction µ at x.

In the queue theory ϑo, momenta become higher-stratum representatives of underlyingswaps. The infinitesimal displacements on M are singular limits of representatives of ele-ments of so[C] in so[E]. The queue correspondents of the total, orbital, and spin momenta,represented by ∂, D, and Γ, can be called swaps of strata F, E, and C, represented byω[F], ω[E], ω[C].


The queue correspondent of the Minkowski coordinate xµ is an E-stratum representa-tive of a C-stratum swap ω[C]µ ∈ so[C].

C also indexes the complementary momentum(-energy) coordinates and the angularmomentum of the quantum. The queue correspondent of a function of x is an operator onstratum-E vectors, represented by a matrix of the type M e′

e with two vector indices e, e′

labeling a basis se ∈ S[E] of stratum E. Therefore the presumptive queue correspondent ofthe total momentum ∂µ is a swap tensor ωCC′e

′e representing how the cumulated swaps of

stratum C act on the vectors of stratum E.Now the total swap tensor ωCe

′e corresponding to Dµ is to be decomposed into parts

corresponding to borbitalc and bspinc parts ∂µ and Γµ. The basis vector 〈e probablyconsists of many D-stratum factor vectors, all contributing to the moment ωCe

′e . Probably

a logarithmically small number of these are C-stratum factors. Their contribution to thetotal moment will be identified with the bspin partc; the rest with the borbital partc.

To consider whether the spin and orbital parts of the usual connection can be singularlimits of the stratum-C and stratum-E swaps we must examine how they transform underthe queue transformation that corresponds to a diffeomorphism of the event manifold M.

General relativization includes the familiar replacements

g0 → g, ∂ → D = ∂ − Γ, (1.52)

designed to convert a Poincare- invariant theory into a nearby UDiff invariant theory thatis equivalent in the limit G→ 0 of flat space-time. To fully quantize the process of generalrelativization one first fully quantizes these groups.

The process of general relativization was already generalized within canonical physicsto the heuristic process called bgaugingc. Gauging converts a theory invariant under agauge Lie group of stratum C (“of the first kind”, as Dirac calls it) into one invariantunder a gauge group of stratum E (“of the second kind”), indicated in (4.33). A gaugegroup acting on elements of stratum C will be called a bC-gaugec group, and similarly forother strata.

There are two famous ways to gauge, here termed the Weyl (§1.6.3) and Kaluza (§1.6.4)gauge strategies. Both set out from general relativity.

1.6.3 Weyl gauge strategy

The bWeyl gauge strategyc adjoins to the space-time tangent space at each point an “in-ternal” gauge vector space V , on which the gauge group G acts through its defining rep-resentation G → V D ⊗ V . This enlarges the fiber of the physical bundle but not its basespace of events. It accounts for the gauge vector field Cµ

G(x) — here G is an index for abasis of the Lie algebra dG — as a connection form ΓµG(x) defining the parallel transportof new gauge variables of various kinds from event to event:

CµG(x) = ΓµG(x) (1.53)


carries new things in V in old directions labeled by µ.The Weyl theory of electromagnetism, the Dirac theory of electromagnetism, the

bYang-Millsc theory of the isospin connection, and the standard model gauge theories ofhypercharge, electroweak, and color gauge fields, are instances of the Weyl strategy.

1.6.4 Kaluza gauge strategy

In the Kaluza gauge strategy, four-dimensional space-time manifoldM is directly multipliedby a Kaluza space that is isomorphic to the n-dimensional gauge group manifold G itself.This supplements the old space-time coordinates xµ by new group coordinates xG — hereG indexes the group parameters — on the group G. Their differentiators ∂G(1) at theorigin 1 ∈ G span the Lie algebra dG.

For Kaluza the tensor C that multiplies the currents in gauge interactions is not aconnection at all, but a sector of the causality form,

CµG(x) = gµG(x). (1.54)

CµG(x) is the element of the metric tensor that couples the external differential dxµ

with an internal differential dxG. In Kaluza’s original model, the internal Lie algebra wasthe one-dimensional so(2) = u(1), so the index G had only one value.

Now the gauge vector field appears as an off-diagonal block gµG in the chronmetric,coupling the usual macroscopic or external space-time dimensions of M with new micro-scopic dimensions of space-time forming the manifold of a Lie group G in an enlargedspace-time

M ′ ∼= M × G (1.55)

The same metric also has a gravitational block gµ,ν , and a block gG′G, the Killing form ofthe internal Lie algebra.

Kaluza strategy enlarges both the base space and the fiber of the physical bundle. Itsconnection carries new things as well as old to new places as well as old.

Since Einstein calls gνµ the gravitational potential, its bordering block, the gaugevector field C, can be called the gauge potential in the Kaluza strategy. The Kaluza strategybuilds the strong bgauge equivalence principlec into the foundations of the resulting theoryas part of an extended Einstein equivalence principle. The bKaluza-Kleinc, bDe Wittc [22],bMacDowell-Mansouric [54] theories and many others use the Kaluza strategy.

One defect of the original Kaluza strategy is that for the extension to quantum fieldtheory the internal manifold must be a compact Lie group, closed on itself, so that integra-tions over it converge, and huge energy is required to close it, but none is provided by thetheory. This is the bcompactification problemc. It arises in Kaluza theory because, beingclassical, this theory can account for a Lie algebra only by providing a physical continuumfor the algebra to act on.


1.6.5 Queue gauge strategy

Yet molecular biologists do not see a compactification problem in DNA, because theyunderstand its quantum structure and do not invent internal continua. In a queue theorythe internal Lie algebra can arise from a finite-dimensional vector space instead of aninfinite-dimensional function space over a classical manifold. To sum or average over suchquantum degrees of freedom, one does not integrate over the group, one traces over theLie algebra. The non-compactness of the group is irrelevant. In quantum theory regularity— convergence — is assured by simplicity and the ensuing finite-dimensionality, not bycompactness of the group. In the theory ϑo, the Kaluza classical internal manifold istherefore replaced by an internal finite-dimensional vector space supporting the simplegauge Lie group.

One problem of quantum physics today is the non-simple nature of the external partof the system, which gives rise to infinities. In a queue space-time, the external manifoldof Kaluza too is replaced by a vector space, whose Lie algebra includes the quantized co-ordinates, momenta, and angular momenta. That is, instead of compactifying the internalspace of Kaluza, a fully quantum theory quantizes both the external space of Einstein andthe internal space of Kaluza. The external and internal groups are both simple Lie sub-groups of one larger simple group, acting on unified vectors for the combined space. Sinceone does not integrate over vector spaces but merely traces over them, the compactifica-tion problem is replaced by the bgrowth problemc: to account for the great growth of sixdimensions relative to the others, and the freezing of two, that results in a crystal film. Itis natural to use the large ratio between the multiplicities of the low and high strata of thequeue to account for the Large Number.

Full Fermi quantization represents the system as an organization of a great many simpleunit cells without any external imbedding space or manifold. What are usually regardedas internal dimensions are represented by the thickness of the crystal dome, perhaps onecell thick. Cosmic cortication presumably results from anisotropic crystal formation, likethat of a snowflake.

In the Weyl gauge strategy, the internal and external groups act on spaces of supposedlyseparate origin and nature; in the Kaluza and fully quantum strategies the spaces areunited.

The canonical quantization process worked well enough for photons but poorly forclassical point electrons. The spin of photons is already represented in classical physicsby the tensorial nature of the electromagnetic field, and so does not need to be inventedfrom scratch during quantization, but electrons were represented in classical physics aspoint particles, with neither spin nor tensorial nature. This problem is evaded here bystarting from the standard model and its spins, which are readily accommodated withinthe self-Fermi algebra S.


1.6.6 Fully quantum gauge group

Here a fully quantum gauge strategy is executed.The equivalence principle of Einstein and its gauge generalization are local. An in-

finitesimal acceleration generator, for example, is an infinitesimal translation generatorwith time-dependent parameter, 1

2at2∂x. In full quantization, space-time coordinates and

momenta become binducedc representations on stratum F, of rotation generators, spins,ωβα ∈ so(n) on stratum C..

The idea of a function of non-commuting variables is not well defined, but the constructof ordered polynomial expression in non-commuting variables is unproblematical. This willbe called a bpolynomialc for short.

A conspicuous gauge algebra that has bdiffc (the Lie algebra of Diff) as a singular limitis the Lie algebra of all polynomials in the E matrix representatives of the C gauge Liealgebra. This is a sub-Lie algebra of so[E] := so(SE). For simplicity it is taken to bethe entire so[E]. That is, the fully quantum correspondent of diff := dDiff is taken to beso(S[E]). This symmetry ignores the deep structure of the vectors in S[E], associated withthe general event.

This makes it possible to fully quantize the process of general relativization, and ofgauging in general. In ϑo six quantum strata A – F , each an exponential of the previous,are used for this. Three quantum strata D – F correspond to the familiar classical strata D– F. The bC gauge groupc is so[C] = so(10, 6), the kinematic group of stratum C, and hasinduced representations in so[L] of every higher stratum L = D, E, F. The F gauge groupis the representation of so[C] in the orthogonal group so[F] of the vector space of stratumE.

General relativity is a special case of a gauge theory, and suggests the following defi-nitions:

A bgeneral coordinatec of an event is a normal operator on the event vector spaceS[E]. Therefore the queue correspondent of the group Diff(M) is presumably the specialorthogonal group of the vector space of the event:

SO(S[E])⇒ Diff(M) (1.56)

The bqueue gauge groupc of the F stratum is then

GF := Π SO[E], (1.57)

the group on stratum F induced by SO(S[E]. It is to be shown that the gauge transforma-tions of the standard model too are approximated by elements of the group SOE, modifiedby the assumed vacuum organization. GE is necessarily non-local; simple quantum space-times do not admit the construct of locality. Full quantization of the local equivalenceprinciple, however, provides a quantum bequivalence principlec:

Assumption 6 (Quantum equivalence principle) The interactions of nature are e-quivalent in each queue cell to a fully quantum gauge transformation.


Assuming Fermi statistics, the low-stratum quantum gauge group is a linear group.The gauge group of the standard model with gravity is

Gg,sm = ISO(3, 1)× S(U(2)×U(3)) (1.58)

taking into account a discrete central correlation [62, 63]. This is not even a simple group,let alone an orthogonal one, so in the canonical theory one assumes that Gg,sm is a singularlimit of an actual simple group GE of the event stratum E. A gauge transformation is nowa group action on the internal coordinates by an amount that depends nonlinearly on theexternal coordinates. In the classical limit this does not affect the external coordinates andso the theory is singular. In full quantization the external coordinates are affected too andthe theory is regular.

The usual Dirac Clifford algebraic representation of the causality form through thespin form γµ puts a Clifford algebra on the tangent vectors at each event. The γµ(x)encode the causality form as the square of a Clifford vector. Therefore they also encode aclassical gravity field. At the same time they are spin operators of a spin 1/2 quantum atthe point x, and so are already quantum variables from birth. They therefore need not bequantized in the usual sense but can merely be quantified, aggregated, an easier problem.

The spin system at an event has only four dimensions to its vector space, those whichin Dirac one-electron theory give rise to spin up or down, energy positive or negative. Thequasi-continuity of gravity arises from the many events supporting such four-valued spinvariables, and this number already becomes finite when only the xµ are fully quantized.

Segal and Vilela-Mendes quantized the coordinates and momenta xµ, pµ on one stratumwithin an orthogonal Lie algebra (§5.2.5, §5.2.8). A similar one-stratum quantization withina Clifford algebra is described in §5.3. This is imbedded in a full quantization in §6.3.

1.6.7 The space-time truss

The spectral spacing for the quantized time coordinate can be regarded as a quantum oftime or bchronc X, with the understanding that times add in series but average in parallel,leading to finer spectra. It is sometimes heuristically useful to picture an binputc vector ofa quantum mechanical system as a “bcellc of size ~n ” in a classical phase space. It mayalso be useful to picture binputc vectors of the individual quantum event as n-dimensional“cells” of size Xn in a classical tangent bundle to event space, with a preferred vertex ofeach cell as its origin, and elements of a lower stratum as edges or struts at the origin.Similarly, the vectors of the quantum space bsetc are cells with events as their edges. Thecell of 16 vertices has the kinematical group SL(16), which accommodates the stratum-Cgauge groups of gravity and GUT.

The fully quantum ambient space-time would then have a highly asymmetrical struc-ture like that of a fullerene or a btruss domec: long in some directions, short in others.The so-called external and internal dimensions of the standard model are the longitudi-nal and transverse directions of the cosmic shell, the blongc and bshortc dimensions. The


long dimensions have many struts extending along the dome hypersurface for macroscopicdistances, and for many purposes can be handled classically, because they contain manyquantum units. The short dimensions are transverse spreaders only one or several cellslong. Therefore they require description at the quantum stratum of resolution. There arewell-known problems arising when classical and quantum systems are coupled. The validityof such a semiclassical description is limited. The standard model is of this kind. Here welook a little beyond the semiclassical stratum.

Possibly the short dimensions of the set belong to lower strata but the long dimensionscertainly require higher strata.

For example, one might consider a btrussc composed of uniform basic quantum cells,each composed of 16 independent struts. One such cell is first found on stratum 4, whoseGrassmann algebra S[4] has 16 monadic generators. A truss of such cells can then beconstructed on stratum 6, which accommodates about

(216

16

)such cells. The SL(16) would

then provide both a bGUTc group SO(10) of the short dimensions [43, 42] and a simpleLie group SO(3, 3) of the long dimensions [66]. The truss is best not assembled cell by cell,however, but by the groups that act on its swaps and monads.

Gauge theory is singular due to the unstable commutation relation between the gaugedifferentiator and the space-time coordinates:

[Dν , xµ] = δµν , [δµν , Dλ] = [δµν , x

λ] = 0. (1.59)

The assumption of zero torsion is also unstable. Like the commutativity of space-timecoordinates, these commutativities must be artifacts of the classical space-time limit, anddo not hold exactly in the fully quantum theory. Full quantization eliminates these singu-larities.

Curvature, the classical non-commutativity of the covariant differentiators, is thena classical vestige of the quantum no-commutativity that results in a discrete uniformlyspaced spectrum for basic coordinates and set variables. This classical non-commutativityis the gravitational field.

Gravity is a quantum effect.

1.6.8 The gravitational and gauge potentials

Fermi algebras accommodate antisymmetric tensors readily but not symmetric ones likethe gravitational potential gνµ(x). However the first-grade differential dxµ becomes dxµ5

in quantum space-time, part of a swap dxα2α1 . This can be assumed to be anti-symmetricin its two indices. In ϑo, the space-time tangent space of stratum D corresponds to stratum4, the lowest with enough dimensions. This vector space supports the Lie algebra so(10, 6)and its associated swaps

ωβα =14

[γβ, γα] =:14

[γβ, γα]+ (1.60)


and metric form

gβα =12γβ, γα =:

12

[γβ, γα]− (1.61)

This spin serves as a prototype quantum space-time cell. Each such cell carries its ownClifford bilinear form. Beneath it lies stratum 2, with a Minkowskian vector space and aLorentz Lie algebra so(3, 1).

The operator-valued causality form gνµ(x) of canonical bgravityc is understood tocorrespond to a single operator g ∈ FS ⊗ FSD. The space-time coordinate xµ becomesa representative on stratum E of a grade-2 swap γµ5 of so(6;σ), part of a form γα2α1 .The tensor gνµ(x) becomes part of a form gα4α3α2α1(x), symmetric with respect to theinterchange of collective indices α4α3 ↔ α2α1 because it is totally antisymmetric. Thex-dependence becomes a large number of further indices, approaching ∞ in the limit ofclassical space-time.

This means that the bgravitonc, as well as and the gauge vector bosons of this modelare not true bosons at heart but bpseudo-bosonsc with fermionic cores, analogous to two-neutrino models of the photon and four-neutrino models of the graviton considered by deBroglie, Feynman, and others. There need be only one space-time coordinate in g, whiletwo or four neutrinos would provide two or four coordinates. The catenation that formsthis graviton occurs at stratum E, a deeper stratum than the set stratum F where fermionsare found. This core structure might show up at high energies but it may be invisible atlow energies.

1.6.9 Vacuum

bNewtonc and bFresnelc reasoned that their vacuum, the betherc, is crystalline because itpropagates transverse modes of light with great transparency, unlike fluids. This argumentapplies to the vacuum today. Nothing stiffer than the vacuum is known in nature if onejudges stiffness by the speed of waves, and also nothing more transparent, judging by themean free paths of photons and neutrinos. These observations indicate that the vacuum ishighly organized and quite cold at the ambient temperatures common today, compared toits melting point. On the other hand, it seems plausible that space-time meltdown occursnear central quasi-singularities of black holes.

The vacuum of the standard model is a bhigh-temperature superconvectorc of colorand weak currents in that some of its gauge symmetry is broken, such as the symmetriesbetween the electric and weak gauge dimensions. According to GUT there is a similar gaugesymmetry breaking between the hypercharge, electroweak, and color gauge dimensions ofGUT. In a Kaluza-style theory like ϑo there is another symmetry breaking between gravityand the other gauge dimensions. Only the gauge invariances of electricity and gravity areultimately considered to be unbroken in the present vacuum.

In the classical limit with finite c and with ~,X → 0, the vacuum experimentallydefines a light-cone and an electromagnetic axis at each point. On a laboratory rather

1.7. UNIFICATIONS 101

than cosmological scale, Poincare symmetry also survives. In the even more singular limitc→∞ the vacuum defines a Galilean instant, still without absolute rest.

1.7 Unifications

Canonical quantization unified energy and frequency, and other ~-related pairs, almost asan incidental by-product. Further simplification results in further unification. The follow-ing units of this section summarize unifications that arise in this work. One unificationguided the construction, that of

• the bracing ID of classical set theory, Peano’s ι,

• the bracing of spin (§3.3.4),

• the Dirac construction of the space-time Clifford algebra

• the Fermi-Dirac construction of fermion algebras

• hopefully, the rank-raising operator connecting the three successive strata of leptonsand quarks

are unified here into one bbracingc operator I. The other fusions that this requires aresomewhat unexpected outcomes and are listed in advance to alert the reader.

1.7.1 Being and becoming

There are well-known unifications of being and becoming, or essence and existence, in bothrelativity and quantum theory. In classical thought, space is a pattern of relations betweenstates of being and time is a pattern of relations between becomings. In special relativitythat partition between space and time is relativized, and relations that are purely spacialfor one experimenter have a temporal component for most others.

In classical physics, again, the points of the coordinate space describe states of be-ing and tangent vectors represent modes of motion, becoming. In quantum physics eachoperator is both a coordinate and a generator of a mapping, describing both being andbecoming.

In quantum space-time both unifications are unified. The idea of a space-time pointis unified with that of a tangent vector into one construct called here an event or Ewhich decomposes into coordinates and momenta only as the result of a symmetry-breakingorganization. In this respect the event E resembles a point of phase space more than apoint of space-time. The Lorentz group that mixes space and time is enlarged here to anorthogonal group on the event space that mixes space and momentum, time and energylike the canonical group.


During cosmogenesis, one of the early phase transitions was a quantum organization ofevents that formed the space-time quasi-continuum, centralizing the position variables andfreezing momentum variables 0 by a quantum organization. Local space-time meltdownmust be expected in sufficiently hot fireballs today.

1.7.2 Gravity and quantum theory

bPeanoc introduced a bsuccessorc operation ι, first in his theory of the natural numbers,to generate the natural numbers from 0, and then into his set theory, where it generatedall sets. The name suggests that ι determines a temporal succession, though perhaps onlymetaphorically. His ι was soon drowned out by the weaker notation

s := ιs, s, t := ιs ιt, . . . , (1.62)

resulting in the inconvenience of a function with a variable number of arguments and nofunction symbol.

I generates a Clifford algebra S of fully quantum probability amplitude vectors in muchthe way that Peano’s ι generates classical set states. S includes Dirac’s other Cliffordalgebra too, the Clifford algebra of Fermi-Dirac creators, as a limit.

Dirac quantum spin theory and the classical ι of Peano are unified in one constructhere, the brace I. The adjoint of I is the bde-bracec

ID = IH = HTH−1 (1.63)

It is posited that for vectors v of a certain bcharge stratumc C, any spacelike v is unfeasibleand any timelike v is feasible. as Hvv > 0 or < 0 (Assumption 8), and so the probabilityform serves as a causality form. The quadratic causality form v 〉g〈v = gµνvµvν = g v vof Einstein gravity theory is to be a singular limit in stratum F of a form induced by thefully quantum probability form v 〉H〈v of stratum C.

The existence of a three-fold hierarchy of bstratac or bgenerationsc of quarks is still anoutstanding puzzle. Perhaps queue theory can generate such a hierarchy in an ad hoc way,using the operator I invented for other purposes.

1.7.3 Products

Canonical quantization unifies two seemingly independent products of classical mechanicsinto one of quantum mechanics: the classical commutative product is the ~0 term in apower series, and the classical Poisson Bracket is the ~2 term. Similar product unificationsoccur during full quantization and help as guides along the way.

The bClifford productc famously unifies the bGrassmann productc and the scalar orinner product of vectors. Full quantization uses this Clifford balgebra unificationc processto unify the Grassmann product of Fermi statistics with both the inner product of Hilbertspace and the scalar product of space-time vectors on a deeper stratum than space-timeevents (§3.3.3).

1.7. UNIFICATIONS 103

1.7.4 Non-commutativity and granularity

Canonical quantization expands the classical construct of random variable to a later con-struct of quantum variable. These variables need not be numerical; the system itself isreferred to as a random variable in probability theory. A random variable is described on-tologically, by saying what it can be. The space of the possible states of a random system iscalled its bstate spacec (or bphase spacec, from its use by Gibbs in the statistical mechanicsof phase transitions).

An bontologyc is a a theory of what it means to “be”. Here the term is applied toan analysis of Nature into beings, things that exist. Calling something ontological is anemphatic way of saying that it exists as a physical object, in contexts where there also thingsunder consideration like probability distributions, that have a more abstract nature. It isexplicitly understood that these “beings” can be associated with symbols that completelydefine them, on the grounds that this belief is implicit in most ontologies. The integers ofmathematics and the coordinates of classical mechanics are symbols in this sense. Thus“bontologyc”, like “reality”, is used here somewhat pejoratively, in a naive pre-quantumsense relevant to classical physics, where it works well enough. By bontologismc is meantthe belief, tacit or explicit, that such an ontology exists.

Theories that work in the quantum domain do not deal with pure unobserved being butwith experimental procedures. They describe a quantum not by its possible states of beingbut by actions that can be carried out with it, especially how to prepare, sort, and registerit, by input, throughput, and output processes. This formulation in terms of feasibleoperations, such as filtrations and sortations, makes possible kinds of non-commutativitynot easily imaginable within classical mechanics, which left filtrations out of its early formalstructure but tacitly assumed that they commute.

One may therefore speak of quantum theories, say of geometry, simply as non-commutativetheories. Mere non-commutativity is not what this means to convey, of course. There are agreat many non-commutative groups in classical theories too. The name “quantum theory”originally meant a corpuscular or granular theory, and noncommutativity does not meangranularity. It was a remarkable intellectual leap to go from the grains of light and the dis-crete atomic energy strata of the bBohr atomc to the non-commutative complex algebra ofHeisenberg. This is not a mere change in a theory, but a change in what is understood by atheory, in what the theory is about, and in the theorist’s bstrategyc and philosophy. Whatis non-commutative in non-commutatuve physics or non-commutative spaces are filtrationactions representing predicates, or combinations of predicates and numbers representingcoordinates. When that non-commutativity is of the kind that can be represented by asimple Lie algebra, as assumed here, it leads to total granularity, and to a unification ofthe strata of physics.


1.7.5 System and metasystem

The bmetasystemc has degrees of freedom relating one experimental reference frame toanother. Such degrees of freedom of the metasystem are represented in the system aswell. For example, in the 19th century, when space was flat and canonical mechanicsreigned, every system in nature had to have a momentum operator p generating the changesin the variables of the system generated by a uniform translation of the experimentalreference frame in the metasystem. Such operations in the metasystem will be calledımetasystemic. System operators that represent metasystemic operations have long beenused. In the last century they grew enormously in mathematical complexity and theoreticalimportance. The story can begin with Einstein’s treatment of time in special relativity,which assumes that the experimenter can assemble a rigid lattice of clocks and rulers in themetasystem. Such frames were supposed to form a 10-parameter family: 4 coordinates forthe origin, 3 for its velocity, and 3 orientation angles. In general relativity the rigid lattice isreplaced as framework by a fleet of experimenters smoothly distributed over space-time inan otherwise arbitrary way. Such a frame is specified by an infinity of parameters, definedby 4 arbitrary smooth functions of time-space. In canonical quantum theory this timeframe is accompanied by a fact frame, each experimenter having a maximal collection ofmutually exclusive input or output possibilities for the system, related to others by a unitarygroup. The metasystems in gauge theories, including general relativistic spin theory, whereindependent transformations of the gauge frame are permitted at each space-time event inthe metasystem, have the most complex structure so far.

The relation between system and metasystem is not a symmetrical or transposable one.The metasystem belongs to a higher stratum than the system in several senses. A personcan know more about an atom than the atom can know about the person. The metasystemis thus on a higher epistemic stratum than the system. When the bmetasystemc inputs asystem it can be represented as a source by a class or virtual set of systems, from whichone is withdrawn by any input, and into which any output is deposited. A class belongsto a higher stratum than its elements, and the same stratum as its subclasses.

Under closer inspection, if necessary by another experimenter, any part of the meta-system always reveals its own quantum structure. In the small we are as quantum asany system we study. bBohrc suggested that the canonical quantum strategy might needto be revised to take into account the quantum structure of the experimenter [13]. Thiswould seem to require extending the quantum theory from the system stratum to a higherstratum. In this work an exponential algebra S with an infinite hierarchy of nesting stratais constructed, of which a finite number are used.

The interface between quantum system and metasystem is not located or shifted by amere mental decision or shift of attention. It is a controllable physical gate, closed duringthroughput and open during input-output. Its construction may require vacuum Dewars,lead walls, superconducting cages, or physical cuts. To move it may be a major enterprise.

Nature sometimes provides the insulation needed for the system interface. At the clas-

1.8. OUTLINE 105

sical level of resolution, the planets of the Solar System are surrounded by natural vacuumisolation, fortunately making each nearly a one-body problem. In the Malus experimentthe photons in the beam are effectively insulated from each other and the apparatus whilethey are in flight.

In particular, dispensing with the interface, regarding the metasystem as part of thequantum system, seems absurd at the quantum level of resolution. This is not the unifica-tion discussed here. Some processes cannot be watched. A maximal observation overridesthe dynamics of the system with a system-metasystem interaction, greatly changing thebehavior of the system. An observation process that is observed closely cannot observereliably.

To be sure, sufficiently small and isolated chunks of the metasystem can be transferredto the quantum system without making the experiment impossible. This may be a way toinfer the dynamical law of the metasystem, piece by piece, Then one can require that thedetermination processes on the system conform to the dynamics of the metasystem. Thisis the extent of the unification contemplated here.

1.8 Outline

The job at hand has at least four interlocking phases, taken up in as many Parts in whatfollows:

Syntactic Set up the queue algebra.

Semantic Translate the concepts of the standard model and gravity into the queue algebraas closely as possible, including the dynamics.

Logistical Deduce experimental consequences of the regularized theory and compare themwith those of the previous theory and as far as possible with experiment.

The succeeding Parts of this treatise undertake these respective stages.Chapter 2 introduces probability vectors and takes up their addition. It reviews the

praxics of random or quantum individuals, including single-quantum kinematics.Chapter 3 adjoins the operation of vector multiplication. It reviews the classical and

quantum polynomial praxics of a random set or queue of individuals, often called secondquantization or field quantization.

Chapter 4 adjoins the operation of vector bracing. It develops classical and quantumexponential (power set) praxics, the theory of a random set of sets of . . . sets of individuals;in which any assembly of one stratum can serve as an individual of the next stratum. Thebrace I generates a self-Grassmann algebra S, a Grassmann algebra over itself:

S = Grass S = 2S. (1.64)


Chapter 5 reviews increasingly quantum space-times that have been proposed, whichguide and inspire the present fully quantum constructions.

Chapter 6 constructs queues in general and corresponds them to variables of the stan-dard model and gravity.

Chapter 7 specializes fully quantum dynamics to bgravityc, giving a quantum originfor the chronometric.

. . . . . . [080331: Update]Chapter 8 sums up.

Chapter 2

Linear praxics

of hits and misses, superposition and complementarity.

2.1 Praxics in general

This chapter deals with entire systems. Grade structure is added in Chapter 3 to describecomposite systems, and stratum structure is added in Chapter 4 to describe stratified orranked systems.

Boole characterized the classes of a logic by elective actions, a kind of mental filtration,and represented serial action as a product and parallel action as a sum [14]. He left systemsources and targets implicit. It suffices to introduce an ideal universal source and universalcounter, for then the most general source and target can be made by catenating filters afterthe source or before the target.

Praxics treat of physical filtrations rather than mental elections. The filtrations com-mute, AB = BA, for classical logics and not for quantum praxics.

One critical difference between classical and quantum theories is the quantum super-position principle: a quantum physicist sums complex transition probability amplitudesin situations where classical thinkers like Pascal, Boole, and Markov would sum positivetransition probabilities, as in a two-slit experiment. Another well-known difference is thatclassical filtration operations or predicates all commute, while no non-trivial quantumfiltration commutes with all the others, leading to quantum complementarity. The super-position principle leads to a matrx algebra, implying the complementarity principle andmore.

For example, linear polarizersX, Y , andQ oriented normal to the z axis with polarizingdirections along the directions of x, y, and x + y filter photons, and one can see with thenaked eye that one order XQY passes some photons, one by one, and another order XYQstops them, one by one:

XYQ = 0 6= XQY . (2.1)

107

108 CHAPTER 2. LINEAR PRAXICS

Classical probability theory deals with both state probabilities (like Boltzmann’s) andtransition probabilities (like Markov’s). Quantum probability amplitudes like QHX arealways for transitions like X → Q, from an input to an output, never for states of being.Tables of such transition amplitudes are square tables or matrices that almost never com-mute. That the squares of the vector components α〈ψ are relative probabilities followsfrom the quantum law of large numbers.

The quantum principle implies the integrity of the system under measurement. Whenv′ ø v does not hold, the theory does not predict whether the output process v′ counts asystem or not. It predicts, however, that the count will be 0 or 1, not a fraction, just as inclassical probability theory. The quantum does not divide when the vector does.

The quantum principle can be stated succinctly, if cryptically:

A quantum praxic is a projective geometry.

The elements of the projective geometry are the elements of the Galois lattice of the relationø (or of the relation Hv2 v1 = 0) [40].

For example, the line uv determined by two inputs u, v is defined as the set of allinputs w that miss whatever u and v both miss, and corresponds to the plane of the twovectors u, v. Its elements are the quantum superpositions of u and v. It is then empiricalwhether any two points of a line determine the same line, an axiom of projective geometry.The quantum superposition principle — that distinct input processes have a quantumsuperposition distinct from both — is then the postulate of projective geometry that thereare at least three points on every line, another empirical matter.

2.2 Heisenberg and von Neumann praxics

bBohrc and bHeisenbergc introduced non-commutative bpraxicsc when they invented thequantum theory. Even in classical physics, assertions about a physical system are binaryvariables of the system with the value 0 for “False” and 1 for “True”. The matrix productBA represents doing B after A. In classical mechanics predicates are binary functionson the state space of the system, and commute. In quantum theory they are projectionoperators and mostly do not commute.

Later bvon Neumannc extracted a bquantum logicsc of ∩ (and) and ∪ (or) from thematrix praxic of Heisenberg, using a blatticec of projectors within the matrix algebra. Hisbackground uniquely qualified him for this work. His 1925 doctoral thesis had reconstructedlogic and set theory as a theory of mappings rather than propositions. It was quite unwieldybut its functional foundations foreshadowed both non-commutative quantum observablesand categorial algebra [77]. Then he had been assistant to David Hilbert, whose program toaxiomatize physics still required axioms for quantum mechanics. Thus bvon Neumannc wassingularly prepared to invent a logic whose assertions are functions, now from vectors tovectors. In at least one discussion he spoke of a “quantum set theory”, without constructingone.

2.2. HEISENBERG AND VON NEUMANN PRAXICS 109

On the other hand, bvon Neumannc concerned himself less with the operational mean-ing of quantum theory than with the mathematical structure. His bquantum logicsc wereblatticesc with mathematical operations ∪ and ∩ designed to resemble the ∪ and ∩ of earlierlogics, not any laboratory operations. ∪ and ∩ have the advantage over + and × of beinginvariant under unitary transformations of their arguments and so being implementable op-erations on projectors, at least as limits. But they are further from laboratory operationsthan Heisenberg’s ×. It takes an infinity of ×’s to make one ∩. They are also structurallyunstable. And they are also unwieldy for theorists, no practical physical theory, classicalor quantum, has ever been invented in the lattice language. Only the language of linearalgebra has been fertile for new quantum theories that work. We use it here.

In the language of linear algebra, a predicate is a binary-valued observable, representedby a symmetric matrix obeying the projector condition (2.12), which restricts eigenvaluesto 1 and 0, standing for true and false. The physical operation it represents is a filtrationfor individual systems with the value P .= 1. The logical complement is ¬A := 1−A. Thefundamental dyadic operation is the matrix product BA, representing B after A.

Predicates are not closed under this operation however. The praxic algebra is bestregarded as the entire matrix algebra, provided with the probability form H.

The famous bnon-commutativityc of physical variables is equivalent to the bnon-commutativitycof the predicates, and the filtrations they represent. Quantum praxic is non-commutativelogic as quantum geometry is non-commutative geometry. It is related to the usual com-mutative logic in the way that Galilean relativity is related to special relativity: the oldertheory is a singular limit of the newer that is an adequate approximation in much ofordinary experience.

Matrix addition too can be carried out in the quantum laboratory, though only as alimiting case. The problem was solved in principle when projective geometers of the 19thcentury showed how to pass from the axioms of synthetic projective geometry, which use∪ and ∩ only, to those of analytic projective geometry, which concern vectors defined bya sequence of coordinates, numbers in some field, and use + and ×. If a vector has Ncoordinates, the ray through it has only N − 1. To represent the vector itself by a ray,one uses a ray in an enlarged vector space of N + 1 dimensions [4]. Physically, one mayincrement the number of possibilities of the quantum system by adjoining a vacuum mode.

In an alternative thoretical approach based on the Lie group of the system insteadof the predicates, one defines the addition of operators by multiplying operators near theidentity in one-parameter groups:

A+B = limε→0

[d

dε

(eεAeεB

)]. (2.2)

The same arguments that had been raised against the relative time and non-com-mutative boosts of relativity theory were also marshalled against the relative states andnon-commutative predicates of quantum theory. The reconstruction of the theory of physi-cal truth has proceeded more slowly than the reconstruction of our theory of physical time,


even though it had the the earlier reconstruction as a guide and enormous technologicaladvances as a by-product. Kant said that Euclidean geometry is a necessity of thought,given apriori, and so could never be revised. Poincare said that Euclidean geometry is alinguistic convention, and so need never be revised. Einstein cut through this fog by talkingabout two experimenters and their experimental operations in plain language.

Similarly, some said that bclassical logicc is a necessity of thought, so that it can neverbe revised. Others said that bclassical logicc is a linguistic convention, so that it need neverbe revised.

bMalusc ignored such philosophical arguments and talked about transition probabil-ities between two polarizers in plain language, although the epochal significance of theMalus Relation was not realized until long later. First bPlanckc, bHertzc, bEinsteinc, andbComptonc had to re-establish the existence of the photon that Malus had accepted on theauthority of Newton, even though Young had just demonstrated the interference fringes oflight.

Nor can classical logic be an absolute necessity of human reasoning. If it developsthat we are biologically constrained to prefer bclassical logicc, as is plausible on Darwiniangrounds, we nevertheless communicate our measurements with symbols, and so we canreason about them with classical symbolic logic, without straining our brains more thanby learning matrix algebra. In this way we use bquantum praxicsc for systems by usingbclassical logicsc in the metasystems; just as we use Einstein relativity for fast electronsand Galileo relativity for the electron accelerator. This is not a contradiction, only a usefulapproximation.

Nor can a physically useful logic be entirely conventional. To be sure, there is muchconvention in any language, considering all the words that could be invented and arenot. But in practice the domain of discourse imposed important constraints too. Forexample, once we establish a language to represent physical filtration operations, and definethe product AB = AafterB among these filtrations, whatever we call them, the non-commutativity AB 6= BA is learned by experiment, not imposed by convention. One couldsay that only polarizations that commute are predicates, thereby resurrecting classical logicby casting out quantum superpositions, but this would break the experimental symmetrybetween the X and Y of one experimenter and those of another, and so violate rotationalrelativity. Filtration non-commutativity is about as factual as sunrise whatever we call it.Similarly for the non-distributivity of the ∩ and ∪ of quantum praxics, which representmuch the same facts about polarizers as the non-commutativity of after.

It seems clear to many, including founders of the quantum theory, that the problemsof present-day quantum physics, especially the infinities, call for a post-quantum physicsthat departs even further from classical physics in its praxical structure than the theoriesof Bohr, Heisenberg, and von Neumann, and is less commutative. This is the directive towhich the present work responds.

2.2. HEISENBERG AND VON NEUMANN PRAXICS 111

2.2.1 The system itself

Quantum theory is an exercise in humility. It is the first mathematical physical theory toaccept that the entities of nature have no mathematical physical model. As Bohr put it,quantum theory is not about nature, it is about what we can say about nature. Put in moredetail, quantum theory does not model quanta mathematically, it models processes per-formed by huge aggregates of quanta like ourselves that create, transmit, or register quantawithout determining all their properties. Mathematical modeling sometimes becomes pos-sible for quanta organized in masses even though it is impossible for the individual.

The minimum condition for a mathematical model is that the physical entities shouldbe distinguishable when their symbols are. If A and B are symbols, it is assumed that thetransition probability for writing an A and reading it as a B is 0 or 1:

B〈A = δBA. (2.3)

On the other hand if b < and 〈a are processes for “reading“ and “writing“ a quantum, likepolarizing and analyzing processes, then they can be represented by vectors so that thetransition probability is

|b〈a|2 = cos2 θ, (2.4)

which ranges continuously between 0 and 1. It is easy to accomplish the yes-or-no trans-mission probability for symbols and impossible for quanta. On the other hand, there isno physical obstacle to distinguishing quantum vectors b〈 and 〈a, and so they can bemathematically modeled in quantum theory.

Just as one can explicitly distinguish between a numerical variable and any one of itsvalues, an individual quantum system can be distinguished from any one of its states oractions. This rather nondescript kind of quantum entity admits a mathematical model. Inprobability theory, the entire state space S, not one of its points, serves as a mathematicalmodel for the random variable, the individual system itself, abstracted from its contingentfeatures. This is the conventional zero-point of information about the system, or thepredicate of mere existence of the system, without further specification. The unnormalizedconstant probability distribution function, the bunit functionc

1S : S → 1, S 3 v 7→ 1, (2.5)

represents mathematically the mere existence of the system. 1S is also used here to repre-sent the individual system with state space S.

Similarly a quantum individual is the general case where the special case is what iscreated or destroyed by the process represented by any one binputc or output ray in anassociated vector space. The entire binputc vector space V , or the associated ray space,not one of the vectors or rays, can be used as a mathematical model of the hypotheticalsystem itself, the individual that the system counter counts, abstracted from its contingentfeatures. Therefore the unit operator on V , which is the projector

1V : V → V, v 7→ v, (2.6)


can also be used as a representative of the quantum individual (system) I[V ]. The operator1V corresponds to multiplication by the unit function 1S in the classical theory. Thequantum individual can be modeled in this way because the model says so little about thequantum.

Example: Since 2 := 0, 1, I[2] is the binary random variable, with values 0 and 1; 2is a two dimensional space with basis 2 = 0, 1; and I[2] is a hypothetical quantum withvector space 2.

The bmultiplicityc of a random system I [S] counts mutually exclusive possibilities forthe individual, and is the cardinality of its sample space: Mult I [S] = CardS.

The bmultiplicityc Mult I[V ] of a quantum individual I[V ] is the dimensionality DimVof the binputc vector space of the individual and the number of mutually exclusive, togetherexhaustive, possibilities for the individual.

The bdual systemc I[V ]D to the system I[V ] is the hypothetical system whose inputvector space is the output vector space V D of the system IV . Therefore an binputc of aI[V ] is an output of a I[V ]D.

The banti-systemc I [V C] is the dual of a system of the negative energy, defined morefully in §6.4.2.

2.2.2 The equatorial bulge in Hilbert space

Almost all the area of the unit sphere S∞ in Hilbert space H = ∞ · C is at its equator.Almost none of the area of S∞, proportionally speaking, is near its North Pole. Moreexactly put: Let δθ > 0 be any fixed angle, no matter how small, As the Hilbert spacedimension D →∞, the area of the belt about the equator of SD−1 of width δθ approachesthe area of the entire unit sphere, while the equator of a zone of width δθ about the NorthPole approaches zero in relation to the total area. The probability of finding a randomvector within δθ of the equator approaches 1 as D →∞. This geometrical phenomenon isthe “equatorial bulge” in Hilbert space.

If we combine N systems of multiplicity M , we need a vector space V of dimensionD = MN to describe the composite. In any vector space of high enough dimension,however, almost every pair of vectors is as close to orthogonal as desired, due to theequatorial bulge in Hilbert space, and their projectors, together with all the variables thatcan be formed from them, are as close to commuting as desired.

Therefore two directions chosen at random in infinite-dimensional Hilbert space arealmost always arbitrarily close to being orthogonal; their projectors are almost alwaysarbitrarily close to commuting, and as predicates are almost always arbitrarily close toobeying Boolean logical laws. It is highly improbable to see noncommutativity amongtypical variables of macroscopic systems. Classical commutativity emerges for sufficientlycomplex quantum systems as a result of this equatorial bulge in Hilbert space.

2.3. STANDARD SEMANTICS 113

2.2.3 Commutative reduction

The process that converts a quantum system I[V ] into a random object I [S] whose statespace S is the set of rays of the vectors of some orthonormal basis B ⊂ V is calledbcommutative reductionc here. The predicates of a bcommutative reductionc are all repre-sented by diagonal matrices with respect to the chosen basis B for the quantum system.

The operator methods of quantum physics also apply to classical physics in a singularlimit. One can reduce a quantum system to a classical one by reducing the operatoralgebra A of the system to a Cartan (maximal commutative) subalgebra Ac ⊂ A. This isa commutative reduction. Classical logics are diagonal parts of quantum praxics.

Changes being off-diagonal operators, it is paradoxical that change still occurs in theclassical limit, where all variables are diagonal. The well-known resolution of this paradoxis discussed in §2.4.

2.3 Standard semantics

By the semantics of a physical theory is meant here mean the two-way connection betweenits operators and laboratory actions. Almost all working physicists use the quantum se-mantics presented in the rest of this section. It includes a correspondence principle: Itreduces to the classical semantics under a commutative reduction of the quantum theory.

By a “bcompletec representation” of a system is meant here one from which answersto all meaningful experimental questions about the system can be computed. This usageis consistent with the logician’s, and in this sense quantum theory and number theory areboth incomplete, one physically and the other mathematically, though in different waysand for quite different reasons.

Bohr called a quantum representation “complete”, however, to express something else:that even though a quantum representation of a system leaves most questions about thesystem undecided, merely assigning probabilities to the possibilities, it does not follow froma complete representation of some possibly larger system by deleting some information.Such representations of a system are called “maximal” by von Neumann and here. Theterm “maximal” too should be used carefully. A quantum theory often gives birth to amore informative theory when new degrees of freedom are discovered in nature, and thenboth mother and daughter can have maximal representations.

The ontological quantum mispresentation leads to misunderstandings of the dynamicalprocess. Some infer from it that the quantum correspondent of Newton’s equations, orof their Hamiltonian form, is the bSchrodingerc equation. Actually it is the Heisenbergequation; the Schrodinger equation is the correspondent of the Hamilton-Jacobi equation,not Newton’s. The system changes in time, its vectors represent actions that we simplydo or not. They are not measured, they measure. To speak of them changing or notchanging with time during the development of the system confuses the system with aninput-output operation for the system, a photon spin for a polarizer, a product for the


process. When a vector 〈 t is understood to be a process carried out to produce a system,the equation 〈 t′ = 〈∆t〈 t does not mean that 〈 t is changing, it means that as far as anylater output operation o〈 is concerned, the process 〈 t is indistinuishable from the laterprocess 〈 t′ = 〈∆t〈 t, the predicted transition probability amplitude being o〈 t′ = o〈∆t〈 t,where ∆t = t′ − t labels the dynamical transformation 〈∆t〈 .

Similarly the unitary operator 〈∆t〈 is not what actually happens to the system norwhat the system actually does. Its matrix elements are transition probability amplitudesfor what might happen, not what does happen, but nevertheless determining the proba-bilities for all possibilities. It is a coherent quantum analogue of the table of transitionprobabilities of a Markov process, which is also not what actually happens but only astatistical description. What actually happens is not completely describable in quantumtheory, any more than what actually is. Nevertheless it works.

2.3.1 The orthogonal group

is used to projectively represent the kinematical group of I[V ], the group of the re-versible transformations of the system, including both coordinate transformations,which act in the bmetasystemc alone, and dynamical transformations, which are actionson the system.Example: Passing a photon of monochromatic visible light through a cell of sugar watereffects a dynamical transformation of its spin, a rotation through an angle θ about the beamaxis. Rotating the binputc polarizer by −θ effects a coordinate transformation representedby the same operator.

The identity operator I[V ] on V represents the quantum system better than V itself,since I[V ] = I[V D], simply acting from the left or right on the two spaces.

2.3.2 Probability vectors

The contraction [o < i] is a relative transition probability amplitude. This is expressed by

Assumption 7 Malus-Born Probability Principle There is a bprobability formc | > h〈 onprobability vectors defining the probabilities for the transitions o〈 × 〈 i by

P =o〈 i i〈oo〈o i〈 i

, where i〈 = i〉H〈 , 〈o = 〈h−1 〉o. (2.7)

Example: The two-dimensional space 2R and its dual 2RD are used to represent binputcand boutputc (processes) for photon spins using linearly polarizing filters at the binputc andoutput ends of the experiment. If the filters are oriented along vectors 〈 i ∈ 2 and o〈 ∈ 2D

then (2.7) becomes the bMalus Relationc P = cos2 θ.In classical thought an input-output process of maximal resolution defines a state of

the system. If Alf prepares one system in a state s, then Bea can determine that state in


a single measurement on that same system. The state is observable in that sense. A statecan be determined from a single system.

A vector that inputs a system cannot be found from that system, which it describesonly statistically. If Alf inputs one system with a vector 〈 i then Bea can never determine〈 i, or its ray, by any measurements on that system whatever. In order for Bea to learn 〈 iwith arbitrary precision, Alf must supply arbitrarily many systems with vector 〈 i. In thisregard a vector is like a probability distribution function, not a classical state. The classicalcorrespondent of quantum theory is not classical mechanics but statistical mechanics [64].Variables of the system that may be observable in a single measurement are representedby operators, not vectors, with the possible values as eigenvalues. The classical state isa variable of the classical system. One classical system has a state just as it has (say) aposition or a momentum; one quantum system does not.

The term ‘bstate vectorc seems to facilitate a misunderstanding of the quantum theoryon this point. It leads the unwary to suppose that every quantum system has a state vectoras every classical system has a stste, an error of level.

Heisenberg’s term “probability function” avoids this reification but also omits the quan-tum aspect: the components of the vector are probability amplitudes, not probabilities.

Dirac’s terms “bra” and “ket” remind us appropriately of how these vectors figure inexpectation values, designated with brackets by Dirac; unfortunately expectation valueshere have the non-bracket form ψ〈Q〈ψ, so the bra-ket terminology is no longer a helpfulmnemonic.

On the other hand, there are quantum constructs that can well be called states withoutviolating the correspondence principle (§2.9).

A probability vector has a dual role, with both a macroscopic and a microscopic aspect.On the one hand its ray may be a macroscopic state variable of the experimental apparatus,as the polarizing angle is for a polarizer. On the other it represents statistical informationabout the quantum system. This double role is familiar from thermodynamics. The pistonposition for a steam engine also has two such roles: it is a macroscopic variable of the engineand a parameter in the distribution function of the steam molecules in the cylinder of theengine. A major difference is that the piston’s position does not give maximal informationabout a water molecule, but the polarizer’s angular position gives maximal informationabout the photon polarization.

It is the custom in physics to keep the pre-evolutionary terminology as long as it canbe made to work. We still speak of the time of an event in special and general relativity,knowing that a reference frame must be specified to give this term meaning; in principle wecan do the same with state in quantum theory, but it has not worked as well. The evolutionfrom classical to quantum physics required a relativization of state that seems deeper thanthe relativization of time and simultaneity in special relativity. The quantum theory notonly relativizes but also dualizes the classical concept of state. Each quantum experimenterdefines and is defined by a class of input actions and a dual class of output actions, properto the experimenter, for which classical logic works. The probability vector is relative to


the experimenter in that the class of accessible actions varies with the experimenter, anda probability vector is determined by examining the experimenter, not the system. It isdualized in that quantum theory has both input and output vectors, used in mutually dualways. In a general experiment one experimenter carries out the input process and anotherthe output process; the experiment is a quantum communication channel. Vectors of Vand V D represent the actions chosen by the two experimenters, not state variables of thesystem, which are represented by operators.

2.3.3 The probability form

A probability vector space V requires a symmetric form 〉H〈 : V ↔ V D to represent assuredtransitions and system similarity. 〉H〈 then also induces experiment transposal, transposingthe order of all operations and interconverting input and output processes.

H is also the form used to compute transition probabilities. The transition from aninput 〈 i to an output o〈 is bassuredc, has transition probability 1, if o〈 = i〉H〈 .

A form H with these physical interpretations is called the bprobability formc. It is,to be sure, completely determined by the inclusion and occlusion relations for the systeminput-output processes, which mention probability 0 or 1 only; but these relations are stilldetermined by statistical sampling.

Queues require a probability form for each stratum. It follows that if

o〈 :=[〈o]†, 〈 i :=

[i〈]†. (2.8)

theno〈 i = i〈o. (2.9)

(In a complex theory a complex conjugation C is required on one side.)

2.3.4 The linear operators

The dynamical development from one time t to another t′ defines an isometry 〈 t′, t〈 inquantum mechanics as in classical. In a relativistic theory this extends to a representationof Poincare transformations T ∈ ISO(3, 1) by isometries 〈T 〈 and one postulates invarianceunder ISO(3, 1). This postulate is structurally unstable and implies that time is not sim-ple. Since it has worked rather well it is not dropped here but changed slightly, perhapsundetectably, just enough to make its group simple and structurally stable.

It is possible to hide the coordinates and work with invariants. Then instead of vectorsone speaks of points of a bprojective geometryc, which correspond to rays of vectors, andinstead of operators one speak of projective transformations, as in perspective theory.Vectors and matrices are more powerful tools for theory construction, however. Every usefulphysical quantum theory was discovered in the language of matrices or linear operators;none in the language of projective geometry or lattice algebra.


Spectral linear operators on V → V are used to projectively represent dynamical vari-ables, ideal multichannel sortations of the system by the bmetasystemc. Linear operatorsare also used as infinitesimal generators of one-parameter groups of system transformations.

Sometimes one treats matrices themselves as vectors in a higher-dimensional vectorspace. The matrix 〈A〈 can be written with both brackets on one side, representing acomposite index:

〈A〈 ≡ A〈〉. (2.10)

The contraction of two such vectors is the trace of the operator product:

Tr 〈A〈B〈 = A〈〉B. (2.11)

2.3.5 The projectors

A predicate or class is defined as Boole did, by a filtration of the system under study.Projection operators, or projectors, of V , the idempotent symmetric operators P with

P = P 2 = P †, (2.12)

represent such filtration operations on I[V ]. The lattice they form corresponds to thebclassc or bpredicatec logic of a classical system. They are partially ordered by the inclusionrelation

P1 ≤ P2 := P1P2P1 = P1. (2.13)

The supremum P1 ∨ P2 and infimum P1 ∧ P2 of projectors in this partial order representsthe Boolean operations P1 or P2 and P1 and P2 on predicates. The product operationP2P1 represents an and then operation; the predicate algebra is not closed under thisoperation. If P1, P2, and P3 are projectors then P3 is called a bsuperpositionc of P1 andP2 if for all projectors P ,

P1P = 0 and P2P = 0 : implies P3P = 0. (2.14)

The above list of standard usages is redundant. With reasonable definitions, the usageof §2.3.1 probably entails all the others. It excludes superselection (centralization) lawsand bcompoundc — non-semisimple — Lie groups; but these still arise as singular limits.

The semantics of the quantum predicate algebra can be considered known. Its com-mutative reduction to the classical theory requires a special limit (§2.4).

The system under study is here a set, either c (sea) or q (queue). The bracing operatorI on probability vectors, together with addition and multiplication, results in a bquantumset theoryc, or bqueuec theory, in that the resulting theory has classical set theory as acommutative reduction. While some combinations of I and ID are given empirical meaning,further clarification of their meaning is still needed.


2.4 Change is a quantum effect

The commutative reduction of quantum predicate algebra is classical predicate algebra butthat of quantum kinematics. is not classical kinematics. A quantum theory uses the samenon-commutative algebra to house its projectors and its dynamical processes. Classicaltheory is somewhat weird in that its predicates commute and so all its variables commuteyet its dynamical processes do not. Diagonal operators generate no change in diagonaloperators. There should be no dynamical evolution after a commutative reduction toclassical physics, because the the rate of change

i~dQ

dt= [H,Q] (2.15)

is a commutator and all classical variables commute.Motion remains in the classical limit because the limit is singular. The time-develop-

ment operatorU(∆t) = e−iHt/~ (2.16)

involves 1/~. This amplifies the effect of the dwindling commutator so that change canstill occur as ~→ 0. The commutators [q, p]→ 0 vanish but −i~−1[q, p] does not; insteadit approaches the Poisson Bracket. All change, according to canonical quantum theory, isa vestige of quantum change that survives the singular classical limit. The split betweenobserving and acting is an artifact of the classical limit and does not occur at the quantumlevel of resolution.

2.5 Simple systems

By a bsimple systemc is meant one with a simple kinematical Lie group. Nowadays onerecognizes several simple systems, such as the spin, isospin, and color of a quantum.

A simple kinematics is quantum, without superselection laws. No classical space-timeis simple, since its coordinates are central.

By the simplicity strategy Cartan’s Table of the Simple Lie Groups is also the Tableof Simple Systems. Here the possibility is explored that actual systems have SO(n;σ)kinematical groups, in the D family of the table.

The usual quantum theory represents the dynamical development as a one-parametergroup of isometries with time as the parameter, applying the first construct in the abovelist. In bqc theory, however, time enters more deeply, as one of the coordinates of thespace-time events to which the field is attached. The continuity of time leads to an infinitedensity of field variables and to all the important infinities in present physical theory.

Usually the event of space-time is treated as a classical random object E with statespace S[E] composed of space-time events. Here the event is treated as a physical systemin its own right, with a vector space S[E], though each event is probably strongly entangled

2.6. PROBABILITIES 119

with other events, and the fully quantum strategy is applied (Assumption ??): The genericevent is represented with a simple vector space V [E] = Grade1 S[E]. A coordinate of theevent I[SE] is then a normal operator in the operator algebra Alg(V [E]). A quantumcoordinate transformation is an automorphism of this algebra, which is inner, generatedby an operator in SO[E].

Then a fully quantum time cannot have a continuous spectrum and cannot be theparameter of a Lie group. It is an operator on the vector space V E associated with the event,with a discrete bounded spectrum. This time construct, like most quantum constructs, isrelative to the partition into system and metasystem.

Since i generates a radical, on the grounds of simplicity we use a real quantum theoryhere in the sense of bStuckelbergc, who also showed how to reconstruct a complex quantumtheory within the real one [72]. It is likely that Stuckelberg’s study of real quantum theoryinfluenced his early theory of the massive vector boson [?], as it did later work [?].

2.6 Probabilities

Any variable of a system is represented by an operator on a vector space for the system.To input a photon (say) through some chosen polarizer is an input action 〈 i. To output aphoton through some chosen analyzer is an output action o〈 . The input and output actionscan be chosen independently by the experimenter at each end of the optical bench. Thereis no question of one developing or collapsing into the other; that idea is a vestige of thenon-quantum theories of de Broglie and Schrodinger. Then the Malus-Born Principle ofAssumption 7 and equation (2.7) gives the transition probability. The transition probabilityis invariant under the group SO(V ) that respects H.

2.7 Mixtures

Each bvectorc and its negative represent the same input process but give different results insuperpositions. Therefore quantum superposition does not represent a physical operationon input processes alone. The invariant operational content of the phase of the vectorand of quantum superposition is discussed elsewhere, for example in [40]. The process ofrandom bmixturec of the inputs from two separate input processes is physical, though itloses information; It is also called bincoherent superpositionc. In contrast, vector additionis then called bcoherent superpositionc. To represent mixing one associates with each vector〈 i an input bprobability tensorc (“bdensity matrixc”) and a dual output bprobability tensorc

〈I 〉 :=〈 i⊗ i〉i〉i

, 〉O〈 := 〉o⊗ o〈 . (2.17)

The common term “bdensity matrixc” is rooted in ontology. The ontologist reads itseigenvalues as material densities, although they are actually probabilities. Such mispresen-


tations of physical interpretation are avoided here. The rule in quantum theory is to nameoperators after the physical meaning of their eigenvalues. The diagonal elements of one ofthese tensors are probabilities, so it is properly termed a bprobability operatorc.

Input processes represented by vectors 〈 i are called bsharpc or coherent. Those repre-sented by a projector of any dimension are called bcrispc. The most general input process iscalled bdiffusec and is represented by a more general input bprobability tensorc 〈I 〉 =: 〈〈I,operationally defined by the condition that the transition probability for an output vectoro〈 is

P = o〈I 〉o (2.18)

Output probability tensors 〉O〈 = O〉〈 are defined dually. The btransition probabilityc

when both input and output processes are mixtures is

P = Tr 〉O〈I 〉. (2.19)

The definitions have been chosen so that the bdualityc H is not brought in until it isneeded. The empirically forbidden transitions alone, with o〈 i = 0, determine the vectorspace structure, including the dimension and the ring of coefficients. The empiricallyassured transitions then determine the bprobability formc b〈H〈 c so that if o〈 = 〈H〈 i, thenthe transition o〈 i happens in every trial.

A more common convention converts these second-grade symmetric tensors 〈I 〉 and〉O〈 to linear operators 〈I〈 and 〈O〈 by appropriate factors of H = 〉H〈 . Then thereis no difference in appearance between the input and output process symbols, and for atime people spoke ontologically of “the” bdensity matrixc when there are two in everyexperiment. The explicitly operational analysis of bGilesc restored this duality [45].

Then bsuperpositionc adds vectors, 〈a+〈b = 〈c, while bmixingc or bincoherent superpositioncadds probability tensors, 〉A〈 + 〉B〈 = 〉C〈 .

2.8 Transformations

A general bexperimentc can be idealized as a three-stage transaction o〈 ×〈 t〈 ×〈 i consistingof input, throughput, and output (or emission, transmission, and admission) of a system.This usually employs two experimenters, one at each end of the experiment, each with anexperimental frame that includes the action performed. The expression

A = o〈 t〈 i (2.20)

for the btransition probability amplitudec is also a diagram of the experiment. If both o〈and 〈 i have unit norm then A2 is the transition probability.

Sharp experiments require a bmetasystemc much more complex than the system stud-ied. Merely recording the outcomes requires the bmetasystemc to have a separate subsystemfor each possible system input process; for example, a photosensor for each channel of a

2.9. STATES, PROPER AND COORDINATE 121

spectroscope. Even if these parts are merely binary, the number N of independent possibil-ities for the bmetasystemc must then be at least exponential in the number n of possibilitiesfor the system:

N>∼ 2n. (2.21)

Determining all the properties of a quantum system together is impossible, according tothe quantum theory, since determining one property is found to undetermine others, in theway described by quantum bnon-commutativityc. This makes the quantum representationof experiment bincompletec. It is still possible to extract statistical information aboutpractically all the properties of practically all the systems from a given source of statisticallyindependent systems, however, by determining the mean of each property on a small butnot too small sample of all the packets from the source. Operations on an individual donot change the averages over such an ensemble by much.

2.9 States, proper and coordinate

The bstatec of a classical particle moving on a line is a point in a phase space with acoordinate pair (q, p) of a position variable q and a momentum variable p. It providesa complete description of the particle for the purposes of mechanics. The classical state(q, p) has a quantum correspondent, the Manin state (q, p) [56, 57]. The Manin state clearlysatisfies the bBohrc bcorrespondencec principle. Since q and p do not commute, the Maninstate is not observable.

In classical physics it is common to use the same name for a variable and one of itsvalues. For example, one can say that the momentum is p or that the momentum is 22kg.m/sec. The former declares a variable, the latter gives a value. At least four kinds ofthing can be called the bstatec of a classical particle on a line without solecism: the pair ofvariables (q, p), a pair of values (q′, p′), the variable point of phase space with coordinates(q, p), and the fixed point of phase space with coordinate values (q′, p′).

Generally the correspondence principle is used to fix our quantum terminology, but inquantum theory the variable and the value have such different commutation relations thatit is helpful to distinguish them more clearly than classical usage.

The classical pair (q, p) is both a maximal set of independent variables and a maximalset of commuting independent variables. In quantum theory, (q, p) is a maximal set ofindependent variables while q by itself is a maximal set of commuting independent variables.(q, p) is covariant under canonical transformations but not observable; q is observable butnot covariant. Which shall be called the quantum bstatec? One condition that is imposedhere is that it is something that can be determined from the individual system, or is acollection of such things.

A vector is not such an entity; it is found either by a single inspection of the metasystem— a polarizer, for example — or from statistical studies of many experiments on likesystems.


This recalls the relativistic choice between proper time, which is invariant but notintegrable, and coordinate time, which is integrable but not invariant. Indeed, the propertime is not an observable coordinate of the particle, but of a history segment. Both timeconstructs are useful and are used. This relativistic policy is adopted here for the followingtwo state constructs also:

2.9.0.1 Proper state

A bproper statec of a quantum system is a maximal set of independent generating variables.Example: (q, p) is a bproper statec.

This is the bManinc concept of bstatec. It is bcompletec, in the sense that its bcommutantc((6.59)) is trivial, but it is not observable, since its variables (q and p in the example) donot commute.

2.9.0.2 Coordinate state

A bcoordinate statec of a quantum system is a maximal set of independent commutinggenerating variables. Example: q is a bcoordinate statec of the same system.

A quantum bcoordinate statec is an observable like the classical state; but it is notcomplete or unique. It defines a frame, and is defined relative to that frame.

Notice the trap that awaits one who calls the vector a state vector: A property of asystem is a variable whose different values correspond to disjoint sets in classical phasespace, or orthogonal subspaces in quantum theory. The state is a property of the classicalsystem, whose values correspond to distinct points of phase space. If the state vector weremistaken for a property of the quantum system, as the term suggests, then its differentvalues would correspond to orthogonal vectors of the vector space. This would imply thatall vectors that are not parallel are orthogonal; which actually holds for classical systemsin a vector description.

2.10 Praxiology and its singular limit

bHeisenbergc called quantum theory “non-objective physics” presumably because his op-erator formulation worked not with nouns standing for objects but with verbs standingfor actions, especially on quanta. “Action physics” would also have been a reasonableterm, but action painting was still two decades in the future. A theory of what exists iscalled an ontology, a theory of physical actions can be called a bpraxiologyc.1 Then the

1Not to be confused with praxeology, a general theory of human actions from the viewpoint of economics[76]. The praxeology of von Mises has elements in common with the present concept, but differs substan-tially. For example, it explicitly assumes that we have apriori knowledge and in particular that only onelogic is possible for the human mind, and it does not undertake to analyze objects into actions. Praxiologyseems consistent with the pragmatism of Peirce and the process philosophies of Whitehead [?] to the extent

2.10. PRAXIOLOGY AND ITS SINGULAR LIMIT 123

Bohr-Heisenberg theory is bpraxiologicalc, not ontological. Since the state space of everyobject has a transformation group and semigroup, which characterize it and can be used torepresent operations on the object, every ontology implies a praxiology, but one of a spe-cial kind, in which the sorting actions associated with observable properties of the systemunder study commute, as Boole noted [14]. Almost no praxiologies in the general sensederive from ontologies in this way. In generic terms: An ontology is a singular limit of apraxiology.

The proposed fully quantum theory is praxiological in that it does not assume absoluteobjects as ontological theories do. There is no reason to suppose that this theory is theend of the line, however. It gives its actions an absolute identity in that it allows one tospeak of the same actions acting in a different order, but the empty set associated with thevector 1 may be relatively rather than absolutely empty, and all other vectors are definedrelative to 1.

Some of the founders of the present quantum field theory spoke in operational terms,praxiologically, and others attempted to interpret the theory ontologically. Much of thelanguage in current use was formulated and promulgated by ontologists. As a result, agree-ment is excellent about how to use the quantum theory in the laboratory, but not abouthow to present it. Sometimes such presentations are called interpretations of quantumtheory, as if quantum theory did not already have one. By the interpretation of a theorywe mean rules about how to use the theory. But a physical theory less its interpretationis not a physical theory but a formalism. Quantum theory already has its interpretation,discussed in §2.3.

Bohr and Heisenberg do not posit systems with complete representations but concernthemselves with operations on systems. A presentation that accurately describes theirquantum theory is necessarily praxiological. An ontological mispresentation strives to fitsuch theories into the classical state strategy, which contradicts them.

Today the name “Copenhagen interpretation” is often used for various mispresen-tations of the quantum theory that attempt a classical ontology, avoiding the quantumconcept of physical entities that have no complete mathematical representations. Thesemispresentations attempt to remain within the domain of classical logic and probabilitytheory, while quantum theories modify the predicate algebra and the probability algebraof the system.

Ontological mispresentations all share one tactic: A construct that is known onlystatistically, from many experiments, and is used to compute probabilities of future exper-iments, is nevertheless given ontological status, and said to be physically present in eachexperiment. Being statistical, the chosen construct is not appreciably affected by its deter-mination, which can be carried out on a small subsample, and so it can safely be treatedas a classical object. On the other hand, the statistical construct is no more present in the

that these focus on processes rather than their products, but is applied here just to the interpretation ofquantum theory.


individual case than a probability.Furthermore, the attempted ontological theories never close, in that they do not ex-

plain how an object that is alleged to be physically present in the individual experimentdetermines probabilities as if it were a probability distribution across many experiments.Some of the terms that are found in ontological mispresentations are “the vector of thesystem”, “the collapse of the vector”, “many worlds”, “the vector of the universe”, and the“quantum potential”.

The purpose of this section is to prepare students for the mutually inconsistent pre-sentations of quantum theory found today. This section is not necessary for the formaldeductions in this work, nor for those already aware of the inconsistencies in current usage.

The earliest such ontologization is incorporated in the terms “wave function” and“state vector”. Heisenberg called such vectors “probability functions”, not state vectors,and properly they are probability amplitude functions, but these names are unwieldy.The term “vector” used here makes it plain that the meaning intended is statistical, notontological.

Some say that every individual system has a state vector, just as every individualsystem has a state according to classical physics. They imagine a random variable I [V ],the wave function, which ranges over a vector space V . The praxiological presentation,on the contrary, concerns a quantum system I[V ] (both defined in §2.2.1). A vector of Vrepresents a quantum process and I[V ] designates the variable quantum produced in suchprocesses; ontologists confuse the process and the product. Both praxiological presentationand ontological mispresentation accompany the same statistical practice, but one presentsthe practice accurately and praxiologically, and the other reifies probability distributions.This discrepancy does not affect the application of quantum theory in atomic physics,which mainly ignores the ontological presentation and carries out the praxiological one,but it affects the research strategy, and hinders attempts to extrapolate quantum theoryinto new domains.

Ontological mispresentations also lead many to see mere consequences of this misp-resentation as difficulties of the quantum theory. The ontological mispresentation of thenon-commutative quantum theory is so absurd that it makes it seem that we must havegone astray, and must go back and start over. This diverts valuable research resources fromthe well-known genuine problems of quantum theories, and spreads public misinformationabout nature. The purpose here is not to enshrine the canonical quantum theory but to goforward into physics that is even more quantum rather than less. The term “Copenhagenquantum theory” seems mired in ontological mispresentation, and permanently associatedwith a concept of wave-function collapse alien to what was taught in Copenhagen. Its useis therefore avoided here. One may speak instead of the canonical quantum theory; and ofits praxiological presentations and ontological mispresentations; and of the fully quantumtheory worked out here.

Ontological theories proceed as if the theorist has a representation of what things reallyare, perhaps by seeing them as they really are, perhaps by gnosis. Leibniz gave a source of


such ontological knowledge explicitly in his monadology: Monads are informed about eachother and the universe directly by God.

While the canonical theory is praxiological, ontological presentations continue to sup-pose the main purpose of science is ontological, to mirror nature, to represent the worldand its history and relations by sets as faithfully as possible. Even quantum theories allowstatistical distributions or collective variables of macroscopic apparatus to be completelyrepresented, at least in a useful limit. Ontologism assigns to these statistical constructs anontological status that they do not actually have, so as to limit discussion to constructs thathave complete descriptions. Ontologism still allows relativity, different observers can makedifferent complete representations of the same object, but it assumes that there is a 1-1transformation relating these representations, and that representations can be complete,answering all well-formed experimental questions about the object or its history. Thecommon method is to imagine that the statistical construct is present in the individualexperiment, ignoring how it is actually determined experimentally.

A photon cannot be completely represented by symbols but a photon vector can, forexample, so an ontologist might teach that “the vector of a photon” is the state of thesystem, or is even the system itself.

When ontologism effectively mispresents every system as a random object of the formI [S] for some state space S, it represents every transformation of the system as a permu-tation of the points of S. It tells us what “is”: a point of S. The quotation marks areto remind us that this existence is supposed to include unobserved existence. Presum-ably ontologists think that knowing what “is” can help us to understand and control ourexperiences with nature, but actually a statement of what “is” is useless for predicting ex-perience unless it is backed up by theories of perception and dynamics, relating what “is”to to what we experience. From an operational viewpoint, the assertion that somethingexists is meaningless without such a context.

The context may be a naive theory of perception, claiming that what we see is whatthere is. Since the process of perception was often unmentioned in the past, some mayhave believed that somehow we see things as they “really are”, as if by a form of mysticalgnosis. It is not easy to believe this today. A typical one-mol-kg object has on the orderof 1024 molecules. The eye, however, has only 108 receptors, so we have access to buta tiny fraction of the properties of such an object; the rest is missing but not missed.For celestial bodies our ignorance is even greater. The construct of complete knowledge isuseful when a high organization among all these variables leads to a few collective variableswhose future behavior can be usefully predicted from their present value, leaving the vastmajority of variables relatively unknown but permitting effectively complete knowledgeabout the collective variables.

The success of macroscopic ontology, furthermore, tells us nothing about the validityof some submicroscopic ontology. We know few of the variables of an “object” system,and we also change some of them as we perceive others. The argument is based on aprinciple already known to bBerkeleyc: There is no fundamental physical difference between


perception and other physical interactions. Light reaching us from a system causes changesin us, since we perceive it. The common sense that action accompanies reaction implies thatthere must be comparable reactions in the object from which the light comes. Because wesee so little of the system under study the changes produced by light leaving it are usuallyignored. The changes occur in relatively few variables, leaving the collective variableseffectively constant. For sufficiently small systems, the total number of variables is smallenough that changes in them cannot be neglected.

Most of the cosmos is outside the system under study and almost unobserved. Theeffects of its many active elements on the system is small but certainly exists and appearsas random noise. To be viable, a theory must not be too unstable against such noise.

As a result of all these effects, the idea of an exact complete description of a system isnot useful for quantum physics. As bNicolas of Cusac taught, we can know mathematicalobjects exactly because we make them, but not physical ones because we do not.

The quantum strategy takes the main purpose of science to be praxiological or func-tional knowledge as opposed to ontological knowledge; know-how as opposed to know-what.Each quantum theory starts from a repertory of actions represented by operators, and de-duces their experimental relations from their algebraic relations. Quantum physics omitsthe hypothesis of the unobserved object and the state space, and cuts directly to the pro-cesses that they were invented to handle. In doing so, Bohr and Heisenberg intentionallyapplied the practice that Newton and Einstein promulgated:

Omit the unobservable.

Quantum theories represent a quantum system by a class of acts that we can carry out onit, rather than a class of states of being, unobservable by postulation. Namely — to repeat— it represents each system as an I[V ] with a bvectorc space V , a dual vector space V D,and an operator algebra V ⊗ V D. Its presentation is the minimum required to express thequantum semantics (§2.3): The probability amplitude for the experiment consisting of theactions o〈 × 〈 t〈 × 〈 i is o〈 t〈 i.

This principle is not an addition to the classical theory but a viable descendant of it,it actually holds in the classical theory, only superpositions are forbidden in the classicaltheory, so interference terms are missing, and the probabilities add as well as the probabilityamplitudes. It is therefore not necessary to explain this probability amplitude formula inthe quantum theory any more than one explains the corresponding probability formula inthe classical; we must only explain why classical theory omitted superposition, and this isclearly a matter of large numbers, an economic limitation.

The quantum theory does not describe our experience with the system by a state in asystem state space and a naive theory of perception, but by an operator in a system algebra,representing the entire process, including the perception and recording of the result.

Since this distinction between the praxiological and ontological is often understated,it is exaggerated in the next two sections, leaving out many moderating overlaps andvariations.


bNewtonc (in Query 34 in one edition of his Opticks) supposed that light consisted of acorpuscle stream accompanied by transverse waves. The companion waves were supposedto influence optical surfaces and gave rise to apparently random “fits and starts” of trans-mission or reflection of the photon at these surfaces. He postulated a crystalline ether toprovide a medium to carry these transverse companion waves for the photon. Since herecognized both wave and particle phenomena in beams of light, I think it fair to regardhim as the initiator of quantum mechanics as well as classical mechanics, though bMaluscdid not discover the Malus Relation, the prototype for the bBornc probability law basic toquantum theory, until about 1805.

When Newton introduced force — a push or a pull – into natural law, he was proceedingpraxiologically. When he assumed that a light beam consists both of photons and guidewaves, however, he was still ontological.

Malus did not model the photon itself but only polarizers, and discussed only thetransition probabilities between polarizers, not the trajectory of a polarization.

Boole explicitly based bclassical logicc on an algebra of human actions of mental se-lection from a population. He postulated that these actions commute, but declared atthe same time that this evidently praxiological postulate was an empirical conclusion, andenvisaged with some drama a logic of a more general kind [14].

This puts Malus and Boole at the head of a procession of praxiological quantum the-orists that includes bBohrc, bHeisenbergc, bPaulic, bFeynmanc, and bSchwingerc. Bothparades can validly claim to follow Newton.

Quantum theory was invented twice, some say, once by Bohr and Heisenberg, andagain by bSchrodingerc. This version of history is part of the problem today. It obscuresvital differences between two theories, differences which were small at the time compared tothe new achievements of the theory, but which seem to have grown during the interveningdecades. These differences, it should be understood, have nothing to do with the laterdifference between Heisenberg and Schodinger pictures, which are equally valid when bothexist, and equally praxiological. Rather, the quantum theory of Bohr and Heisenberg waspraxiological and worked while the attempt of Schrodinger was ontological and, strictlyspeaking, did not.

Heisenberg and Bohr assumed that the atomic domain was granular and obeyed non-standard probability laws. Matrix mechanics replaced the multiplication and addition oftransition probabilities that occurs in a Markov process by the multiplication and additionof probability amplitudes.

bSchrodingerc originally assumed that the electron in a hydrogen atom was a waverunning around the nucleus, implicitly retaining classical probability laws for states of thewave. Since his wave equation was of the first degree in the differentiator with respect totime, the state of his wave was a bwave functionc evaluated at one time. His theory is asontological as the Newtonian particle model of nature, it merely made the atom a waveinstead of a particle. This lead to the following immediate difficulties for the bSchrodingerctheory that are not problems for the bBohrc-bHeisenbergc theory. Some of these trouble


ontologists to this day:

2.10.1 Ritz combination rule.

bSchrodingerc found a well-known discrete spectrum of possible vibration frequencies fn =f1/n

2 for his hypothetical wave. Had his wave theory been right, these would also havebeen the frequencies that the hydrogen atom radiates, as a vibrating cymbal radiates soundof the same frequency as the vibration. Instead the bSchrodingerc frequencies agree withthe empirical spectroscopic terms for the hydrogen atom; the atom radiates some of thedifferences fn−fm between terms and not the terms themselves. This disconfirms the wavetheory.

2.10.2 Probability Principle.

Moreover the electron does not divide when the bwave functionc does, as in electron diffrac-tion. In the ontological theory this definitenesss of events is a problem because according tothe bSchrodingerc equation the bwave functionc usually spreads out during the dynamicaldevelopment and never comes back to one point, while the entire electron is often registeredby a detector much smaller than the alleged wave. The praxiological theory has no suchproblem. The classical correspondent of the bSchrodingerc equation, the Hamilton-Jacobiequation, also describes a principle function that may be concentrated at one point of con-figuration space initially and then spreads out, generally never converging to a point again,and this is no problem for classical physics because the principle function is not a physicalobject. The Boltzmann equation has similar properties; its probability distribution doesnot reconcentrate on one point either. Nevertheless in these theories there is no problemdeducing that definite events happen; one simply assumes it from the start as a matter ofdefinition. A probability is the probability for a definite occurrence. A probability p = 1/2does not mean that half an event occurs; it means that the event in question tends to occurabout half the time in the long run. Quantum praxics make the same assumption for aprobability amplitude. A probability amplitude o〈e = 1/2 does not mean that 1/4 of anevent occurs. Nor does it give the amplitude of some physical wave. The experiment actson a quantum, not on a bwave functionc. A probability amplitude is by definition a proba-bility amplitude for a definite occurrence to the quantum. It therefore cannot be measuredwith one quantum but requires many trials and a fixation of a frame, like a probability.

One cannot learn the probability interpretation from the equations governing the prob-abilities but must provide that interpretation at the start, in both classical and quantumtheories. In the quantum theory, the definiteness of experience was even made explicit, asthe Quantum Principle:

The quantum remains whole when the probability amplitude divides.

Here the name “Quantum Principle” is a monument to early ontological confusion. Aprobability distribution for Mars also spreads out, and Mars too does not split, but we do


not call this the Classical Principle, it is part of the operational meaning of probability.Likewise the Quantum Principle is merely part of the operational meaning of probabilityamplitude. It could better be called the Probability Amplitude Principle. There is no morepossibility for events to split or “collapse” or become “indefinite” or fuzzy in quantumtheory than classical. A fuzzy prediction is not a prediction of fuzz.

There is still another parade in town beside the praxiological one. Following Newtonis a second procession of many ontologists including Einstein, de Broglie, bSchrodingerc,bWignerc, and bGell-Mannc, who believe that we can do better than a theory of ineffableentities, with no complete description in symbols. Ontology can be attractive.

2.10.3 System catenation

Again, if the bwave functionc were the state of a wave, in the Helium atom it would still bea function of one point of space. Instead a probability amplitude distribution for a Heliumatom depends on the coordinates of all its electrons, just as a probability distribution does.When we combine several uncorrelated electrons we multiply these probability amplitudedistributions, each with a separate argument, just as we combine planetary probabilitydistributions, except that the product is anticommutative for electrons. If one has maximalinformation about a solar system one also has maximal information about each planet, whileone can have maximal information about an aggregate of electrons and know nothing aboutany variable of any electron in the aggregate. The electrons of such an aggregate are saidto be entangled.

2.10.4 Relation to probability

Finally, if the “bwave functionc” were indeed a wavy object, it would not be a probabilityamplitude. A probability for an object is not an object. An object does not have aprobability as one of its properties. Objects and transition probabilities belong to differentlevels of discourse.

Unlike classical probabilities, quantum probability amplitudes can interfere. Sincematerial waves like sound can interfere, and probabilities cannot, ontologists, who insiston classical probability theory and logic, are liable to regard probability amplitudes asmaterial waves. Then, however, to close the theory, they must account for how allegedlymaterial waves can give correct probabilities for experimental transitions. No ontologicalformulation seems to have such closure.

The wave theory has the right terms, but it has the spectrum, the Quantum Principle,the rules for composing systems, and the probabilities wrong. When we call a vector abwave functionc we memorialize Schrdinger’s original missing of the center of the mark.

bHeisenbergc blocked such bontologismc by using the term “probability function”, notbwave functionc. The term “transition probability amplitude function” would have beenmore accurate but longish. It is understood that probabilities do not exist in the way that


pretzels do; they cannot be eaten. The theory of collapsing wave-functions is often calledthe Copenhagen interpretation, but it is opposed to the interpretation of Heisenberg andBohr.

The term vector has an appropriately praxiological root sense, that of carrier:“1704 . . . A Line supposed to be drawn from any Planet moving round a Center, or

the Focus of an Ellipsis, to that Center or Focus, is by some Writers of the New Astronomy,called the Vector; because ’tis that Line by which the Planet seems to be carried round itsCenter.” Oxford English Dictionary .

Here it is used in the praxiological sense too, indicated with the modifiers “input” and“output”.

The praxiological bBohrc-bHeisenbergc theory of non-commuting variables automat-ically gets these four matters right from the start. Its primary vocabulary consists ofmatrices representing operations. Vectors are optional luxuries; and they do not representthe system under study in the way that a bwave functionc represents a wave, they projec-tively represent ways to input or output a system. by assigning probabilities to all ways ofoutputting the system. They are praxiological, not ontological.

This would seem to settle the question of praxiology versus ontology. Then Diracshowed that Schrodinger’s equation was mathematically equivalent to Heisenberg’s. Thetwo differed merely by a choice of representation of the invariant formulation set up byDirac, a quantum relativity transformation. This made it possible to transfer the Bohr-Heisenberg semantics to the Schrodinger mathematics.

It is a poor road that cannot be followed both ways. The Dirac dictionary made itpossible to translate the praxiological quantum theory of Born-Heisenberg in the ontologicallanguage of Schrodinger. One could speak bSchrodingerc and practice bBornc-bHeisenbergc,despite the serious mismatch between the two theories, by permitting as many non-standarduses of words as this requires. What Bohr-Heisenberg called a bprobability functionc, toindicate how it is measured and used, the ontologist must call a wave, and consider it tobe present in the individual experiment, and subject to changes by measurement. Thenormal switch in probability distributions when one transfers attention from one ensembleto another, and in particular, from input source to output sink in a quantum experiment,the ontologist must rename a “collapse” of this wave during measurement, and may thencriticize the spookiness thus created.

2.10.5 The von Neumann ambiguity

The ontological mispresentation already appears in the seminal work of von Neumann [78]side by side with the quantum logics of non-commuting projection operators. Von Neumannintroduced formal quantum logics closer in form to Boolean algebras than the matrixpraxics of Bohr-Heisenberg but also less practical. He also permitted some ontologismsto enter his formulations of quantum theory that have been canonized and stilll haunt ustoday, and are therefore discussed here.


The book of von Neumann is ambivalent about the semantics of the theory. In thefollowing excerpt U designates a probability operator, not a unitary one:

“We therefore have two fundamentally different types of intervention which canoccur in a system S or in an ensemble S1, . . . ,SN. First, the arbitrary changesby measurements which are given by the formula

(1.) U → U ′ =∞∑n=1

(Uφn, φn)P[φn]

(φ1, φ2, . . . a complete orthonormal set, see above). Second, the automaticchanges which occur with the passage of time. These are given by the formula

(2.) U → Ut = e−2πi~ tHUe

2πi~ tH

(H is the energy operator, t the time; H is independent of t). . . .

([78], page 351).The count “two” in the first line of this excerpt is inexact. There are three modes of

physical intervention in a quantum experiment, input 〈 i, throughput 〈 t〈 , and output o〈 .Each has its own large building in many particle laboratories, making it harder to missone today, and the accelerator itself, part of the input process, dominates them all. But allthree are present in quantum experiments since Newton’s prisms and polarizing crystals.The miscount suggests that the attention of the writer was on the mathematics, not thelaboratory. Evidently U itself was taken as a given, when actually it describes the firstintervention. In astronomy, to be sure, the input planet seems given. In the quantumdomain, however, input is as drastic an intervention as output, and U itself represents thatfirst intervention. In an accelerator laboratory, U includes the whole accelerator. If thereis an external target, U describes the beam extractor too. For colliding beams, U includesboth beams.

Moreover, although the author speaks of interventions in the system, he actually at-tends only to changes in U . U represents an ensemble, not an individual system understudy. In this quotation the author seems to take the probability operator U as the sys-tem, ignoring its statistical nature. This may be because the author seeks to deal withwhat is completely representable, and U is completely representable while the system isnot; but that is an ontologism.

The “or” reveals a significant ambivalence. One atom is physically very different fromN atoms and yet only one theory is presented, allegedly applying to both N = 1 andN = 1028. The “or” seems to mean that the difference between the system and theensemble is immaterial here. The author has elsewhere made it plain, however, that U isstatistical and concerns ensembles, although quantum determinations – interventions —are performed on each individual in the ensemble, one by one.


This inferred ontologism seems to recur. (1.) is said to describe the “changes bymeasurement” in S or S1, . . . , §N. But it what it gives is a change in U , not in S.It is a property of probabilistic ensembles that a change in one element does not affectthe ensemble distribution. Thus despite the preamble, (1.) cannot describe the changein S but only the change in the ensemble suggested by S1, . . . , §N. The changes in Sare described, incompletely but maximally, by the diagrammatic formula o〈 t〈 i giving theinput, throughput, and output process, and by stating whether this transition actuallyoccurred.

Besides this recurrent ambiguity there is a significant lapse in (1.) itself, undoubtedlypointed out many times. A measurement results in a value of what was measured, but U ′

is an unsorted ensemble of many different values, so the process producing U ′ is as mucha non- measurement as a measurement. Newton’s measurement of the physical colors ofphotons with a prism suffices as an example to illustrate the difference between (1.) anda measurement. There are two typical ways to use a prism to measure color that must bedistinguished, and neither is represented by (1.): bfiltrationc produces a beam of photonsof one color, as in a monochromator, and bsortationc produces a plurality of distinct beams,each with one beam color and one beam direction, with a 1-1 correlation between color anddirection, as in a spectrograph. A filtration is represented by a projector, and a sortationis conventionally described by a normal operator, a sum of orthogonal projectors, one foreach output channel, with numerical coefficients giving the results of measurement in thatchannel.

If the momentum of a photon is p and the velocity vector is v then the photon massis their ratio, p = mv. The photon mass m is the photon color: blue photons are abouttwice as heavy as red photons. The prism accepts a collimated beam with one v and manym’s, passes them through a glass that couples m and v, and delivers a beam in which mand v are correlated. The prism does not change the color of the beams that pass throughit, Newton showed, but it changes all their velocity vectors, and the velocity vectors serveto register the colors. If photon color is the system measured in this example, therefore,the direction of v is a registering variable in the metasystem, like a meter needle. No suchcorrelation exists in U ′; all the beams of different colors have been mixed. Measurementis supposed to increase information of the experimenter about the system; U ′ has lessinformation than U . The imprecise description (1.) should be replaced by two moreprecise descriptions, of filtration and sortation.

These are given elsewhere in the same work, however: a puzzling inconsistency untilone is given to understand that while the rest of the book was written by bvon Neumannc,the famous paragraph quoted was contributed by bWignerc [82].

It was suggested that the existence of “two fundamentally different types of interven-tion” is peculiar to quantum theory and problematical. In classical physics too, however,experiments have a distinct beginning, middle, and end as they do in quantum physics;and the input and output processes, in which the system is open, are not different in kindfrom the throughput process, in which the system is closed. Galileo drops two weights


in the input process, they fall during the throughput process, and they hit the groundloudly in the output process. The input and output acts are often left out of the theoryin classical physics because they often have no measurable effect on the system and cansafely be treated as purely mental acts with no physical consequences.

The Correspondence Principle points to important cases where measurement cannot beleft out of the theory. The classical correspondent of the statistical operator U is a classicalprobability distribution function U . To make the discussion comparable to Wigner’s, letus omit the input process that produces U in the classical case too. Suppose U = U(n)is a function of a variable n with integer values, as if the system were a digital computer,and that classical closed-system development proceeds in a way similar to (2.), conservingthe number of possibilities for the closed system. One typical measurement is a classicalfiltration for the value 0:

U(n)→ U ′(n) = U(0)δn0. (2.22)

The experiment starts with a probability distribution function U spread over many values ofn and ends with a distribution function concentrated on the value 0 with probability U(0),which may be 0. This process reduces the number of possibilities for the system, unlessU happened to be concentrated on the value n = 0. Therefore closed-system developmentcannot carry U into U ′, and a filtration process is required separate from the dynamicaldevelopment, classically as well as quantally.

Perhaps the misimpression that measurement does not need separate mention in clas-sical physics has to do with the misimpression that (1.) represents a measurement; for theclassical correspondent of (1.) would be the identitity mapping U ′ = U .

The two processes (1.) and (2.) — more accurately, the input-output processes andthe throughput process — differ in that (1.) operates on an open system and (2.) on anclosed system. Some accept the mispresentation of collapsing wave functions and attemptto account for the alleged collapse with mental activity. This is wide of the mark. Thereis no collapse phenomenon to explain, there are merely two vectors for two ensembles,one for input and one for output. The throughput process occurs in the closed system,after input and before output, and conserves the number of possibilities. The input andoutput process are informational and change the number of possibilities for the system.Compensating changes must occur elsewhere, in both classical and quantum physics. Themajor difference between the classical and quantum praxics is the emergence of filtrationcommutativity in the classical limit. This makes it seem that classical experiments cangive complete information about the system, including the result of subsequent filtrationsand sortations. Actually the classical filters can appear commutative only because theyare far from maximally informative.

The author invokes a meta-experimenter II who can regard the input-output processesof I on the system S as parts of one throughput process on the combined system I × S.For II the distinction can disappear between processes 1, 2, and 3 can disappear. But thismerely shifts the distinction up one stratum. II still distinguishes between the throughput


and input-output processes of II. Here the author commits the grave error of imaginingthat the unobserved acts of I are somehow the same as the acts that I can carry out undermaximal determination by II, which is generally lethal.

This mis conception that acts of knowledge are merely mental and not physical recurswhen von Neumann speaks of the bcutc between metasystem and system, here called thesystem interface. The cut is a physical information valve, open during input and output,but closed during throughput, when it acts as an adiabatic wall. A single uncontrolledquantum passing through the system interface during throughput can destroy quantuminterference. For example, the interface for some systems might be a superconducting leadvacuum chamber with mirrored walls close to absolute zero, while for the spin of a photonin flight, clean air will suffice to maintain the isolation of the photon spin for some metersof flight path. Von Neumann speaks of placing the cut between the retina and the brain,so that the eye becomes part of the system while the brain remains in the metasystemobserving the eye; as though such an operation were a mere change in point of view andnormal eye function could proceed undisturbed. The problem is not merely that this is adifficult operation, but that it must interfere with the function of the eye in order to carryout its function. To measure the eye maximally is to freeze it.

Much the same error arises when von Neumann assumes that the observer has a stateand that “this state is completely known” ([78], page 439). The observer would then bein the cold state of Schrodinger’s cat, discussed in §2.10.6. The physical process that the“state” represents has again been left out of the picture, as it is in classical physics. Thisfrequent omission is good reason for discontinuing this misuse of the classical term “state”in quantum theory. Safer quantum state constructs are suggested in §S:STATES.

bWignerc [83] called his ontological mispresentation of the quantum theory, with itscollapsing wave functions, the “orthodox” quantum theory, and it is often called the“bCopenhagen quantum theoryc”, despite some sharp conflicts of formulation between itand the canonical theory of Heisenberg and Bohr. In view of this confusion, both terms“orthodox” and “Copenhagen” theory are avoided here.

The ontological mispresentation made it possible to present the mathematics of thequantum strategy without dealing with its more challenging conceptual revisions in philos-ophy, probability theory, and predicate algebra, as in the above quotation. Nevertheless,it misuses ordinary physics language and creates diversions from the actual problems ofquantum theory. It is widely written, taught, and discussed today, at the same time thatthe praxiological theory is widely used.

The ontological mispresentation claims to describe what “is”, not what goes on.In an experiment where a photon is emitted, flies along an optical bench, and is counted

or not, the ontological mispresentation says that in the real world a wave function prop-agates down the optical bench to the analyzer, and there “collapses” to another wavefunction or not. This is understandably considered to be problematical, even by the ontol-ogist who introduced it. It is a position like solipsism, invented to attribute to others. Aphoton that we find in raw nature does not come with a sharp ensemble of like photons.


To speak of “its vector” is therefore operationally meaningless in principle and leads toconfusion in practice. The system does not carry a vector as much as a vector carriessystems. The vector represents a process, a system is one of its products. The term “thevector of the system” or “wave function of the system” confuses a process with its product,and is a reversion to the early wave theories of de Broglie and bSchrodingerc, in which thestate of the hypothetical wave system was indeed a wave function. Today speaking of the“bcollapsec of the wave function” is a fortiori an ontologism.

Despite the well-known uncertainty of quantum prediction, an ontologist might stillassert that the dynamical law of quantum theory determines the future behavior of thesystem from the initial conditions [85]. He knows perfectly well that usually the orientationsof two given polarizing filters do not determine whether a photon from one will pass throughthe other, so it must be the development of “the state of the system” that is being describedas deterministic, not the history of the photon, and the ontologist must again be identifying“the state of the system” with some wave function, as if the system were a wave. Thedevelopment of any statistical distribution is “deterministic” in this mathematical sense,even in classical statistical mechanics, where the development of the individual system isnot determined.

This common ontological mispresentation of a praxiological theory, not the transitionfrom ontology to praxiology, is what causes many to suppose that quantum kinematicscannot be taken seriously even though it works. Here it is taken seriously, which meanswithout ontological post-editing. To be sure, there are still troublesome infinities in quan-tum theory, but they arise from classical vestiges in the theory, were worse in the classicaltheory, and disappear altogether in fully quantum theories.

In both classical and quantum physics an ideal filtration for a property of a systemdoes not change that property. This is expressed by representing the filtration process bya projector, obeying P = P 2. This idempotency was one of the first facts that Newtonverified experimentally for both refracting prisms and polarizing filters, exactly to convincehimself that they did not change the color or polarization of photons but merely filteredor sorted photons by these properties.

In the classical case, a sharp measurement operation is represented by a diagonalmatrix P of 0’s and one 1. But the ontological mispresentation postulates that what thequantum theory calls a measurement P is actually a “collapse” process C : ψ 7→ ψ0. Cclearly changes what the ontological mispresentation treats as the system, that is, “thewave function”. Whatever the wave function ψ was said to be before the measurement,if the system passes the test, “the wave function of the system” is said to become theeigenfunction ψ0 associated with the test.

Such a collapse C is certainly not a measurement of ψ in the terms of the ontologicalmispresentation itself. A measurement of ψ would not change ψ. To be sure C : ψ → ψ0

is idempotent, C2 = C. But it lacks the symmetry required of both classical and quantummeasurements. Since the alleged collapse C is a many-to-one relation, its dual C† is a one-to-many relation, and C 6= C†. To measure the ontologist’s “wave function” would require


a filtration process that passes a certain wave function ψ0 with certainty and rejects anyother wave function with certainty. This process would violate the superposition principleof quantum theory. What the ontologist could call a true measurement is impossibleaccording to quantum theory.

Since the ontologist claims that the system is a wave function, and that quantummeasurements change it, he or she must claim that what quantum experimenters callmeasurement is not truly measurement. Thus the ontologist’s mispresentation violatesStandard Semantics for the term “measurement”.

Nevertheless in practice even ontologists use the two vectors o〈 and 〈 i of every experi-ment to compute the expected mean count just as the quantum theory does: as probabilityamplitude distributions, not as physical waves. And they find these vectors as the quan-tum theorist does: not ontologically, by measuring a wave function, point by point, butpraxiologically, from an inspection of the measurement action that the experiment carriesout on photons in general during input and output, repeated many times.

Thus the ontologists speak ontology but practice praxiology. They speak of an imagi-nary random system I [V ], of which ψ0 is one value; but in practice they use vectors o〈 , 〈 ito represent processes carried out on the actual quantum system I[V ], the photon, just likequantum theorists. By criticizing the ontological mispresentation of the canonical quan-tum theory, ontologists uphold the standards of physics. But a mispresentation is not goodreason to modify the quantum theory; there are better reasons for that.

Roughly speaking, the quantum formulation approaches a classical one as ~ → 0;this refers to the Correspondence Principle of Bohr. While Segal [66] examined both thesingular limits c → 0 and ~ → 0 as examples of Lie algebra homotopy, the study of thismathematical process by bInonuc and bWignerc [47] significantly ignored the limit ~ → 0.By putting c and ~ on the same footing we explicitly accept that both relativity andquantum theory might be irreversible advances in physics.

Some, however, expect that the quantum non-commutativity of filtrations will disap-pear and classical commutativity will re-emerge, for example when a suitable theory ofconsciousness is found. This leads to two conflicting teams in theoretical physics: a prax-iological one working to eliminate residual commutativity from the canonical quantumtheory, and an ontological one working to restore classical commutativity. Here we join thepraxiologists.

In the canonical quantum theory, as opposed to the ontological mispresentation, aquantum system carries no vector, any more than the planet Mars carries a probabilitydistribution for the planet Mars. There is no vector present in the system to evolve orcollapse during the experiment, what evolves is the system.

The temptation arises to replace the system by “the” wave function because each wavefunction is a completely describable mathematical object, and the system is not. As longas we work with wave-functions they make no demands on us to change our logic, and theyremain unchanged by our mathematical study of them.

The quantum system is not completely describable and obeys a non-commutative


praxic instead of a commutative logic. To accept quantum theory requires us to reviseour logic and probability theory.

Many experience an urge to explain “why” we add probability amplitudes instead ofprobabilities in quantum physics. This too is a symptom of bontologismc. People didnot explain the addition of probabilities back in the days of classical mechanics. Theypostulated it, presumably because it worked. Therefore, by correspondence, the additionof probability amplitudes in quantum theory must not be explained but postulated, becauseit works better .

The laws of probability and logic are so basic that they defy deduction from somethingmore basic, since the deduction itself would use them, but must be gleaned inductively fromexperience. If adding vectors — as in quantum interference — seems to require deduction,as opposed to induction from experiment, then one is attached to the classical laws ofprobability, and has not replaced them by the quantum ones.

On the other hand, if our praxic includes vector addition, it is legitimate to ask howbclassical logicc emerges as an approximation. But this is clear. The difference betweenthe classical and quantum kinematics is commutativity of the operator variables. Classicalcommutativity emerges from quantum non-commutativity through the a bLaw of LargeNumbersc: If two quantities q and p are sums q =

∑qi, p =

∑pi of N independent terms

from N independent systems, then in the limit N → ∞, the commutator [q, p] qp+ pqbecomes negligible compared to the product. Briefly put, if an experiment involves actionS, an associated action quantum number S/~ becomes large as ~ → 0, and the bLaw ofLarge Numbersc takes over.

Non-commutativity, however, is structurally stable, and can be verified with three po-larizing filters, which also demonstrate the role of probability, granted the existence ofphotons. Quantum theories are innately stochastic as opposed to complete. The “deter-minism” of the Schrodinger equation resembles that of the Liouville equation, which alsodescribes the time development in a stochastic theory.

Non-stochastic mispresentations for the quantum theory abound. The ontological mis-presentation reifies “the” vector and supposes it to be carried by the quantum, like theclassical state. Then one accounts for the vector either by assuming two such objects, orby having the first “collapse” into the second, usually — for no good reason — during theoutput process rather than the input process.

Such a belief in a “collapse” in the absence of experimental evidence is another symp-tom of bontologismc. In the Copenhagen theory, one probability-amplitude vector rep-resents a certain ideal input process, another an ideal output process, in the way thatprobability distributions do in classical theory. They are needed to describe the two ar-bitrary choices, the input and output process, that we make in setting up an experiment.One does not develop into the other as the system develops. Nothing that could be called a“collapse”, with all its ontological implications, occurs in the experiment or in the canonicalquantum theory.

Classical probability theory also has a probability distribution for both input and


output processes, and few imagine that one collapses faster than light into the other. Aconsistent replacement of probabilities by probability amplitudes on the dynamical resultsin the quantum theory of Heisenberg and Bohr, equally free of “collapse”, and calledcanonical here. “Collapse” is an artifact of the ontological mispresentation. Attempts toeliminate a collapse that does not exist are symptoms of ontologism.

For example, there is a bMany World Theoryc of bEverettc whose mispresentationterms a “world” what is merely a possibility in classical and quantum physics. This toois a reaction to the contradictions of the ontological mispresentation, not to the canonicalquantum theory, which Everett may never have encountered, since it is rarely taught.Since the bMany World Theoryc attempts to work with one vector when there are twoindependent mutually dual processes, and attempts to avoid mentioning the experimentinterface, it is yet another ontological mispresentation. Its “vector of the universe” isthe state of the universe, regarded as an absolute ontological being. Since no one inputsor outputs the universe sharply from outside it, there is no physical need for a universevector. If the Many World Theory theory were taken literally it could make no experimentalpredictions. Since the “vector of the universe” is taken as the system state, it cannot be aprobability amplitude, which is not a system state. It is then inconsistent to use the “vectorof the universe” to compute probabilities in the Many World Theory. Its exponents actuallycontinue to use the canonical quantum theory in practice but describe their practice usinga non-standard semantics.

2.10.6 Schrodinger’s frozen cat

To indicate what he felt was an absurdity in the quantum theory, bSchrodingerc proposeda famous thought-experiment: A cat is caged with a lethal device that may be triggeredby the decay of a radioactive nucleus. After a nuclear half-life, according to Schrodinger,“the wave function of the cat” would be in a quantum superposition of alive and deadstates. This is presented as an affront to common sense, which tells us that a cat is eitheralive or dead and not in limbo waiting to be observed. Actually it illustrates two basicmisunderstandings by the narrator:

In the first place bSchrodingerc spoke of “the wave function of the cat”, as thoughevery cat had one. This is an bontologismc, consistent with his original approach to theatomic spectrum problem. If the cat were a classical wave it would always have a state,and the state would indeed be a wave function. Actually the cat is composed of quanta,and what would a cat vector would actually describe is a coherent source of well-isolatedcats. No comon-sense cat ever came from such a source. To be associated with an exactvector each cat in the emitted cat beam must be insulated from interaction with any otherquantum and have entropy S = 0. Therefore the cat beam has absolute temperature T = 0.Its cats are frozen, not alive; unless some experimenter can produce an isolated live catreversibly from its chemical elements at an absolute zero temperature. The bSchrodingercatc is therefore not the cat of common-sense every-day life as he suggests but a cryological


fantasy that will presumably never come true.Incidentally, the bAharanov limitc on coherent superposition, set by vacuum fluctu-

ations in the gravitational field, requires anything that exhibits quantum interference tohave mass much less than 10 micrograms, unless the preparation of the cat is accompaniedby a special maximal preparation of the vacuum gravitational field, which is another fan-tasy that will presumably not be actualized. It would have been surprising if Schrodingerhad known this limit to coherence.

Similarly, the cat must be electrically neutral; a charge of 137 electrons would seriouslydamage the proposed interference. Superpositions of macroscopically distinct vectors aretaken for granted today for quantum computers of much less than 10 microgram mass andless than 137 e charge, and are called “Schroedinger states” in his honor. They are indeedcold.

In the second but more fundamental place, even if a cat is described by such a super-position vector, this does not mean that the cat is neither dead nor alive, or that it is both.Assuming as Schrdinger did that these predicates correspond to projection operators L(for living) and 6 L (for dead), even in quantum theory L∪ 6 L = 1 is true, the cat is livingor dead, and L∩ 6 L = 0 is false, the cat is not both living and dead. Every time we do theexperiment the quantum theory predicts that we find a live cat or a dead one, just as inreal life. The coherent superposition does not make a definite prediction about L but givesprobabilities for L and for 6 L. An indefinite prediction, however, is not a prediction ofindefiniteness. The probabilities provided by quantum physics are probabilities for definiteoutcomes. The quantum theory predicts that when we measure L for a cat from such asource, we will always find one of L or 6 L, never a fuzzy cat. The flaw that Schrodingerreveals is not in the quantum theory but in the ontological status he gave to wave functions.Heisenberg called vectors “probability functions” to forestall such ontologisms.

The question is not a matter of convention or philosophy; it is a question of how onedetermines wave functions in actuality. Determinations of wave functions of the systemunder study are made through the Malus Relation for transition probabilities, not bymeasuring a wave amplitude.

Normal practice is to reserve such ontological mispresentations of quantum theory forpublic or classroom occasions, not for doing experiments, not even thought experiments,where it is a hindrance. In this thought-experiment, for example, Schrodinger does not tellus how to determine the entity he describes as “the wave function of the cat”.

If one were given a living cat rather than a frozen one and were required to find“its wave function”, much interpretation of this request would be required. It might beinterpreted as a request to determine the values of a maximal set of observables of thecat. If the observables are chosen to be the positions of the particles in the cat, the resultof such a determination would be an explosion of nuclear magnitude, by the uncertaintyrelation Most choices of complete sets of observables are similarly catastrophic.

Schrodinger’s discussion shows that at least as of that writing he had not yet acceptedthe praxiological nature of vectors and the responsive nature of quanta but regarded “the


wave function” as a physical property of the cat, like the state of a classical object, givingit ontological status.

bWignerc elaborated bSchrodinger’s catc to what has become known as bWigner’sfriendc, who can be asked about what happened in the experimental chamber before themeasurement. The treatment compounds both of Schrodinger’s errors. Again it is sup-posed that a living creature can be assigned a vector without disrupting its life processes; asthough the friend actually carried a vector with his identity papers, and we merely had toread it. Again it is supposed that an indefinite prediction is a prediction of indefiniteness.

2.10.7 Fundamental law

Quantum dynamics assigns probabilities to individual experiments, and in exceptional butimportant special cases the probability is 0 or 1. If quantum dynamics does not causediscomfort in a physicist then it likely has not been understood. Quantum dynamics doesnot fulfill the Laplacian dream of a fundamental law. For most experiments, no matter howideal, it fails to predict the outcome except statistically and with large dispersion. Fur-thermore a theoretical probability statement relates what happens in an actual experimentto experiments of the future that may or may not be performed. Quantum dynamics givesan operator 〈T 〈 that relates an early input action 〈 to a later one 〈T 〈 i, earlier probabilityto later probability. It never tells us if the experiment will actually be carried out, andit rarely tells us what will actually happen if it is. Therefore quantum theory does notcontrol the future in the way that Laplace demanded of the law of nature. But clearlyLaplace too could not actually predict such human phenomena, and his idea of a funda-mental dynamical law that could was a great extrapolation beyond experimental evidence,and has failed, while the Malus Principle continues to work for us today without a viablecompetitor.

Empirically, these conceptual evolutions seems easier for younger physicists than old,and some physicists regress to the classical metaphysics after productive years of quantumpractice, with no experimental support for this decision. Two strategies can be suggestedfor reducing this conceptual dissonance between the quantum dynamics and the classicalideal of dynamics.

First one might accept that the quantum theory does not deal with certainties alone.When we enter the quantum world we leave behind the ideal of fundamental universal law.This explains much of the pain that Bohr mentioned. Moreover, a quantum dynamics is notabsolute but contingent on the metasystem, being disrupted by measurements. This makesit necessary to weaken the concept of fundamental symmetry too; if the dynamical law iscontingent, then likely its symmetry is too. Both the dynamical law and its symmetriesare contingent on the metasystem.

Second, one might note that the classical principle of deterministic dynamical lawemerges as singular limit of the Malus Principle and quantum dynamics or approximationsthereof. This anticipates the many-quantum logics of the next section. When o〉 and 〈 i are


normalized and o〉i ≤ 1, then the many-quantum vectors (o〉)n and (〈 i)n for a sequence ofn trials of the experiment o〈 i are nearly orthogonal, and transmission is nearly forbidden.By taking n large enough this can be made as close to the deterministic case of probability0 as desired. In this way one can deduce the full Malus Principle from the following WeakMalus Principle:

For a given number of trials N , when |o < i| 1 is sufficiently small, the transitiono < i almost certainly does not occur.

This reduces the case of general probability to the case of arbitrarily small probability.The dynamical law of canonical quantum theory deals with vectors that depend on

time. This goes outside the first-order language of linear quantum logics, since it uses theconcept of function. It is taken up after the appropriate foundation is laid.


Chapter 3

Polynomial quantum logic

which deals with pluralities of individuals.The system of the classical random set is discussed next (§3.1.1) to guide the discussion

of the queue that follows (§3.1.2).The terms bclassc and bpredicatec are reserved here for collections of possibilities for

one system, represented by projectors in a space of monadics. The terms bsetc and bqueuecare reserved here for a classical or quantum system composed of lower-stratum systems.

3.1 Set algebras

3.1.1 The random set

Consider first the states of a random classical object, supposed to form a semigroup Sof random sets. A space S of statistical distributions is introduced here that is in thefirst place a polynomial algebra. Its + is statistical mixing. Its × is direct multiplication,combining descriptions of n1 and n2 distinct bodies into a description of n1 + n2 bodies.These two operations suffice to express the most general statistical description of a randomset of individuals as a polynomial in states of the individual.

For example, if a and b are states of an individual, ab is a pair state, while a + abstatistically describes an object that is either in state a or ab with equal likelihood.

The number of factors in a monomial in S is called its bgradec g and is indicated byterms like (bmonadc, bdyadc, . . ., bp-adc, . . .), or bpolyadc, or by a subscript g. The 0-ador cenad, the product of no factors, is 1, representing the empty set. The dimensionalityof a subspace of S is called its bmultiplicityc and is also expressed by terms like (bsingletc,bdoubletc, . . ., bm-tupletc). Thus the suffix -et indicates a count of terms, and the suffix -adindicates a count of factors. The 0-plet, the sum of no terms, is 0 and represents nothing,certainly not the empty set, which 1 represents. The operation × distributes over additionbecause of the statistical meaning of addition.

143

144 CHAPTER 3. POLYNOMIAL QUANTUM LOGIC

To avoid multiple occupancy of a state, one posits that individual states obey a2 =1 and commute. Lacking these postulates the polynomial algebra is that of a randomsequence; with them, a random set.

This algebraic language may not seem natural from the viewpoint of standard classicallogics. It is a back-formation from quantum algebras in present-day use.

To generate nontrivial objects or states themselves takes at least another operation.One monad-forming operation I of bbracingcwill do. For any monomial state p, of any grade,Ip = p is the state of a monad, different from p, whose sole element is p. Polynomialsand bracing suffice to generate all finite abstract classical sets; the empty set, to start with,being the product of no factors.

For example, a field state f(x) is a set of field event states fx, each constructed bymultiplying a field value state f by a space-time event state x and bracing the productf × x.

The random sets with no more than L nested braces form bstratumc L of S, designatedby S[L]. Monomials may be classified by grade g and rank r.

I is assumed to be additive because of the statistical meaning of addition: If a set iseither a or b with equal probability, then its brace is either a or b with equal probability.The infinite-dimensional algebra of random sets generated by summation, multiplication,and bbracingc is designated here by bSc. The subspace of S consisting of homogeneouspolynomials of grade g is then Gradeg S. Elements of S with no more than L nested bracesform its stratum L, written as bS[L]c. In general, restriction of any construct to stratumL is shown by a superscript L. Numerical properties of a random set can be representedby elements of the dual space SD, maps v : S → R, the value v(p) being the expectationvalue of v on p.

Thus I converts any polyadic set s of one stratum into a monadic set Is of the next:

I : S[L] → Grade1 SL+1 ⊂ SL+1. (3.1)

In the present terminology, classical blinear logicsc represent random individuals usingthe linear operations +, ×R. Classical bpolynomial logicsc adjoin the operation × torepresent random first-order sets. Classical bexponential logicsc further adjoin I to representrandom sets of all finite orders.

The multiplicity of each stratum in the exponential logic is exponentially larger thanthe previous, so stratum multiplicity grows superexponentially with stratum number L:

MultSL+1 = 2MultS[L]=: 2L+10. (3.2)

3.1.2 The queue

The above classical logics have fully quantum correspondents. Canonical quantum theoryrevised the coefficient system and the two operations + and ×. The complete description by

3.1. SET ALGEBRAS 145

basis vectors in the algebra S turned into merely sharp but incomplete statistical descrip-tions by general vectors in the same algebra S. A bvectorc represents a sharp input-outputoperation, like a source or counter. Positive real probabilities became complex transitionprobability amplitudes. Real polynomial addition a+ b representing mixture became com-plex polynomial addition α + β, representing quantum superposition. The commutativelaw ab = ba split into an anticommutative one αβ = −βα for fermions and a commutativeone αβ = βα for bosons. The unipotency law a2 = 1 survived as an exclusion principleα2 = ±1 for fermions but was dropped for bosons.

The real algebra S of statistical descriptions becomes an algebra S, here taken to bereal. Thus the vectors in S do not correspond to states of a classical system, which do notadd, but to statistical distributions, which do.§2.3 gives further principled reasons for replacing the common term “state vector” by

“vector” in this work. One major difference between the classical and quantum theories isthat the classical vector space S has a preferred set of basic rays S representing completedescriptions, states, while no vector in S gives a complete description of what actuallyhappens, and there is no absolute preferred basis. There are no states in S; they are foundelsewhere for quantum systems.

Random set addition + and multiplication × are modified by the quantum evolution sothat it is not possible for the brace I that they support to survive unchanged. To describehow I enters the quantum era it is helpful to review how + and × did, returning to I in§1.2.1.

3.1.3 Input-output processes

Modern quantum theory began with a finitization, Planck’s prescription for regularizinga singular integral, one that that gave a cavity an infinite heat capacity. Since finitude isstill far off it can continue to serve as our pole star.

Yet effectively infinite groups of actions like translation and rotation occur everywhereand must be represented in our theories.

Classically, the world cannot be both finite and round. Any circle has an infinity ofpoints, except the trivial one of radius 0. One can turn a body through an infinity ofangles. To realize such a continuous group faithfully a set of classical states must haveinfinite grade. But then it also supports unbounded variables. Classically, a potentialinfinity requires an actual infinity.

Quantum theory passed between the horns of this dilemma. Queues of finite grade canstill realize a continuous group. A finite quantum world can be round. For example thevector space of multiplicity 2, describing a quantum with two independent input modes,supports the defining representation of the infinite group SO(2R). Queues do not transformas random sets do, a different kind of relativity comes into play, the bDiracc btransformationtheoryc, allowing a finitude of vectors to have an infinitude of frames.

Instead of building from states, quantum theories build from ideal binputc and boutputc


processes of maximal resolution, such as sources and counters, collectively called bioc pro-cesses. An input process accepts a signal and puts a system with specified properties intothe experiment chamber; an output process takes a system with specified properties outof the experimental chamber and puts out a signal. Unlike classical theories, quantumtheories do not predict the result of every such experiment, but give the transition proba-bility, the expected value of the ratio of output counts to input counts. Quantum theoriesrepresent input-output processes statistically by vectors in mutually dual vector spaces ofthe system, input and output, provided with a non-singular bprobability formc, that mapseach vector into the other partner of an assured transition.

The prototypes are the input and output actions of the Malus experiment (1805) withtwo linear polarizers. There the vectors in question can be etched on the polarizer andanalyzer filters, and the probability form transforms any input action into the output actionthat uses the same polarizer as an analyzer.

States constitute a preferred basis for S. S has no such preferred basis, until one isetched on the filter by arbitrary choice. The classical construct of state is relativized bythe quantum theory.

If the transmission probability is 0, as for orthogonal polarizing axes, the transition iscalled bforbiddenc; if it is 1, as for parallel, bassuredc. For orthogonal or parallel vectors theMalus Principle gives a transition probability 0 or 1, as does a deterministic law. Grantedthe existence of the bilinear space V , with definite probability form, V is completely definedas vector space by its forbidden transitions and then as a bilinear space by its compulsorytransitions. This result of the 19th century culminated centuries of study of bprojectivegeometryc. It is presented more fully elsewhere [40].

Quantum theories represent bthroughputc processes including symmetry transforma-tions, by square matrices, which accept an input vector on their right and an output oneon their left. An operator defines a matrix in each reference frame and is defined by anyone of them, so we need not pay much attention to the distinction. A matrix is a morphismof a vector space, and so is represented by a labeled arrow. The arrows

A→ ≡ A← ≡ 〈A〈 ≡ 〉A〉 (3.3)

all represent the same matrix A, the arrows define their own directions.An input and an output vector combine into an arrow representing a throughput

operation. Therefore they are best represented by semi-arrows like 〈 i and o〈 . Thesecombine to form the throughput arrow 〈 i⊗ o〈 .

As matrices, an arrow is square, an input vector is a column, and an output vector isa row.

The widely used symbols |i〉 (kets) and 〈o| ( bras) for for input and output vectorsclash with the usual arrow notation for mappings or operators and are not used here.

Let V D designate the bdual spacec to V . The Statistor Principle implies that in a realquantum theory there is a fixed symmetric bilinear form 〉H〈 : V ↔ V D, the bprobability

3.2. CLIFFORD ALGEBRA 147

formc, such that when i〉H〈 i > 0 the transition i〉H〈 i from input 〈 i to output i〈 := i〉H〈 isassured. In complex quantum theory the form 〉H〈 is Hermitian symmetric and sesquilinear.Then 〈 i and i〈 can be regarded as input-output processes of opposite polarity for likequanta. When o〉H〈 i = 0, 〈o and 〈 i have zero probability.

A positive definite probability form is a real Hilbert-space metric form in its mathe-matical properties; its physical interpretation, is crucial for the development.

3.2 Clifford algebra

The linear operators on a Grassmann algebra form a special case of Clifford algebra, Con-cepts of Clifford algebra are summarized next without reference to the quantum interpre-tation.

A Clifford algebra C is an associative distributive linear algebra consisting of allpolynomials — that is, finite sums of finite products — in given variables, with real scalarcoefficients, subject to the Clifford Clause:

The square of a vector is a scalar.

The real numbers then form a central subalgebra R = C0 ⊂ C. The linear combinationsof variables are called the bvectorsc of C and form a real vector space C1 ⊂ C.

0 is the monomial with no factors; 1, the polynomial with no terms.The Clifford ring over a an arbitrary ring of scalars and module of vectors is an obvious

generalization that is sometimes useful. It includes binary, integral, real, and complexClifford algebras, with scalars 2, Z, R, and C, respectively .

The Clifford Clause defines a bilinear form on the vectors C1 in terms of the Cliffordproduct:

u · v =uv + vu

2, ‖v‖ = v · v = v2. (3.4)

The linear transformations C1 → C1 preserving this bClifford formc and having determinant1 make up a special orthogonal group SO(C1). If the Clifford form u · v is bregularc theClifford algebra is called regular.

A Grassmann algebra is a most singular Clifford algebra, with v2 ≡ 0 for vectorsv ∈ C1. The Clifford algebra of queues constructed here will have a regular bilinear form.

Every Clifford algebra has a natural Grassmann algebra structure as well, with Grass-mann product u ∨ v defined by setting

∀u, v ∈ C1 : uv = u ∨ v + u · v. (3.5)

One then defines a Grassmann derivative Dv : C → C with respect to a vector v relativeto a given vector basis B ⊂ C1 as usual; and a basis-independent bgrade operatorcbGradec:C → C that acts on Grassmann polynomials P (v) ∈ C by

Grade =: sumv∈BvDv (3.6)


Eigenpolynomials of Grade .= g are said to have bgradec g and are called bg-adicsc.They form a subspace of C designated by Gradeg C. Those of odd (or even) grade formthe subspace C− (or C+) and are called boddc (or bevenc, respectively). C+ is a subalgebraof C.

3.2.1 Clifford semantics

Every Clifford algebra is isomorphic to a matrix algebra over some vector space, called aspinor space for the Clifford algebra.. Therefore a given Clifford algebra has at least twoconceivable quantum interpretations, besides all the usual classical ones. Being a matrixalgebra, it can be provided with an adjoint operation H and used as the kinematical algebraof a quantum system, to represent throughput actions, Being a vector space, it can beprovided with a probability form H and used as the vector space of a quantum system, torepresent input-output actions. Both interpretations occur in standard quantum physics:

The bDirac algebrac is the Clifford algebra over a Minkowski tangent space, alge-braically isomorphic to Alg(4R), used as the kinematical algebra of a spin 1/2.

The Clifford algebra over V ⊕ V D is used as vector space for a Fermi catenation ofreplicas of the individual system I[V ], and is then called the Fermi (-Dirac) vector algebra(over the individual quantum system I[V ]).

Any element of a Clifford algebra Cliff(nR) can be interpreted as a vector of a queuecell on n quantum vertices (§4.1.3).

The Clifford algebra of a Fermi system is exhibited in more detail next.

3.2.2 Fermi Clifford algebra

A classical bsetc can contain an element 0 or 1 times only and is unchanged by exchangingany two of its elements. bFermi(-Dirac) statisticsc defines a quantum assembly that alsohas these properties. The Fermi relations (1.35) define a bClifford algebrac over the duplexspace W = V ⊕ V D with neutral duplex norm.

In quantum logics it is ordinarily pointless to form the direct sum of an input and anoutput vector space like V ⊕ V D. Such a sum assumes a coherent phase relation betweeninput and output vectors by allowing their quantum superposition. This assumption is du-bious because each input vector refers to a prior interaction with a macroscopic source andeach output vector refers to a later interaction with a macroscopic sink. Such macroscopicinteractions destroy coherent phase relations.

Yet a rather similar superposition is commonplace in canonical quantum quantification(§3.3). Quantification converts any one-quantum input vector 〈 i into an input operator〈 i〈 , and any one-quantum output vector o〈 into an output operator 〈o〈 , both acting onthe many-body vectors. These operators are freely added even though they are linearlyrelated to a vector and a dual vector, which ordinarily cannot be added.

3.2. CLIFFORD ALGEBRA 149

It seems that I and quantification shift the system interface to enlarge the system. Actspreviously regarded as input-output across the interface are now represented on the systemside of the interface. Instead of saying that an electron is input from the metasystem, afterquantification one can say that the electron is produced by a certain beta decay within thesystem.

This does not eliminate the input but enlarges it. It removes the electron input butreplaces it, for example, with the pregnant neutron that gives birth later to the electron andits sibling. Maximal determination of the macroscopic external source is still inconceivable,but now a microscopic internal source, the neutron, can be maximally determined.

3.2.3 Fermi vectors

Queues, as correspondents of classical sets, are represented here vector spaces that areGrassmann algebras. Their operator algebras are Clifford algebras over the duplex spaces.Evidently we come closer to the standard quantum physics if we use Clifford algebras asoperator algebras rather than as vector spaces.

To make a kinematical algebra A like Dirac’s spin algebra from a Grassmann vectorspace G = 2V = GrassV , one merely “squares” G, forming A = G⊗GD.

Any vector s ∈ V defines linear operators of bleft-multiplicationc Ls, right-multiplicationRs, and commutation 4s = (L− R)s on G→ G.

Commutation 4 defines a representation of the commutator Lie algebra of Grade2C,called the bcommutator representationc (“regular representation”, “adjoint representa-tion”).

4[s, s′] = [4s,4s′] (3.7)

is a form of the bJacobi identityc.

The vector space V [F] of stratum F requires a probability form for quantum use. If aprobability form H is given on V then it defines one on 2V in a well-known way, designatedagain by H when context prevents confusion.

There is a natural norm on R: ‖r‖ = r2. This can be propagated up the ladder ofstrata by imposing the canonical relation (1.15) and supposing that the norm of a productof orthogonal factors is the product of their norms. The resulting norm will be called thebmultiplicative normc. It is positive definite, making Lorentz invariance impossible. Itseems that two indefinite forms are required: H on V [F] for the probability form, and g onV [F]⊗V [F]D for the causality form. If they are not to clash, then they must determine oneanother on some stratum.

If v, w are vectors then the Clifford, Grassmann, and inner products are related by

vw = v ∨ w + v · w. (3.8)


3.2.4 Grade operator

The Grassmann grade g defines a symmetric operator Grade : C → C with respect to themean square form. Grade : C → C is the linear operator that when c is the Grassmannproduct of g vectors, obeys

Grade c = gc. (3.9)

Such a c is called homogeneous of grade g, or g-adic. Clifford elements of different gradeare orthogonal with respect to the mean square form.

Grade has a spectral resolution Grade =∑∞

n=0 nGraden with eigenvalues n and associ-ated eigenprojections Graden. The operator Graden takes the grade-n part of its operand.Grade0 c is called the bscalarc or cenadic part of c; Grade1c is the vector or monadic partof c; Gradenc is the nth-grade or n-adic part of c.

3.2.5 Mean-square form

Just as any Lie algebra has a natural bilinear form, its Killing form, any (finite-dimensional)linear algebra A has natural bilinear forms invariant under inner automorphisms of A. Inthe case of a unital algebra A, a unique invariant form is defined for all α, α′ ∈ A by

αHα′ :=Trαα′

Tr 1. (3.10)

Its normalization was fixed so that the norm ‖v‖ := vHv on a Clifford algebra agrees up toa constant factor with the Clifford square Q(v) = v2 for first-grade elements. This will becalled the bmean-square formc. It is designated by H, or 〉H〈 , because when H is definiteit is a Hilbert space metric form. When the elements of A are used as vectors 〈α, themean square form will be interpreted as a relative transition probability amplitude like theHilbert space form.

The symbol 〉H〈 accepts two vectors and delivers a real number.A vector 〈v = 〈α ·vα ∈ V is represented isomorphically in the Clifford algebra C = 2V

by the element v = sαvα ∈ C. Similarly a linear operator 〈q〈 = 〈β qβα α〈 : V → V is

represented isomorphically by the element

q = sβ qβα s†α ∈ AlgC = C ⊗ CD. (3.11)

where s†α ∈ CD is the dual vector to sα with respect to the mean-square norm (1.28).A metric form on a Clifford algebra cannot be represented within the Clifford algebra

but it can be represented within the algebra of linear operators on the Clifford algebra. Letγα := Lsα stand for left multiplication by the basis element sα ∈ C, and let the invertedcomma ‘ stand for right multiplication: γα‘ = Rsα. Let us lower and raise the index α withthe bmean square formc Hβα and its inverse.

3.3. QUANTIFICATION 151

Then any bilinear form g : C ⊗ C → R, v = sαvα 7→ gβαvβvα, with coefficients gβα,

can be faithfully represented by the element

g = gβαγβγα‘ ∈ AlgC. (3.12)

3.3 Quantification

The passage from a yes-or-no property of a individual to a how-many property of an aggre-gate of such individuals will be called bquantificationc. A quantum theory of quantificationis called a bstatisticsc because it first appeared as the problem of how to weight the termsin a statistical average over all possibilities of a quantum catenation.

3.3.1 Choosing a quantification

We choose a candidate quantification for a fully quantum theory as follows. For an indi-vidual with vector space of D dimensions, Fermi statistics leads to a complex Grassmannalgebra of 2D dimensions as vector space of the catenation. This is structurally unstable,however, due to the central i. The prime candidate for its stabilization is a real Cliffordstatistics, based on these inconclusive indications:

1. The central quantum i of the quantum theory from which we start is structurallyunstable, though this leads to no bsingularitiesc.

2. The algebra of Fermi statistics is a Clifford algebra.

3. Clifford statistics accounts for Bose statistics as well, through bPalevc statistics.

4. The fundamental representations of the classical groups are Fermi catenations of the“atomic” representations at the terminals of their bDynkin diagramsc.

5. bFermi statisticsc is applied to produce spinor fields, and accords with the empiricalbspin-statisticsc correlation (§3.4.3).

6. Random finite set theory, the prototype of exponential logics, can be constructed byiterating a real bClifford statisticsc and restricting vectors to the bstandard basisc ofmonomials.

7. Clifford statistics is a generic variant of bFermi statisticsc and therefore structurallystable.

8. The Clifford ring of classical bgravityc is a singular limit of a Clifford algebra.

Therefore attention is turned first to basic quanta with Clifford statistics (Assumption??).


For any vector space V ,I’V := Iv : v ∈ V (3.13)

designates the space of braced V vectors.

Definition 1 (Stratum L) The vector space of stratum L is

S[L] := (PI’ )L R, (3.14)

for both individuals and pluralities on that stratum, where P is the power space operationdefined in (??).

The vector space S[L] for any stratum L has 2L dimensions.The self-Clifford algebra S is the Clifford product of its ranks S(r):

S =⊔r

S(r) (3.15)

in which the first-grade vectors of each Clifford algebra in the product anticommute withthose of any other.

Klein and bDeWittc assumed that Kaluza’s extra dimensions were compact, closed onthemselves. This created a bcompactification problemc: What compactifies these classicaldimensions? In ϑo both the internal and external groups are orthogonal groups of subspacesof the event vector space, and they have enough finite-dimensional representations whetherthey are compact or not. The charge spectra like the coordinate spectra are discreteand bounded merely because simple Lie algebras have finite-dimensional representations;compactness is not required. This replaces the compactification problem by the bgrowthproblemc:

What causes the queue to grow to macroscopic sizes in four dimensions while all otherdimensions remain microscopic?

As with snowflakes, graphenes, and soap bubbles, this enormous anisotropy may be aconsequence of the structure and dynamical interaction of the structural elements. Per-haps — speculating freely — the four-dimensional structure of stratum 3 triggers a four-dimensional crystallization on stratum E. At first the extension problem is approachedsemi-empirically. Hopefully it will then become possible to approach it deductively.

Simple bquantizationc eliminates non-generic bsingularitiesc like the Wronskian bsingularitiescof bgaugec theories and the singularities of propagators on the light cone. Instead of infiniterenormalization constants, simple bquantizationc has finite quantum constants. bSegal-Vilela-Mendes spacec has three new homotopy parameters and quantum constants: in thepresent symbols, a space-time quantum X with time units, an energy quantum E withenergy units, and a large quantum number N with no units. The usual quantum of actionand angular momentum is ~ = NEX. Since a non-relativistic limit is not considered it isconvenient to set light-speed c = 1.


The scalar meson in Minkowski space-time and general relativity both have infinite-dimensional Lie algebras whose elements depend on arbitrary functions, for example func-tions of time. Full quantization shrinks these Lie algebras to high but finite dimensionality.

Let A = Alg NR be the algebra of N ×N real matrices, with dimensionality N2. Thebmean-square formc 〉H〈 of A has signature N . For a familiar example, it is a Minkowskianform of signature 2 on the 2× 2 matrices, which represent the monadic vectors of S[3]. AsN →∞, therefore, the signature N ∼ o(DimA), and in some weak or statistical sense theform approaches neutrality, signature 0, as for a Fermi algebra. Thus the normed algebrasA = Alg NR interpolate between, and unify, the algebras of Dirac spin for stratum L = 3and bFermi statisticsc for L→∞, where with high precision ∞ ≈ 6.

3.3.2 The cumulator

Operations on one stratum have consequences on all higher strata. A one-quantum operatorq of one stratum defines a cumulative many-quantum operator Σ q of the next stratum.In canonical theories, one writes the many-quantum operator as ψ†qψ, where ψ is a vectorof annihilators. This must be slightly modified here to incorporate the stratum-raisingoperator I.

Suppose that sl are generating monadics for S[L], so that a typical polyad of S[L+ 1]is a polynomial s = Poly(I sl) ∈ S[L+ 1] ∈ S[L+ 1]. Then Σq acts on s by applying q toeach sl in s in turn, and summing over l:

Σq =∑l

d

dsl (qsl) =:

d

dsl (qsl)

by the summation convention.This can be accomplished by replacing each of the outermost braces I in s in turn by Iq

and summing all the results. This is a well-defined polarization process and will be writtenas

Σq = (I q) ID. (3.16)

Here ID is a Frechet differentiator with respect to the outermost operator I in itsoperand. In general x ID replaces each of the outermost Is in turn by the operator x andsums.

In particularΣ1 = Grade . (3.17)

The one-quantum operator q can be expressed with basis vectors em ∈ V and dualbasis vectors en ∈ V D, en em = δnm:

q = em ⊗ qmnen (3.18)


For Fermi statistics, a creator em : 2V → 2V is defined as the operation of left-multiplicationby the basis vector em:

em := Lem.

Dually, a Fermi annihilator en is defined with a bGrassmann differentiatorc with respect toen and written as

en := en LD =d

den,

so that[em, en]− = δnm

(an anti-commutator). Then to form Σ q, one replaces dual basis vectors en in q byannihilators en and basis vectors em in q by creators em:

Σ q = em ⊗ qmnen = L q LD. (3.19)

This way of writing Σ emphasizes its basis-independence.Σ : so[L]→ so[L+ 1] is a Lie-algebra homomorphism:

Σ [a, b] = [Σa,Σb].

It constructs the action of a one-quantum operation on a many-quantum collection, andconverts one-quantum coordinates into cumulative many-body coordinates.

Σ sends each predicate projector q = q2 = q† ∈ EndoV into a bnumber operatorc Nq :=Sq ∈ Endo 2V giving the number of individuals with the predicate q in any catenation. Inthis way it serves as a bquantifierc; not one of the traditional four A,E, I,O, but a numericalquantifier.

P and S in turn define operators Π and Σ that convert finite or infinitesimal trans-formations Ω, ω : S[L] → S[L] of a parent stratum to induced transformations ΠΩ,Σω :S[L+ 1]→ S[L+ 1] of the next stratum:

ΠΩ = I(PΩ)IH, Σω = I(SΩ)IH. (3.20)

First the operator IH = I† unbraces monads, stripping them to their internal polyads. ThenΩ acts on the internal polyad within the monad by the extension PΩ. Finally I rebracesthe resulting polyad. Similarly for Σ.

There are now two natural Lie homomorphisms from the commutator Lie algebraGrade2 S[L] of the second-grade elements sβα = [sβ, sα] ∈ S[L] to that of Grade2 S[L+ 1]that must be distinguished:

(1) The left-multiplications Lsβα := γβα define infinitesimal operators γβα : S[L] →S[L]. These induce transformations Σγβα : S[L+ 1] → S[L+ 1]; here Σ is the cumulatoralready defined.


(2) Another Lie homomorphism Σ2 is defined by

Σ2[γβ, γα] := [LIsβ, LIsα] =: γβ α. (3.21)

Evidently∀ω1, ω2 ∈ S[L]2 : Σ2[ω1, ω2] = [Σ2ω1,Σ2ω2]. (3.22)

Familiar universal, existential, and numerical quantifiers ∀, ∃,N are easily expressed interms of Π and Σ. Np is a queue variable whose value is the number of monads with theproperty p in the queue. For any monadic projector p, with complement 1− p, one defines

Np := Σp; (3.23)∀p := P [N(1− p) .= 0], (3.24)∃p := 1− P [Np .= 0], (3.25)

where P [. . .] is the projector on the subspace defined by the condition [. . .].Cumulators can be defined for Fermi, Bose, and Maxwell statistics as well; it was a

concept of classical physics before quantum.

3.3.3 Algebra unification

Quantum mechanics has one product xy where classical mechanics had two, a commutativealgebra product xy and a Poisson Bracket [x, y]P, as was emphasized by bGrginc andbPetersenc [46]. Thus canonical quantization unifies two algebras of the classical theoryinto one of the quantum theory. There is material for further balgebra unificationc. Space-time vector fields can be multiplied either as Clifford elements or as differential operators.The non-associative inner product v ·w and Lie Bracket [v, w]L of space-time vector fieldsderive from these respective associative products. The full quantization performed hereunifies these two algebras into one Clifford algebra on the vector space of the quantumbgravitational fieldc.

A Grassmann algebra and a regular symmetric bilinear form on the vectors (first-gradeelements) of the Grassmann algebra define a Clifford algebra. A central i is structurallyunstable, so we begin with the real quantum theory. Then quantum theory provides a realregular bilinear form, the bprobability formc. Therefore ordinary bFermi statisticsc defines areal Clifford statistics, whose Clifford product unifies the Grassmann product of the bFermistatisticsc with the inner product of the quantum theory. This bproduct unificationc is theanalogue for graded canonical quantization of the product unification already mentionedfor ordinary canonical quantization.

3.3.4 Spinor spaces

Clifford elements of C = 2W can be isomorphically represented as linear operators on anassociated vector space, a spinor space SpinorW , defined up to isomorphism. That is,

C ∼= (SpinorW )⊗ (SpinorW )D. (3.26)


Its elements, spinors, are represented by the columns of a faithful irreducible matrix rep-resentation of C.

Aphoristically put, the spinor space of a bilinear space is the square root of its Cliffordalgebra. It is not the square root of the bilinear space itelf, however, as is sometimes said.To be sure, vectors are (represented by) bilinear forms in spinors, but so are forms of everygrade: tensors, axial vectors, and all are bilinear forms in spinors. The Clifford algebracontains them all.

It follows that for the duplex bilinear space W = V ⊕ V D, whose Clifford algebra is2W = 2V ⊗ 2V

D,

SpinorW = Spinor(V ⊕ V D) = GrassV (3.27)

up to isomorphism.Relative to quanta with vector space W = V ⊕V D, a quantum with vector space V or

V D is called a bsemiquantumc, and its vectors are called semivectors. Therefore:

Spinors are the vectors of a Fermi assembly of semiquanta.

The orthogonal group of a vector space V is generated by the grade-two elements sβαof the Clifford algebra C = 2V . When the sβα act on C by left multiplications Lsβα, Ctransforms as a direct sum of spinor spaces of V . Under inner automorphisms 4sβα, Ctransforms as a direct sum of the spaces of multivectors of every grade over V .

Cliff V can then be realized as an algebra of matrices whose columns are spinors of Ψand whose rows are dual spinors of ΨD.

3.4 Fermi algebra

It is particularly easy to factor a Clifford algebra into a spinor space and dual spinor spacefor the special case of a bFermi algebrac, constructed as follows.

There is a familiar process of bBose quantificationc, based on Bose statistics:

1. A vector space V and its dual space V D define

2. a canonical Lie algebra a = V ⊕ C⊕ V D which generates

3. a canonical algebra A that can be written as Alg V where

4. V = Poly(V ) is a higher-stratum vector space.

This process is ripe for iteration V → V → V → . . ., but singular.There is a closely parallel quantification process, bFermi quantificationc, based on Fermi

instead of Bose statistics:

1. A vector space V and its dual space V D define

3.4. FERMI ALGEBRA 157

2. a Fermi Lie algebra f = V ⊕ C⊕ V D which generates

3. a Fermi algebra F that can be written as Alg V where

4. V = Poly(V ) is a higher-stratum vector space.

This process is ripe for iteration V → V → V → . . ., and is regular. The bFermi gradedLie algebrac f := Fermi(V ) is the graded Lie algebra whose elements constitute the vectorspace f := V ⊕ C⊕ V D and obey the graded Lie product relations

∀n, n′ ∈ V D, ∀p, p′ ∈ V :[1, n]g = [1, p]g = 0,

[n, n′]g = [p, p′]g = 0,[n, p]g = p(n). (3.28)

Vectors in V D are assigned grade −1; those in V are assigned grade +1. The numbersz ∈ Z = C are assigned grade 0. The brackets are graded Lie products, symmetric if onefactor has even grade, skewsymmetric if both grades are odd.

Then the bFermi algebrac F := FermiV is the linear associative algebra of polynomialsin the vectors of f modulo the Fermi relations (E:GRADEDFERMI), in which gradedLie products are now read as graded commutators. Grades in f take on only the values±1, 0. Grades in F range over 2m+ 1 values from −m to m. Finally, V = GrassV is theGrassmann algebra Poly(V ) of polynomials in the vectors of V regarded as anti-commutingvariables.

The dimensions of f , V , and F , respectively, are

2 DimV + 1 < 2DimV < 22 DimV , (3.29)

assuming DimV ≥ 3.The above construction lacks two pieces of information vital for quantum application:There is no mention of the coefficient field of V or V The process works equally well

for R, C, or for that matter the binary field 2.There is no mention of the probability form of V and none is provided for V .In both cases, however, if the information is given for V it can naturally generate the

information for V . It is only necessary to provide the missing information at the start ofthe iteration.

This suggests that the physical field is R and that the central i is a singular limit of anon-central operator i in the same way that central coordinates q, p of classical mechanicsare singular limits of non-central quantum operators q, p. This leads to an interestingunification of i and g and is assumed in ϑo.


3.4.1 The duplex space

In standard quantum theories, quantum superposition between input and output vectorsnever arises. They occupy separate spaces V and V D, linked only loosely by an antilinearadjoint operation H : V → V D. When a vector ψ ∈ V is multiplied by i, a dual vectorφ ∈ V D is multiplied by −i. In the bduplex spacec W := V ⊕ V D, the addition ψ ⊕ φis possible, but this sum is not projectively invariant under the usual action of i; duplexaddition is not invariant under i.

In ordinary experiments the suspension of quantum superposition between input andoutput vectors is physically reasonable even if they both belong to some larger vectorspace like W . The input and output processes are widely separated and generally involveinteractions with different macroscopic systems of uncontrolled relative quantum phase,like a polarizer and an analyzer, so quantum superposition of ψ and φ is unfeasible. Thisincoherence, however, is a typical result of large numbers. For experiments of a sufficientlysmall scale, coherent phase relations between input and out processes might conceivablyoccur, of the kind described by vectors in the self-dual duplex space W = V ⊕ V D.

Since the Fermi algebra over V is a real Clifford algebra over W , it loses nothing tosuspend the assumption that the complex number i is in the center of the operator algebraswe consider. In the following the basic quantum system, the queue, has real coefficents,Fermi statistics, no probability form and no central imaginary unit.

This creates an obligation: To account not only for the apparent centrality of i in thecanonical limit, but also for its global universality and its anticommutation with bWignercbtime reversalc T, which maps i 7→ TiT−1 = −1 (Section §6.1.6).

Clearly a Fermi algebra over N is a bClifford algebrac, over the real neutral bilinearspace

V = N ⊕ P,P = ND,

∀n ∈ N, ∀p ∈ P = ND : ‖n+ p‖ := p(n). (3.30)

called the bduplex spacec of N . That is, the anticommutationrelations of FermiN are justthose of the real Clifford algebra Cliff V :

FermiN = Cliff V = 2V . (3.31)

The bduplex spacec W , as the direct sum of complex vector spaces, inherits i fromthem. On the other hand the operator i : N → N, n 7→ in induces the operator i† : P →P, p 7→ −ip, resulting in a combined operator

j : W 7→W,n 7→ in, p 7→ −ip. (3.32)

The operators i and j are both square roots of −1 and commute. Physical operatorsincluding observables are required to commute with both i and j. Thus the physical


operators do not form a simple algebra, because the vectors n and the dual vectors p arenot supposed to have quantum superpositions.

In the present work all such restrictions of quantum superposition are dropped, hop-ing that they are low-energy artifacts. Assuming that multiplication of V by i inducesmultiplication of V D by −i as usual, the duplex space of a complex space is a real spaceprovided with a non-central i. The complex Fermi algebra then becomes a real Cliffordalgebra over a neutral space.

The algebra F = FermiN does not determine the vector space N from which it stems.In particular F is invariant under the exchange of N and P . A one-dimensional projectorΩ ∈ F exists with the property that nΩ = 0 = Ωp for all n ∈ N, p ∈ P , called the vacuumprojector for N . The vacuum suffices to fix the split into positive and negative vectors,creators and annihilators.

Being a Clifford algebra according to (3.31), F is isomorphic to a full matrix algebraover a spinor space Ψ related to the duplex space W , though not naturally isomorphic.

The Clifford algebra defines a Grassmann graded Lie algebra on the same elements,whose graded-commutative product is called the bWick productc of the Fermi algebra.The generators of FermiN obey the anticommutation relations of the real Clifford algebraCliff V ,

FermiN = Cliff V = 2V , (3.33)

where V is the real neutral bilinear space

V = N ⊕ P,P = ND,

∀n ∈ N, ∀p ∈ P = ND : ‖n+ p‖ := p(n). (3.34)

Using the anticommutation relations, every Fermi product can be expressed uniquelyas a bnormally ordered productc, in which all the vectors of P stand on the left of all thevectors of N = PD. Therefore

FermiP = GrassP ⊗GrassPD. (3.35)

It follows from (3.35) that up to isomorphism the spinors of V are the many-quantumvectors of the Fermi catenation over P :

SpinorV = GrassP . (3.36)

Briefly put:

3.4.2 A spin is a queue of semiquanta.

This is emphasized, for example, by Wilczek and Zee [86]. They consider the possibilitythat the three generations of leptons, and also of quarks, are physical elements of the Fermiassembly represented by a higher-dimensional spinor.


Here the possibility is also considered that the leptoquark generations belong to dif-ferent strata of a queue.

The Fermi quantification process is a general way to construct a spinor space SpinorVover any quadratic space V , and so for any simple quantum system I[V ] [16]. This is arare example of stratum structure in present quantum theory, and so merits attention.

bBrauerc and bWeylc liken their spinor construction to “bsuperquantificationc”, bywhich they mean what is usually called“bsecond quantizationc” and is here called bFermiquantificationc. They do not propose a physical interpretation, which would raise hardquestions about the bspin-statisticsc correlation and conservation of angular momentum,but limit themselves to the mathematical problem of explicitly constructing a spinor rep-resentation.

bBrauerc and bWeylc, bCartanc, and bChevalleycconstruct a spinor space in three steps[16, 18, 19]:

Neutralize. Cartan and Chevalley assume that the quadratic form Q of V is bneutralc.One may bring this about if necessary by extending from real to complex coordinates, orby replacing V by DupV with the duplex norm ‖ . . . ‖Dup.

Split . V can then be decomposed into two isomorphic maximal null subspaces P,N ⊂V of half the dimension of V , such that V = P ⊕N and

∀p ∈ P , ∀n ∈ N : ‖n+ p‖ = p · n. (3.37)

We designate one such arbitrary choice (improperly) by

P = Semi+ V, N = Semi− V (3.38)

and call a vector in either of them a bsemivectorc. The hypothetical quanta I[P ] andI[N ] are called bsemiquantac of I[V ] in this context. Possibly Cartan and Chevalley chosethe letters P and N in this construction to recall the positive-energy and negative-energyspaces of Dirac electron theory.)

Power . Then identify V , which we have already factored into creators and annihilators,with the V of (3.34) and form the power space of P ⊂ V , the Fermi algebra

Cliff V = FermiP = GrassP ⊗GrassPD. (3.39)

Therefore GrassP is a spinor space of V . And GrassN is another.

P = Semi+ V , N = Semi− V , SpinorV := Grass Semi+ V . (3.40)

Abspinorc space is a vector space supporting a faithful irreducible representation of theClifford algebra. It is also by definition the vector space of a Clifford assembly. Thereforeone may just as well say that a bspin 1/2c is a Clifford assembly of semiquanta.


3.4.3 Spin-statistics correlation

Let 〈ε] be a probability amplitude vector in the dynamical stratum F. 〈ε] is blocalizedcif for all group elements g ∈ SUF(C) outside a small neighborhood n ⊂ SUF(C) of theidentity,

[ε〈g〈ε] = 0. (3.41)

The exchange parity of ε is the operator X = X(ε) representing a homotopy g(θ) ∈SUF(C) (0 ≤ θ ≤ 1) that exchanges 〈ε] and 〈g〈ε] where they occur as factors:

〈g(θ)〈ε] ∨ 〈g(θ)〈g〈ε] = 〈ε] ∨ 〈g〈ε] for θ = 0,= [〈g〈ε] ∨ 〈ε] for θ = 1. (3.42)

X2 = 1, so X .= ±1.The bspin parityc of ε, is the operator W = W (ε) representing a continuous rotation

of one ε through 2π. W (ε) .= ±1is +1 if the spin of epsilon is even in units of ~/2 and −1if odd.

The observed bspin-statisticsc correlation is

W (ε) = X(ε) (3.43)

for all quanta ε of the dynamical stratum. Since we violate this equality we require separatenames for its terms.

Cartan’s mathematical definition of spinors in terms of Clifford algebra, and the BrauerWeyl construction of spinors by Fermi quantification, both require further physical inter-pretation before they can be taken seriously as a physical model of the phenomenon of spin1/2. The process bSemic is unnatural: there are an infinity of minimal left ideals, and Semiselects one of them. Something in nature must make this selection.

Choosing one ideal out of them all breaks a symmetry. This suggests an organizationon a stratum higher than that of the system.

If V is the vector space of a quantum V; in the case of greatest present interest,V = S[L]. Then a Fermi assembly of V’s has vector space W = 2V .

Suppose that V is half full, in the sense that a subspace N = Semi− V is full and anisomorphic subspace P = Semi+ V is empty. Then V can be represented as a duplex spaceV = DupN . This defines a duplex norm HN on V , in which all the vectors of N and Pare null vectors.

Let C be the Clifford algebra over V with respect to this duplex norm. With respectto left multiplication, the elements of N act on C as creation operators and those of P actas annihilation operators. Then the subspace N generates a Clifford sub algebra 2N thatis a left ideal of C.

For the elements of N adjoin factors in N , and the elements of P remove factors in N ,and both map N → N . Define a “vacuum” vector ΠN ∈ 2N ⊂ 2V , of grade 1

2 DimV , as


a top vector of N . ΠN is an arbitrarily ordered Grassmann product of all the vectors in aminimal basis for N ⊂ V , analogous to the negative-energy Dirac sea.

Let ψ ∈ P and let ψ be the corresponding first-grade vector in 2N . Then any swapω ∈ so(V ) acts on ψ as the inner automorphism

Σω ψ = [γω, ψ]. (3.44)

induced by spin operator γω ∈ Grade2 2V .

Assertion 1 The many-quantum vector ψΠN is a spinor of SO(N).

Argument: 2N is a Grassmann algebra, and Πn is its top element. Therefore, ∀n ∈ 2N :nψ = 0. Therefore nγω = 0. Therefore the inner automorphism generated by γω reducesto left-multiplication by γω, a spinor transformation.

The many-quantum vector ψ ∨ΠN represents one spin in the vacuum.This makes spin 1/2 anomalous in the same sense that spin 1/3 is anomalous in anyon

theory. There is a larger quantum system whose swaps are whole numbers, having thebClifford algebrac Cliff V as vector space.

This vector space factors into two dual subsystems as Cliff V = Cliff N⊗Cliff P . Whenwe observe a spin 1/2, we are observing one of these subsystems and omitting a vaccum-fullof the other because it belongs to the metasystem. bmetasystemc.

If W = DupV is a Minkowskian space then a bspin-statisticsc correlation would seemto imply that a hypothetical quantum entity b with vector space V ⊂W would be a boson;yet to make spinors the cited authors forn the Fermi algebra FermiV of integer-spin vectorsof the boson b, seemingly violating the bspin-statisticsc relation.

Similarly, the components of the Dirac γµ have negative exchange parity and positivespin parity. Evidently the bspin-statisticsc correlation for quanta does not apply to lowerstrata; and indeed Dirac spin matrices are not particle vectors.

If this famous spinor construction has physical meaning — and let us suppose that itdoes if only for the sake of a reductio — then it omits some important physical element thatabsorbs the right action of a Clifford element S(L) associated with a Lorentz transformationL, leaving only the left action. As mathematicians we may evoke minimal left ideals atwill; for the physical interpretation as vector spaces, we need a physical agent to conserveangular momentum or swap-invariance.

The vacuum, the ambient organized stratum-F queue, often serves as this agent. Sup-pose that a one-dimensional vacuum projector Ω = Ω2 = Ω† is invariant under all theantisymmetric generators S(L) = −S†(L):

[S(L),Ω] = 0. (3.45)

ThenS(L)Ω = ΩS(L) = ΩS(L)Ω = −ΩS(L)Ω = 0 (3.46)


andδLΨΩ = S(L)ΨΩ. (3.47)

That is, the natural spinors are not the multivectors of PoV themselves but the elementsof the orbit PoV’ Ω of Ω under these multivectors.

Now each part of the spinor construction can have a physical interpretation.

1. L is a Lorentz transformation that relates two experimenters in relative motion (say)assigning vectors to some quantum entity b of bstratumc 2 with vector in Vb = V ⊂W .

2. S(L) is the induced action of L on the Clifford catenation of b’s, with vector spaceΨ = PV generated by creation operators.

3. ΨΩ is the orbit of a suitable isotropic vacuum Ω under these creation operations.

The real spinor space S = 4R of a Minkowski tangent space M is a square root of theClifford algebra C = PM = S ⊗ SD, 4 =

√16; not merely a square root of space-time as

one sometimes hears. The vacuum takes that square root in step 3, where the dimensionfalls from 16 to 4. In the classic constructions cited, especially Chevalley’s, Ω represents afull Dirac sea.

If this standard theory of spinors is to be taken seriously, as I attempt here, thenelectron spin 1/2 is anomalous like anyon spin 1/3, dependent on an ambient organization.Spin is a relative angular momentum of a Fermi catenation of several even entities relativeto a coherently organized background Ω of many such entities, in an infinite limit.

In the present project, this means that although we build with Fermi catenation, weshould expect to encounter no spinors or spin 1/2 entities below the dynamical stratum,where vacuum organization can occur. And the spinors arising there will be Fermi catena-tions of simpler quantum entities describing “subspins.”

For Minkowski space-time, spinors of 4 components describe Fermi catenations of 0,1, or 2 identical subspins: 4 = 1 + 2 + 1. In a space-time of 6 dimensions, a spinor of 8components describes 0, 1, 2, or 3 subspins: 8 = 1 + 3 + 3 + 1.

SchematicallySpinorV =

√Cliff V =

√2V . (3.48)

bClifford algebracs can be represented as matrix algebras, and then the vectors theyact on are called spinors:

A bspinorc space of a bClifford algebrac C is a module σ provided with an irreducibleisomorphic representation of C in Endoσ.

The Clifford algebra associated with the quadratic form Q of full signature (n+, n−) isisomorphic to a matrix algebra with matrix elements in a ring R(σ) depending only on thesignature σ := n+ − n− mod 8 according to the bSpinorial Clockc [17] of Figure 3.1. Thematrices have size 2D ⊗ 2D/d, where D = DimV and d = DimR(σ) are real dimensions.Also, when the dimension of V increments by 1 the real dimension of the bClifford algebrac


6

&%'$0

R

12R

2 R

3C4

H

52H

6H7

C

Figure 3.1: The Spinorial Clock [17] Find the signature σ mod 8 of the quadratic form Qof a vector space V on the inner circle. Next to it on the outer circle, read the coefficientring R(σ) of the spinors of Q.

2V doubles, starting from V dimension 0 and spinorial dimension 1 at σ = n+ = n− = 0.This leads to the tabulation of the spinor spaces of the real Clifford algebras in Table 3.1,based on the Spinorial Chessboard of Budinich and Trautman [17].

0 1 2 3 4 5 6 7 . . . (n+)0 R R2 2R 2C 2H 2H2 4H 8C . . .1 C 2R 2R2 4R 4C 4H 4H2 8H . . .2 H C2 4R 4R2 8R 8C 8H 8H2 . . .3 H2 2H 4C 8R 8R2 16R 16C 16H . . .4 2H 2H2 4H 8C 16R 16R2 32R 32C . . .5 4C 4H 4H2 8H 16C 32R 32R2 64R . . .6 8R 8C 8H 8H2 16H 32C 64R 64R2 . . .7 8R2 16R 16C 16H 16H2 32H 64C 128R . . ....

......

......

......

......

(n−)

Table 3.1: PERIODIC TABLE OF THE SPINORS. The dimension and coefficient ring ofthe spinor space for the real bClifford algebrac over (n+Rn−R). R2 and H2 are the ringsof real and quaternionic diagonal 2× 2 matrices.

3.5 Clifford statistics

An individual with bvectorc space V is said to have bClifford statisticsc if its quantifica-tion produces a system with vector space Cliff V = 2V . Such individuals may be called

3.5. CLIFFORD STATISTICS 165

bcliffordonsc but they are also fermions and obey an exclusion principle.Clifford statistics has the Wilczek-Zee bfamily propertyc [86]: When the orthogonal

group SO(W ) is reduced to a subgroup SO(V ), V ⊂ W , with dimensional difference δ :=DimW −DimV , then the Clifford representation of SO(W ) reduces to a family of 2δ copiesof the Clifford representation of SO(V ). If these copies are the families of quarks or leptons,then since 3 is not a power of 2, this suggests at least a fourth family. In the present workthe conjecture is natural that the generations of leptoquarks are generated by I acting onsome stratum below stratum F. This is not inconsistent with the Wilczek-Zee hypothesis,and might be a model for it.

3.5.1 Spin-statistics anomaly

Iterating bClifford statisticsc conspicuously violates statististics conservation. The vectorsof each bstratumc are polynomials in the bracings of the vectors of the the previous stratum,subject to the Clifford Clause. An element of even grade has a bracing of odd grade1, as though an even number of fermions could form a fermion. On the other hand,Clifford quantification conserves angular momentum. This flagrantly violates conservationof statistics and any bspin-statisticsc correlation across strata.

Moreover a similarly unnatural re-grading happens in Dirac’s theory of bspinc, whichworks well, and which it is supposed here is an earlier iteration of the same Fermi statistics.Dirac formed a bClifford algebrac over the Minkowski space-time tangent space, therebyassigning anticommutation (Fermi) relations to the four tangent vectors γµ which have spins0 and 1. The conclusion is that the bspin-statisticsc correlation that works on stratum Edoes not apply to stratum C. This suggests that the empirical spin-statistics relation resultsfrom organization. The strong dependence of the spin-statistics correlation on space-timecontinuity also suggests this. For skyrmions, or topological solitons in a tensor field subjectto non-linear constraints, spin and statistics are correlated because a rotation and a pairexchange are connected by a homotopy. No such construction is possible on lower strata.It is a concept of the singular limit.

Moreover the Dirac spin operators γµ transform dually to momentum generators pµ.In a fully quantized theory like ϑo the event coordinates and momenta pµ become singularlimits of swaps like ω(ν5, ων6 of the so(dC) of a deeper stratum C.

C = 3 is the this most economical possibility and is provisionally chosen here.As the Dirac Clifford algebra shows, the usual bspin-statisticsc connection deos not

apply to deeper strata like C, since it concerns physical quanta, and these tangent vectorsare not vectors of quanta. The standard theory keeps space-time and quanta in separatestrata so that these spin-statistics anomalies are harmless.

Here I bridges strata and such anomalies become the norm. The spin-statistics corre-lation at the many-particle stratum F now seems anomalous and requires special handling.The usual arguments for the spin-statistics correlation based on continuity or analyticitywork in a singular limit. They must have correspondents in a fully quantum theory or the


theory must be corrected or abandoned.In the queue theory ϑo spin 1/2 and the bspin-statisticsc correlation are the true anoma-

lies, existing only at stratum F and only as a result of the special organization of thevacuum. This returns to early days of quantum theory when spin 1/2 was still consideredanomalous, but today there is also the example of anomalous anyonic spin 1/3 [87].

3.6 Measurement

Quantum theories swallow their parent classical theories alive (as Teller put it). Classicaltheories continue to work well for some experiments in the quantum universe. To see this,since experiments begin and end with measurements, it is important to recognize that thephysical processes called measurements in classical physics are exceedingly different fromthose called measurements in quantum physics; and yet the quantum theory, if it to becomprehensive, must be able to represent both processes. When quantum theory examinesa measurement closely it sees an interaction. The analysis had to wait until now becausethe linear logic could not analyze an interaction between two systems, but the polynomiallogic can.

The correspondence principle deals with the limit ~→ 0. If the classical measurementprocess is to be represented physically within a quantum theory, however, then the physicalvalue of ~ must be used, not the singular limit.

One difference concerns degree of isolation. Typically the macroscopic system understudy is interacting and exchanging quanta with the extrasystem during the entire experi-ment. The actual elements at any one time are uncertain. One may describe such a processby a statistical operator on a many-quantum vector space of the kind used in quantumfield theory, which allows the number of elements to be a variable.

A second difference concerns the strength of the measurement interaction. An actualmacroscopic measuring process generally has a much smaller effect on the system thanthe input-output processes represented by vectors in quantum theory, one that is usuallyneglected altogether in a classical theory. A visual inspection, for example, couples directlyonly to the tiny fraction of atoms near the surface of the macroscopic system, its skin so tospeak. Such a gentle process may reasonably be called an observation, in the sense that theobserver does not intervene significantly in the development of the system observed. Thenno distinction need be made between input and output classical measurement processes,since what is going on may be a continuing process. In contrast N input process representedby a vector erases the prehistory of the system and reduces the system entropy to 0.

One may represent an observation process by one member of a parametric family ofprobability operators ρ(y), where y = yk designates a family of parameters, much smallerthan the family of all possible probability operators. One then uses the observation resultsto fix the parameter set y. It is supposed here that the parameters y are expectation values

3.6. MEASUREMENT 167

of chosen macroscopic variables collectively designated by Y :

yk = Tr ρ(y)Y k. (3.49)

The method of coherent states [49] is one way to describe classical behavior of quantumsystems. Coherent states are called coherent vectors here for the sake of consistency. Theyform a parametric family of vectors 〈y with a set y of real parameters. For coherent vectors,〈y is bcompletec: ∑

α

〈αα〈 = 〈1〈 , (3.50)

but not orthonormal:β〈alpha 6= δαβ, (3.51)

and bstrongly continuousc in y:

∀y0 : y → y0 ⊂ ‖〈y − 〈y0‖ → 0. (3.52)

The coherent probability operator 〈yy〈 (for any value of y) is then a one-dimensionalprojector, of entropy 0; here this is replaced by the statistical operator ρ(y) of greaterentropy.

One may assume that the classical observation operators too are a spectral family ofprojections ∑

y

ρ(y) = 〈1〈 , ρ(y)2 = ρ(y) = ρ†(y), (3.53)

and strongly continuous in y, but not orthogonal:

ρ(y)ρ(y′) 6= ρ(y)δyy′ . (3.54)

In applying coherent vectors, the approximation is made that the statistical developmentcarries any coherent probability operator 〈yy〈 into another operator 〈y′ y′〈 of the samefamily, with a possibly different parameter value y′. An observation then selects one of theprobability operators 〈y y〈 . The dynamics is then approximated by a time-development ofthe parameter y = y(t). These vectors are coherent in the sense that the wave packet 〈ycoheres — sticks together — during the development.

This classical use of probability operators differs from the quantum use. The coherent-vector probability operator 〈αα〈 , since it expresses maximal information and 0 entropy,does not accurately describe a physical macroscopic input-output process, and cannot bedetermined by inspecting the input-output equipment, as in the usual quantum kinematics.It describes the result of statistical observation, as in classical statistical mechanics. A moregeneral macroscopic probability operator ρ(y), however, may be a microscopic physicaldescription of a macroscopic input-output process.

[Do080318] To be continued


Coherent vectors are often used to approximate a canonical quantum theory by acanonical classical theory. Coherent spin vectors can similarly be used to approximate aspin quantum theory by a canonical quantum theory.

[Do: Extend the method of coherent vectors [49] to approximate the canonical quan-tum Lie groups of canonical quantum physics within the orthogonal group of S.]

Chapter 4

Exponential quantum logics

which concerns iterated quantification.

The maximal descriptions of an isolated physical system are supposed to correspondto rays in a vector space S associated with the system. For an assembly, S has a productrepresenting the union of assemblies. Just as iterated addition is multiplication, iteratedmultiplication is exponentiation. To form polyads of polyads, however, one first convertsa polyad into a monad. The bracing operator I does this. An old example of this processis considered next before taking up the general case.

4.1 Spinors

There are already traces of iterated bFermi statisticsc in present physics that are worthexploring further. The bstandard modelc applies bFermi statisticsc only to spinorial fields,of spin 1/2, not to fields of integer spin. But spinorial fields are themselves Fermi algebrasover deeper spaces [16, 86]. Suppose that the one-particle vector space is

Ψ1 = SpinorV0 = Grass SemiV0 (4.1)

for some quadratic vector space V0. Then the many-particle Fermi vector space is

Ψ2 := Grass Grass SemiV0←Cliff(Cliff SemiV0), (4.2)

a singular limit of iterated Clifford statistics, which we may assume is the more accuratetheory.

We may take SemiV0 to be a Minkowski tangent space M , disregarding functionaldependence on space-time coordinates. But M is itself presented as a thrice-Clifford algebraM = Cliff30 in (4.22). Combined with the previous result (4.2) this gives a five-folditeration Cliff5. We suggest physical interpretations for these lower strata in (4.33).

169

170 CHAPTER 4. EXPONENTIAL QUANTUM LOGICS

A quantum set or bqueuec in a broad sense is a quantum system whose vector spaceis a Clifford algebra with specified interpretations for the algebra operations. Where thequantum element I[V ] has bvectorc space V , the queue of I[V ]’s has the vector space

2V = FermiV. (4.3)

This queue theory includes classical set theory as a commutative reduction (§2.2.3).Classical set theory can also be reconstructed as a stratified Clifford statistics with the

binary coefficient field Z2 = 0, 1 instead of R but the binary Clifford algebra does nothandle classical probability theory as efficiently as the real Clifford algebra and will not beused.

Assertion 2 When V = S the classical power set 2S is a commutative reduction of thequantum power set with vector space 2V .

Argument: Let 〈n : n = 1, . . . , N be a basis for V . The operators 〈1〈 , 〈2〈 , . . . , 〈N〈form a maximal commuting set of independent operators in 2V and generate a Cartan(maximal commutative) subalgebra of 2V , isomorphic to the set algebra generated by theclassical objects 1, . . . , N.

To ease the conceptual transition from classical sets to quantum sets let us begin witha back-formation expressing classical set theory as a commutative reduction of Cliffordstatistics.

4.1.1 Baugh numbers

The sets used here are finite and form an infinite set S generated by forming finite productsof what has already been formed, and by bracing I, which turns each monomial into a first-grade monomial.

One designates the monomial with no factors by 1. It is the multiplicative identity.Recall that if s is any set, 2s, the bpower setc of s, is the set of all the subsets of s.

and 2I [s] is the random object whose values are the subsets of s.If we peel the outermost braces from a set we expose the elements of the set. If we

repeat the process indefinitely, the sets thus exposed are called the bancestorsc of the set.It is helpful to think of a finite set in the way introduced by bBaughc, as a (natural)

number in a positional notation with an expanding base [7]. The usual fixed-base positionalexpansion for a number N s a weighted sum with coefficients cn defined by the number Nand weights bn defined by the base alone:

N = (N1N2 . . . Nν)b =ν∑

n=0

Nnbn, 0 ≤ Nn <bn+1

bn. (4.4)

In the bb-ary expansionc of the numbers,

b0 = 1, bn+1 = bbn, 0 ≤ cn < b. (4.5)

4.1. SPINORS 171

In the bBaugh expansionc, the coefficient Nn in any place n = 0, 1, . . . ranges over thenumbers that can be expressed with the previous places alone, and the weight bn is thesmallest number that cannot be so expressed:

b0 = 1, bn+1 = 2bn , 0 ≤ cn < bn. (4.6)

The Baugh number B(s) of a set s is then determined by the rules that for all disjointsets u, v:

B(u ∨ v) = B(u) +B(v),B(Iu) = 2B(u). (4.7)

The monads of Table 1.1 generate the polyads in the vector space S.Let sb designate the polyad of Baugh number b.Then s0 = 1; s1, s2, s3, and s4 can be identified with the Dirac gamma matrices

γ1, γ2, γ3, and γ4, with γ4 being timelike; and what is usually designated by γ5 is

γ5 := γ4γ3γ2γ1 = s4s3s2s1 = s4+3+2+1 = s10. (4.8)

The set S has no symmetries. No two sets are the same. This is useful, because naturehas no symmetries; an exact symmetry would be unobservable, as Wigner pointed out[84]. The moon disturbs the roundness of the Earth measurably; it disturbs the roundnessof a terrestrial hydrogen atom in its ground energy stratum too, though immeasurablyless. Symmetries in physics result from ignoring some asymmetries, especially of the meta-system. They will be represented by corresponding symmetries in queue theory resultingfrom ignoring some remote ancestors of the queue, terms at the tail of the Baugh expansion.

4.1.2 Random sets

A brandom setc I [S] is a random object with state space S sampled in Table 1.1. Thenits probability distributions form the simplex in S whose edges are the coordinate vectorssb ∈ S.

Thus the state space S of a random set is a set of sets. This is a commutative reductionof a quantum concept:

A b queuec is a quantum entity whose vector space is a (finite) stratum S[L] ⊂ S.

4.1.3 Quantum cells

The general element of 2V is also an element of the Grassmann algebra over V . In anyframe for V , an element of 2V expanded in the induced frame for 2V has the form givenby Chevalley for a chain associated with a simplicial complex [20]. The n basis elementsare the vertices, and the monomials are the simplices. In the Chevalley theory, however,the vertices are absolute, while in queue theory the vertices are relative to a basis, subject


to quantum superposition, and can be transformed into other vertices by arbitrary lineartransformations in GL(n). Such a chain can be regarded as a vector of a bqueuec, a possibleinput process for it. The monomials are basic vectors for the queue.

The Grassmann product abc . . . d of any sequence of vectors a, b, c, . . . d ∈ V cam alsobe regarded as a volume element with edges a, b, c, . . . d. This is the object that Grassmannand Clifford called an element of extension. Its measure is the square root of the Cliffordnorm of the Grassmann product,

µ(abc...d) := [Q(a ∨ b ∨ c ∨ . . . ∨ d)]1/2. (4.9)

It is convenient sometimes to speak of the bClifford algebrac elements as vectors of quantumcells rather than queues because µ(abc...d) is the squared measure of the cell, not the set,defined by the edges a, b, c, ..., d. Subsets of a, b, c, . . . d define subcells or faces of the cellabc . . . d.

To recover the classical set construct from the q, one fixes an orthonormal basis in V ,forms the commutative set S of the projections on the basis vectors, and takes the statesof I [S] to be the elements of S.

To iterate Cliff we require an operation that associates a high-grade element of abClifford algebrac into a first-grade element of the next-stratum Clifford algebra. This isthe bbracec I of the next section.

4.1.4 Bracing

The bbracecI : x 7→ x = x (4.10)

forms monads and is used to express the idea of association, or of one thing being “at”another. In bfieldc physics today, one puts a field variable f , typically an operator, at apoint x by forming the pair IA ∪ Ix. In a physics of atoms, one might describe an atom Aat a space-time event x by forming the pair

A, x := A ∪ x =: IA ∪ Ix. (4.11)

Every physical theory uses I, usually tacitly. Often it is used to couple metasystemicand systemic operations. Time, for example, is measured in a clock of the bmetasystemcand is then used to control measurements on the system, and q(t) associates the meta-systemic variables t and q into the system variable q(t). If the strata of physics are tocommunicate, some such operation must be possible.

Statistors for a queue with Fermi statistics belong to a self-Grassmann algebra S definedconstructively thus:

First one defines a bracing operator in general. Let V be any normed vector space.Let W be the set of all unit classes v for v ∈ V . The bbracec operator I : V ↔W is theunit-set forming operation

Ix = x = x (4.12)

4.1. SPINORS 173

modulo linearity, the identification

(∀x, y ∈ V )(∀a, b ∈ R) : I(ax+ by) ≡ aI(x) + bI(y), (4.13)

This makes the set W = I’V of braced vectors a linear space. I is also designated by abvinculumc as in

I(abc) =: abc (4.14)

to emphasize that it is an associator.In the quantum theory it is also necessary to define the probability form H for W .

This can be done by defining the adjoint operator IH := ID. Two physically equivalentpossibilities have been considered:

(1) I is an isometry S→ S. Then IH := (I)† is the left inverse of I,

IHI = 1, IIH = Grade1 = (Grade1)2. (4.15)

Grade1 is the projector on the first-grade subspace S1 ⊂ S, the range of I.(2) I and ID are related as generalized canonical boson creator and annihilator:

IDI− IID = 1. (4.16)

If I satisfies (1) then I′ := I√

Γ satisfies (2). The matrix elements for I differ between(1) and (2) only by factors depending on the strata. The choice is merely a choice ofnotation. Nevertheless such differences have practical consequences when theories are beingconstructed; a coupling may look natural with one formulation and unnatural with another.

In quantum theory it seems more natural to assume canonical commutation relationsas in (2) between I and ID than to assume that I is an isometry as in (1). The canonicalbrace relations (E:CANONICALBRACE) are adopted here. Then IH := ID has the formalproperties of a differentiator with respect to I. I adds a bar to the stack, I multiplies by thenumber of bars on top of the stack and erases one of them. It annuls an amonadic operand,since the number of top bars is then 0. The analogue of the boson number operator is

〈∆Rank〈 := IID, (4.17)

called the brank jumpc, which counts consecutive bars in its operand, downward from thetop to the first non-monad.

Statistors 〈p ∈ S of the form 〈p = 〈I〈p′ have grade 1 and are called bmonadicc.Statistors orthogonal to all monadic vectors are called bamonadicc and form a vector spacedesignated by S 6=1. They have the defining property

IH〈u = 0. (4.18)

IH is 0 on S 6=1.∆Rank counts ranks above the amonadic, in the sense that:


Assertion 3 If a is amonadic and 〈b = 〈In〈a, then 〈∆Rank〈b = 〈b · n, and 〈b has norm

b〈b = n! a〈a. (4.19)

Argument: Omitted.The canonical relations for creators and annihilators lead to a linear ladder of vectors.

The canonical relations for bracing and de-bracing lead to a fractal network of vectors,with a replica of the entire network growing from every node in the network.

About notation: The most familiar Clifford algebra in physics is that of the γµ of theDirac equation. These operate on the spinorial vectors of the hypothetical single electron.The elements of the Clifford algebra S are used here as vectors of a quantum system, notas operators on vectors. The two kinds of algebra can be isomorphic and can be relatedbut they cannot be identified. To reduce confusion, generic vectors in S will usually havethe root symbol s or ψ, while generic operators on S will usually have root symbols γ andω.

S is partitioned into nesting strata . . . ⊂ S[L] ⊂ S[L+ 1] ⊂ . . . (L ∈ N) defined by

S[L] = [PI’ ]LR, (4.20)

Here R serves as a trivial singlet Clifford algebra S[0] of stratum 0 to begin the construction.A more trivial Clifford algebra S[−1] of one element 0 is stratum −1. The lower strata ofS are shown in Table 1.1.

To show the close connection between Clifford algebra and set theory one may expressthe usual concept of a finite set as a commutative reduction Sc of the bstratified CliffordalgebracS:

Let the classical state space Sc ⊂ S be the set of vectors generated from R ⊂ S by Iand × alone. The brandom setc (understood to be finite) is I [S], where S is the stratifiedself-power set. Sometimes it is useful to extend the term quantum to systems whose vectorsdo not fill out a single vector space but a union of several vector spaces having only theirorigins in common. This is a way of partially suspending coherent superposition but stillallowing bincoherentc or classical superposition, which is mixing.

Since the set of rays in Sc is isomorphic to the self-power set S, I [S] ∼= I [Sc] is also, inthis extended use of the term quantum, the quantum object whose vector space is Sc. Thepredicates of I [S] are those of S that are diagonal in any basis B ⊂ Sc. The transformationsof I [S] are in 1-1 correspondence with the operators on S represented in one such basis bymatrices with elements 0 and 1 and with only one non-zero matrix element in each column.

4.1.5 Critique of the brace operation

The quantum brace operation resembles the pre-quantum one that is still in general use,in keeping with the principle of least change. But the pre-quantum brace is loaded withassumptions that contradict the core ideas of quantum physics.

4.1. SPINORS 175

In the first place, set theory is dedicated to a creation from nothing; I permits all its setsto be constructed from the empty set. This seems to be as much a reenactment of a certaintheological scenario as a representation of known physical processes. In quantum physicsthe creation operators represent multiplication by vectors, which in turn represent inputfrom the metasystem, not from nothing. In particle experiments, every particle creationis balanced by annihilations so that what happens is transmutation, not creation fromnothing. Processes that resemble a creation from nothing in physics might conceivably goon in the regions represented as singularities of the gravitational field in present cosmology,not because there is any evidence for them, but because we have too little evidence toexclude them. It is absurd to take a process that has never been seen as the foundationfor all physics, unless there is no alternative. It seems more likely that all the processesdescribed by bracing are actually results of binding, than the converse.

In the second place, the bracing operation of set theory accepts sets of arbitrarily highgrade and delivers a set of grade 1. It is therefore highly inhomogeneous, and reduces tooperations that accept sets of a given arbitrarily high grade and delivers a set of grade1. The quantum brace I, correspondingly, accepts vectors of arbitrarily high grade andproduces one of grade 1. This reduces into an infinity of irreducible operators that eachtransforms only vectors of one grade, annulling all other grades. There is no evidence at thequantum level of resolution either for such highly reducible operations, or for irreducibleoperations producing such high grade change.

Finally (for now), mathematical set theory does not undertake to represent the meta-system, including the mathematician, as a set. Perhaps cognitive science attempts to modelprocesses of the mathematician; but cognitive science is not considered to be mathematics.The quantum field physicist, however, ordinarily seeks a physical model that works forany system under study, including selected portions of what served as metasystem for aprevious study. The processes of mathematics need not include the life-processes of themathematician, but the processes of physics must include the life-processes of the physicist.There is no reason to doubt that the standard model has this kind of self-consistency; itsregularization can also have it, therefore.

If one were to give up the model of classical set theory entirely and replace all bracing bybinding, the resulting theory would use only one stratum. Then to account for a dynamicsvector space with 2N dimensions, an event vector space would need N dimensions, andN would also be about the number of events in the history of the cosmos. The idea thatsuch a large number is a datum rather than a result of internal combinatorial structureis uncomfortable for someone who believes in a fundamental theory without arbitraryconstants, or in design by an intelligence that cannot remember many digits; but it is inkeeping with the empirical principle.

Matter indeed exhibits a modular structure of several strata: the electron, the atom,the molecule, and so on up to the cosmos, with the human being near the middle. Butnot long ago a physicist as advanced as Mach regarded such structure as in principleunverifiable, and insisted on a continuum model of matter. Perhaps we are in a similar


position in regard to space-time today.If the analogy between space-time and matter is at all valid, then it is significant that

none of the material strata is constructed from lower ones by bracing today. The stratifiedstructure of matter, or of nature above the space-time stratum, results from binding withforces of various ranges, resulting from the exchange of various quanta, not from bracing.Two or three strata seem sufficient for this.

On the other hand, the theory of binding rests on the lower stratum of space-time,which is assembled by set theoretic methods. If space-time itself is crystalline, its structurecannot result from interactions like binding that require a space-time substrate. To accountfor so many interrelated events seems to call for combinatory processes as well as dynamicalones. So does the transition from a one-quantum theory to a many-quantum theory.

In sum, it seems impractical to entirely eliminate combinatory processes in favor ofdynamical ones. The strategy adopted here is the opposite, combined with structuralstabilization and stratification.

4.1.6 Queues

The (finite) queues form a stratified structure in which higher strata are formed from lowerby bIc, b∨c (union) and quantum superposition b+c. The rank of a set in this hierarchy isthe number of nested bracings in its construction. Stratum L is the direct sum of all ranksr ≤ L.

For example, two elements a, b ∈ S[L], the Clifford algebra of stratum L, may havegrade 2 in S[L], and the generators Ia, I ∈ S[L+ 1] have grade 1.

The bqueuec is the quantum system I[S]. The first few strata of the queue are shownin Table 1.1.

The Clifford algebra S[0] of stratum 0 is R as a full matrix algebra of 1×1 real matrices.The Clifford algebra S[1] is a full matrix algebra of 2× 2 real matrices.The Clifford algebra S[2] is a full matrix algebra of 4× 4 matrices over R.The Clifford algebra of every stratum L > 3 is a full matrix algebra over R of 2L-square

matrices, generated by the 2L anticommuting elements proper to stratum L and all lowerbstratumcs. Thus S[4] has 16 generators, all shown. S[5] has 65536 generators, 30 shown.

The SO[L] form a nested sequence of broken groups:

SO(1) ⊂ SO(2) ⊂ SO(3, 1) ⊂ SO(10, 6) ⊂ . . . . (4.21)

This vector space S is infinite-dimensional. Free use of S and operators on S wouldallow infinities into the theory. We therefore restrict ourselves to some finite stratum S[L].L = 6 serves for the present initial study.

For any quantum system I[V ], the queue on I[V ], written as 2I[V ], is the quantumsystem with vector space 2V := Cliff V .

4.2. SIMPLIFYING QUANTIZATION 177

When we wish to emphasize the Clifford algebraic structure of these sets, we call themcells, with vertices of I[V ]. The Clifford norm gives squared Euclidean measures for cells(rectangular parallelepipeds).

We must find bclassical logicsc within the quantum logics, since they work sometimes.Classical sets can be imbedded within queues using the following concept of classical frame:

A classical frame C ⊂ S is a basis of S that includes 1 and is closed up to sign under× and I. C is unique up to the signs of its elements.

To reconstruct the classical set we introduce a superselection law:The classical set variables are those that are represented by diagonal operators in a

classical frame.The classical random set is the random variable object having the classical variables

as its variables. Then the classical random set is I [C]. The quantum theory of probabilityfor the queue reduces to the classical theory of probability for the random classical set.

The queue produced by L recursions is called the queue I[S[L]] of bstratumc L. Wewill use only low strata, L ≤ 6, in the applications that follow.

We define the 0-stratum queue by its vector space, the one-element Clifford algebraS0 = 0. We then iterate the functor Cliff I.

Rank 1 of queue algebra has vector space S1 := PS0 = R, since the product of nofactors is 1. The Clifford norm of 1 is 1.

Rank 2 has as vector space the two-dimensional Clifford algebra S2 = 2R with positivedefinite Clifford norm.

Rank 3 is four-dimensional and has a Minkowskian signature, since the product of twoanti-commuting monadics of positive square is a dyadic of negative square. As a bilinearspace,

Cliff30 = M, (4.22)

the Minkowskian bilinear space. All higher bstratumcs also have indefinite norms.

The signature is the square root of the dimension, σ =√D, for every stratum L ≥ 3.

This follows from the fact that there are√D more symmetric basis matrices than skew-

symmetric, namely the diagonal ones. Thus stratum 4 has dimension 4 and signature 2,like Minkowski space-time, and stratum L as L → ∞ has dimension approaching ∞ andsignature negligible compare to the dimension, σ/D → 0. For the fermionic algebra ofannihilators and creators the signature is 0; in this sense Clifford becomes Fermi in thelimit of infinity dimensions.

4.2 Simplifying quantization

The heuristic process of simplification by quantization might consist of the following steps:


4.2.1 Choose a simple Lie algebra

adjacent to each bcompoundc Lie algebra of the singular theory. This generally requiresappending extra variables: variables appended to the generators of a kinematical Liealgebra so that it has a simple Lie algebra in its infinitesimal neighborhood. A Lie algebrahas no simple one nearby if its dimension is not that of a simple Lie algebra. In suchcases one can adjust the dimension by adding variables, but must then posit organizationsto freeze out these variables in present experiments. One could stay closer to presentexperiment by eschewing such variables and accepting a nearby semisimple Lie algebra.This amounts to suspending quantum superposition at some low bstratumc of catenation,indicating a residual frozen organization. Requiring simplicity forces a resolution of suchorganizations into quantum elements.

4.2.2 Choose a vacuum organization

to freeze any extra variables to constants, like aligned spins in a ferromagnet. One may beable to correlate some structural instabilities of present-day physical theory with vacuumorganizations already posited for the bstandard modelc and inflationary cosmology.

4.2.3 Choose a faithful irreducible representation

(FIR) of the semisimple Lie algebra. Generally there are infinitely many.After these choices the simplified theory is finite, since all divergent integrals have

been replaced by finite traces. Canonical quantization, based on the Poisson Bracket,quantizes only the dynamical variables of one stratum (E or F), leaving the infrastructureunanalyzed, classical, and structurally unstable. Canonical quantization thus disrupts someinter-stratum cell relations. To preserve relations between two strata, one must quantizeboth or neither. Full quantization completes quantization on the dynamical stratum andextends it to the space-time stratum. It can therefore preserve inter-stratum structuralrelations.

The bBohrc correspondence principle gives the action quantum ~ a central role: Itrequires quantum predictions to approach classical as ~→ 0; which means, more explicitlyput, for a sequence of experiments with action scales A/~→∞ .

Full quantization is easier to rationalize than canonical quantization. Since measure-ment always has margins of error, it is reasonable to require a theory supposedly foundedon measurement to be insensitive to small errors; to be structurally stable. A structurallyunstable theory expresses a faith in events of probability 0 that goes beyond experiment.

In addition it leads to dynamical instability and infinities. The simple algebras offully quantum theories, on the contrary, are structurally stable and have complete setsof finite-dimensional representations, which give finite predictions for all observables. Wetentatively restrict ourselves to subsets of S to insure finiteness and reduce the number ofpossibilities.

4.2. SIMPLIFYING QUANTIZATION 179

We sometimes use the tilde x and the circumflex x to designate a skew-symmetric anda symmetric operator respectively.

If Xp is a simplification path in the space of structural tensors, the Xp for differentvalues of the parameter p are isomorphic to each other and not to X0. One often speakselliptically of a path of Lie algebras ap instead of products Xp.

The historic simplification parameter is lightspeed c, the parameter for the path fromthe simple Lorentz Lie algebra to its singular limit, the bcompoundc Galileo Lie algebra[47].

Simplifying quantization is an inverse of a contraction from a simple Lie algebra toa singular limit. It generalizes and extends canonical quantization to strata below thedynamical, and replaces the canonical Lie product X0 by some other Lie product Xp with

X0 =[d

dpXp

]p→0

. (4.23)

Flato (1977) formulated a concept of bdeformation quantizationc that does not takestructural stability and stratification into account but retains a structurally unstable clas-sical manifold with central coordinates [9], [41].

A singular limit is usually called a “contraction” of the regular Lie algebra [47] althoughit is infinite where the regular one is finite. The term originates in the fact that the singularand generic commutation relations agree well only in a neighborhood of the identity; buteach approximates the other and so could be considered a contraction of the other if thatwere the intended sense. Implicit in the name “contraction”, therefore, is a priority of thesingular over the regular. For example, a contraction may convert a compact group to anon-compact, and the term “singular limit” seem more descriptive for the present use.

Simplification is a special case of what is usually called “deformation” or “jump de-formation”, pejoratives expressing a preference for the singular. We simplify in order toreform the theory, not deform it. It is the classical theory that is deformed.

The bcompoundc Lie algebras of physics often contain an element 1 that is representedby a scalar matrix in the algebra of the theory. In the simplified variant it becomes anextra variable. These variables must have hidden from past experimenters. To accountfor this one may also adduce some ambient spontaneous symmetry breaking, degeneratevacuum, or, in the usage of Laughlin [53], organization, analogous to ferromagnetism, ofthe kind already suggested by the Higgs mechanism and Guth inflation. The organizationis required to freeze the extra variables. Disorganization can thaw them and make themobservable.

There are semisimple Lie algebras of every dimension, for one can make a Lie algebrageneric, whatever its dimension, by a generic homotopy, and if a Lie algebra is generic, itsbKilling formc is regular, implying that the Lie algebra is semisimple.

Simple quantization has problems. First, it leaves open too many possibilities. Thesimple Lie algebras near a given Lie algebra are rather few in number, but each stratum


of a theory has such choices, and the number of possibilities multiply. If the choices areindependent, the probability of making them all correctly is rather low.

Second, after choosing Lie algebras must one choose their matrix representations. Arepresentation is defined by several quantum numbers, depending on the Lie algebra, andthe number of possible values they can have is in principle infinite.

Third, the most promising statistics for the elements of the queue, the Fermi-Dirac an-ticommutation relations, define a Clifford algebra over a neutral bilinear space. This is nota Lie algebra but a graded Lie algebra. A Clifford algebra of finite dimensions has a uniquefaithful matrix representation that defines a finite quantum theory generalizing the theoryof spin. If we consider structural stability within the domain of Clifford algebras, thoseClifford algebras with a regular Clifford norm on their vectors are the structurally stableones. The Clifford algebras have unique preferred representations just like the bHeisenbergcalgebras, eliminating the infinite choice left open by the simple Lie algebras in general, andfinite-dimensional. And finally each Clifford algebra C has a useful orthogonal Lie algebrain its second-grade subspace C2, defining a bPalev statisticsc, and another in its Lie algebraof isometries C → C. This points rather clearly to a much stronger quantization processwhich we take up next.

4.3 Fermi full quantization

In bFermi full quantizationc one greatly constrains the choice 4.2.1 by requiring the sim-plified theory to have a finite-dimensional subspace C ⊂ S of the self-Grassmann algebraS for a vector space. This eliminates the choice 4.2.3 altogether, since a Clifford algebrauniquely determines a faithful irreducible representation.

The physical interpretation of such a theory has already been sketched: The additionin C is quantum superposition, already discussed. The scalar element 1 (projectively)represents the empty set of systems. A top element > ∈ C represents the full set. Theproduct operation combines systems into a plurality. The Clifford law ψψ = ‖ψ‖ producesa scalar, a multiple of the vector 1 for the empty set. This gives rise to the exclusionprinciple.

It is natural to hope that C is some entire stratum S[F] ⊂ S. One must still choose thestratum F, but the dimension grows so explosively with stratum that if we take the leastusable stratum, even a rough count of dimensions will fix this choice. In the present modelsstratum 6 suffices for the largest system that we can measure at the quantum stratum ofresolution.

Bose statistics is defined by a canonical Lie algebra and therefore fails the Segal struc-tural stability criterion. bFermi statisticsc is not defined by a Lie algebra in the originalsense, however, but by a Grassmann algebra, a graded Lie algebra and requires separateconsideration:

The Grassmann Relation v2 = 0 for v ∈ V is structurally unstable in the manifold of

4.4. FULLY QUANTUM VACUUM ORGANIZATION 181

Clifford algebras on the vector space V , for which v2 = ‖v‖ can be any quadratic form.The regular Clifford algebras are those with regular quadratic form. Grassmann algebra isa singular limit of the regular Clifford algebra with x2 = C‖x‖ as C → 0.

The Fermi operator algebra on GrassV , however, is stable against these variations,and is used for all physical predictions. Therefore bFermi statisticsc is singular for presentpurposes. While full quantization changes Bose statistics into a Palev statistics, it respectsFermi statistics. . The two main Clifford algebras of the standard model, the Dirac spin1/2 Clifford algebra defined by the time form and the Fermi statistics Clifford algebradefined by the probability form, are both subalgebras of Alg S, one on stratum 4 and theother on stratum L→∞.

Each epoch requires its own stability construct. For example, Segal stabilized theoriesonly against variations within the manifold of Lie products, preserving the Jacobi identityand co-commutativity. Since these idealizations cause no infinities in the present theory,I retain them. Perhaps a closer critique of the operational basis of physics will eventuallyjustify variations that convert Lie algebras into Hopf algebras, and groups into quantumgroups, as in the theory of harmonic analysis on non-abelian groups [67].

Any bgauge theoryc has singular constraints that arise from the vanishing of a bHessiandeterminantc, clearly a structurally unstable condition. A bfull quantizationc removes thesebsingularitiesc too. Because the regular algebra can be arbitrarily close to the singular one,bfull quantizationc can preserve the experimental meanings and agreements of the quantumtheory in the present experimental domain, but the regular and singular theories eventuallydiverge greatly. To be sure, the internal Lie algebra

aSM = su(1)⊕ su(2)⊕ so(3) (4.24)

of the bstandard modelc is only semisimple. While this is sufficient for structural stability,it is natural to consider whether this semisimple Lie algebra results from a simple onethrough centralization — “superselection” — relations resulting from organization; queuetheory is supposed to describe the disorganized vacuum as well.

4.4 Fully quantum vacuum organization

There is evidence of spontaneous vacuum organization that results in a vacuum multipletprojector 〈Ω〈 of multiplicity

Dim 〈Ω〈 = Tr 〈Ω〈 > 1, (4.25)

at least in the canonical limit. To fully quantize the concept of a self-organized vacuum,first the canonical theory is reviewed.

In a canonical field theory, a vacuum is defined by an instant vector ψ0 that is aneigenvector of the system Hamiltonian with minimal eigenvalue.

080522 . . . . . .


One vacuum organization that we require is a non-vanishing vacuum expectation valuefor i2:

Ω〈(ωF56)2〈Ω =: −N 2 6= 0. (4.26)

This implies that the vacuum vector is invariant under neither ωFµ6 nor ωF

µ5:

〈ωFµ6〈Ω 6= 0 6= 〈ωF

µ5〈Ω. (4.27)

It is natural to assume that the dynamics 〈Psi has these symmetries.

4.4.1 Stratum assignments

Some properties found on one stratum derive from lower strata and some from higher. Foran electron in a ferromagnet, an SO(3) symmetry of stratum E is induced from stratumD and broken on stratum F. Gravitation is assumed in ϑo to resemble magnetism in thisrespect: It has symmetries on stratum E that are induced from stratum D and broken onstratum F.

The question then arises for each construct employed on a given stratum whether itis to be constructed from a lower stratum by bracing and catenation, or is proper to thestratum itself, or is a reduction from a still higher stratum. It is possible for some constructA of a stratum L to figure twice in the next stratum L + 1, once in an binducedc versionA′ = ΣA from stratum L, using the bcumulatorc Σ of (3.20); and once in a breducedcversion A′′′ = Av ΣΣA of a construct A′′ = ΣΣA of the higher stratum L+ 2:

Rank L+2 A′′ = A′′

Σ ↑ ↓ Av

Rank L+1 A′ A′′′

Σ ↑Rank L A

(4.28)

A theory keeping only stratum L + 1 loses the physical connection between A′ and A′′′.Then if A′ and A′′′ are too different, they may seem to be unrelated, and if they are tooclose, the difference between them may be overlooked. This is not an academic possibilitybut happened for coordinate time and proper time, which agree for two experimenter eventsbut not for two system events.

A fully quantum correspondent of a canonical bfieldc theory must have a vector space ofdimension great enough to cover the same experimental domain as the canonical theory. Wecannot measure the cosmic queue at the same level of resolution as a small region. All buta logarithmically small part of the cosmos is the metasystem, including the experimenter,and must be described at low resolution to protect it from destruction. In the present workonly small regions are described with maximal quantum resolution.

4.4. FULLY QUANTUM VACUUM ORGANIZATION 183

Each stratum S[L] has a multiplicity (number of dimensions in its vector space) ML

and a maximum grade ML−1, attained by its top polyadic

s> := sMLsML−1 . . . s1 (4.29)

Call the expectation value of the number of events in the vacuum set history the beventnumberc:

NE := Tr[〈Ω〈sHα〈sα〈

]. (4.30)

Since the Planck time is about 10−35 s and the Cosmological Time is at least 1021 s, it isconservative to say that

NE > [1046]4 ∼ 10184 ∼ 2552; (4.31)

this leaves out the field variables at each event. Then there must be at least 2552 kinds ofevents for so many to exist in one set.

The lowest stratum in Table 1.1 with that many different possible events is stratumE = 6, of dimension 264K. The event vector space Grade1 S[6] is tentatively adopted for theevents in the queue. The differential stratum can then be taken to be stratum D = 5. TheLie algebra sl(16) of stratum C = 4 accommodates both the Segal-Vilela-Mendes so(6;σ),including a quantized imaginary

i[C] = ω[C]65 /l, (4.32)

and an so(10), hopefully adequate for GUT, leaving the signatures to be determined. Wetake stratum C = 4 for the sake of the sl(16) invariance of its monadics.

The example bSegal-Vilela-Mendes spacec shows that canonical space-time coordinates,momenta, and i can be represented as singular approximations to high-dimensional repre-sentations of 9 of the 240 generators of sl(16), the group acting as its defining representationon the monadic vector space Grade1 S[4], and that no lower stratum suffices.

In this model the differential stratum D=4 is the highest microscopic stratum andstratum E=5 is the lowest macroscopic stratum.

It is harmless to use a stratum A=1 with vector space S[1] = 2R as the seed of thecomplex plane C for the amplitudes of complex quantum mechanics. The generator ofso[A] can serve as the seed for the imaginary i of canonical quantum dynamics. The vectorspaces S[L] are tentatively used in ϑo to support the following significant groups and vectorspaces, with the given classical correspondents:

L S[L] so[L]

A = 2 2R SO[A] = U(1) = complex phase groupB = 3 4R so[B] = Lorentz group

charge C = 4 16R so[C] ⇒ Poincare, C-gauge groupsdifferential D = 5 216R so[D] ⇒ ?

event E = 6 26R so[E] ⇒ udiff, E-gauge groupsfield F = 7 27R so[F] ⇒ canonical, unitary groups

metasystem G = 8

(4.33)


The canonical group referred to for stratum F is the canonical group of classical gravityand the unitary group of the standard model field system. Rank G is assigned to the]ixmetasystem, discussed in §1.7.5.

This assignment of Lie algebras and therefore of groups has a conspicuous incongruity.The strata of the fully quantum theory nest, and therefore so do their groups and Liealgebras. The strata of canonical physics do not always nest: A coordinate differential isnot an event coordinate, an event coordinate is not a dynamical field. But the Lie algebranestings B ⊂ C ⊂ E ⊂ F agree with the canonical theory, and the others seem harmless.

The Einstein group Diff(4) is too large to be well approximated on stratum 4, whichhas dimension only 216 = 64K . Therefore stratum 4 is used as a filler between strata Cand E. It seems harmless to associate it with the differential stratum D of classical gravity.Then Diff(4) can be an approximate singular limit of SO[E] ∼= SO(2216

; 64) acting on the2216

-dimensional event vector space.A field is usually represented as a set of pairs of events and field values. Following the

bKaluza strategyc the field variable is another event coordinate. It is natural to assign thequeue corresponding to the field to stratum F = 6, the successor of rank E in (4.33).

The canonical group of gravity theory is generated by gravitational potentials andtheir canonical conjugate momenta modulo constraints. It can then be approximated, oneexpects, by a singular limit of SO[S[6]].

Chapter 5

Quantum space-times

which examines early quantum space-times and formulates a fullyquantum one.

Theoretical physics today is a curious sandwich, with a thick upper slice of classicalbread, the macroscopic world of metersticks and stars; a thin slice of quantum cheese, themicroscopic world of quarks and bgaugec quanta; and a thick lower slice of classical bread,the submicroscopic worlds of space-time events and their differentials, classical all the waydown.

The upper disconformity, between the quantum microscopic stratum and the classicalmacroscopic stratum, is understandable. It happens when we pass from fine quantum res-olution to coarse macroscopic resolution, and can be shifted with some effort by improvingour experimental resolution of the part of the bmetasystemc just above the disconformity.

A maximal observer of quanta cannot be maximally observed. To transform changesof one quantum into macroscopic changes, there are necessarily unstable elements in anyobserver of quanta, like the visual purple in the eye. Maximally observing the observerwould cause some of these elements to discharge, disturbing the observation.

Since the bclassical logicsc are reductions of the quantum logics, one can claim tobe using a quantum logic even when one does classical logic, attributing the approximatecommutativity of the macroscopic stratum to the bLaw of Large Numbersc, as one attributesthe approximate absoluteness of time in daily life to our low celerity (cosh−1(v/c)).

It is supposed here that the lower disconformity also results from a lack of experimentalresolution, now of space-time events, but this one is practically immovable at present. Wecan supply reasonable quantum models of classical laboratory instruments, but not ofclassical space-time events.

This is probably a failure of imagination. If there are space-time atoms they are surelyvery small. There is surely no physical reason for smaller things to be less quantum than

185

186 CHAPTER 5. QUANTUM SPACE-TIMES

larger. Classical geometry, whether Euclidean, Minkowskian, or Riemannian, is thereforedubious for microphysics just because it is an exercise in a classical logic, which is amacrophysical logic.

One way to repair the lower disconformity is by a bfull quantizationc, replacing classicallogical elements of space-time geometry by quantum correspondents.

One might consider performing a full quantization by expressing a canonical theoryin formulas of classical logics and reading them in quantum logic. This does not workat all. Much of the logical structure that is used in a systematic set-theoretic model ofclassical geometry lies far below the stratum of observation of classical physics and hasalmost nothing to do with experiment. It is invented merely to build up a continuumfrom the discrete binary elements natural to classical logic. This infinite infrastructure canbe set up in an infinity of ways that differ mathematically in ways that make no physicaldifference, since there is no trace of any such classical set-theoretic infrastructure in nature.From the viewpoint of physics most of the lower strata of the present physical theory arejunk theory, and provide no guide to lower strata of nature.

Conversely, there is no trace of the continuous symmetry of space-time such as Lorentzinvariance in its axiomatic infrastructure, classical set theory. Such invariances first appearin the infinite classical construct.

Thus all this infinite classical infrastructure is inappropriate in a fully quantum formu-lation, which incorporates continuity into its probability amplitudes from the start. Mostof the classical logical infrastructure can therefore be left behind. One point of a full quan-tization is to replace this imaginary infinite classical infrastructure by a finite quantum onethat has at least the potentiality of representing natural processes, which are quantum.The model ϑo replaces an infinity of classical strata to the five quantum strata B – F.

On the other hand, the key classical concept of equality a = b as a relation between twoset variables has no invariant quantum correspondent as a relation between two queues.To be sure, the relation of symmetry, expressed by a symmetrizing projector P+, is usuallyregarded as expressing identity in bosonic statistics but it lacks the important property oftransitivity, and does not imply that two quanta in this relation have the same values fortheir variables. One needs a more knowing guide to the microcosm than classical logic.

Historically, group theory has been used from the start to move the house of physicsfrom its classical foundations to stabler quantum ones. Groups entered physics silentlydecades before physicists began to speak of the “group plague”. Canonical quantizationitself is group-directed. It is set up to preserve isomorphically the 7-dimensional Lie groupgenerated by the three coordinates qk and three momenta pk of each electron, simplyreplacing the classical Poisson Bracket by the quantum commutator times i/~. It furtherreconstructs the larger canonical Lie group, which transforms functions of the classicalcoordinates and momenta, as a singular limit of the unitary group of a Hilbert space.

On the other hand, set theory infiltrated quantum physics almost as early as grouptheory. Quantization was designed to preserve certain membership or product relationsamong sets, simply replacing Cartesian products by Grassmann products. The classical

187

Rutherford atom A is the set whose elements are the nucleus N and the electrons en, asfar as its phase space is concerned:

A = N, e1, . . . , en = N ∪ e1 ∪ . . . ∪ en. (5.1)

It was still modeled as a composite of its nucleus and its electrons in the early days ofquantum theory, with Grassmann products replacing Cartesian products. When we formGrassmann products to make multi-electron orbitals we are still tacitly representing clas-sical set structure within the fully quantum theory.

Every simple Lie group mathematically defines both a quantum praxis and a quantumstatistics, though most of these have had no physical application. The predicates of thefirst-order praxis are the projection operators of the bilinear space on which the Lie groupacts in its defining representation. The statistics is defined by the same commutationrelations that define the Lie algebra. This is the class of bPalev statisticsc recapitulated in§5.2.7.

To be completely constructive, fix attention on the defining representation of each ofthese simple Lie algebras as the isometry Lie algebra of a quadratic space. The quantumpredicates of the praxis are then represented by idempotent matrices in the same matrixalgebra V ⊗V D as the defining representation of the Lie group. The set-theoretic structureand the symmetry group structure are both traces of the same quarry.

Segal too explicitly used group structure as guide to the quantum underworld. Inbsimplification by quantizationc, one replaces every bcompoundc (= non-semisimple) Liealgebra in the formulation of a theory by a simple one of which it is a singular limit, asdid Vilela-Mendez [75]. We describe further examples of bsimplification by quantizationcbelow (§5) and then specialize to full quantization. In these instances of simplificationby quantization, unfortunately, the physicist must not only choose a nearby Lie algebra,sometimes of rather small dimension, but one must also choose a representation of that Liealgebra, typically of huge dimension. The choice of representation is then embarrassinglyrich. Full Clifford quantization narrows the choice to a line of Lie algebras, those of thereal special orthogonal groups, especially those resulting from Clifford bquantificationc.

Here the possibility is explored that all quantum kinematics is derived from quantumstatistics; that the high-dimensional representations of low-dimensional Lie groups thatoccur in particle and space-time physics encode structural information as in atomic physics.A catenation of many identical systems having the defining representation of the same Liegroup as their bkinematical groupcs has a high-dimensional representation algebra of thisgroup as its bkinematical algebrac. Indeed, once one fixes on a family of statistics it can beiterated, for example to form a Fermi catenation of Fermi catenations. This generates a lineof representations of exploding dimensionality, all in the same Cartan family, which maybe adequate for present-day physics, granted singular limits and organizations. Choosinga representation from this line is easier than choosing it from the jungle of representationsthat bsimplification by quantizationc leads us into.


bClifford statisticsc (§3.5) seems especially appropriate for such iteration, since it is astructurally stable variant of bFermi statisticsc, contains both Fermi and Bose statistics assingular limits, and includes the Dirac spin Clifford algebra as a low stratum S[4] in thesequence it generates.

5.1 Problems of classical space-time

5.1.1 Structural instability of time

The bGalileoc Lie algebra of rotations and boosts is the limit (as c→∞) of a sequence ofLie algebras all different from itself as Lie algebras. The commutative algebra of classicalmechanics is also the limit (as ~ → 0) of a sequence of isomorphic Lie algebras differentfrom itself. So is the canonical Lie algebra. Such Lie algebras are said to be bstructurallyunstablec.

Variables that commute with all other variables in the algebra form the bcenterc ofthe algebra and are called bcentralc. In standard field theory, space-time coordinates arecentral. Structural stability requires non-central space-time variables. Central space-timecoordinates are unnatural in the canonical theory of bgravityc too, for what seem to besimilar reasons of structural instability [11, 10].

The structural instability of the Galilean Lie algebra could have told us that it wasalmost certainly an approximation to a more physical stable Lie algebra long before thiswas discovered by more experimental arguments. Now the structural instability of thebHeisenbergc commutation relations tells us that canonical quantum theory and Bose statis-tics are almost certainly approximations, not the end of the quantization trail.

5.1.2 Dynamical instability of time

Structural stability correlates with dynamical stability. If the Lie algebra elements areto be observables, some compound groups of present physics force us to infinite-dimen-sional matrices, most of which are undefined on most vectors. In such theories an energyspectrum can be unbounded below, like those of the classical hydrogen atom and theDirac one-electron quantum theory. Thus structural instability permits the dynamicalinstability of unending radiative decay. Any nearby semisimple group has finite-dimensionalrepresentations in which all observables have bounded and finite spectra. In such theoriesdynamical instabilities are impossible. The work at hand is to present the standard physicaltheories as singular limits of such a stable theory, and of one that is also more physical.

Almost all quadratic forms are regular. Almost all matrices have inverses. Regularityis normal, singularity is singular. Experiments, however, have error bars; experiment isgeneric. Therefore a singular theory cannot be based entirely on experiment, which isalways generic, but must also postulate some structure of probability 0, often an idol in

5.1. PROBLEMS OF CLASSICAL SPACE-TIME 189

the Baconian sense rendered invisible by habituation. This idolization facilitates somecalculations but also corrupts them; infinity in, infinity out.

Present physical spaces have infinitely many infinite tangent planes and produce in-finities. Simple quantum spaces are built from finitely many finite quantum elements likespins and produce finite answers.

Present-day quantum bfieldc theories assume space-time coordinates of infinite preci-sion and range, with no complementary variables. They assume an empty backgrounduniverse with a high symmetry, but a symmetry that is broken by the contents of theuniverse. These assumptions are vestiges of celestial mechanics, where the probe processhas negligible effect on the process probed, and the system has small effect on the metricalstructure of the space-time [25]. There seems to be need for a post-canonical quantumtheory even less commutative and structurally more stable than quantum theory. It isnot yet clear which present experimental processes best expose the hypothesized quan-tum structure of space-time; it is assumed here that all polyadics act on this structure.This gives the physicist license to hypothesize possible experimental operations and fitthem into the framework of quantum kinematics, leaving the manner of their experimentalimplementation for the future; as Dirac did when he invented his theory of electron spin.

Quantization gave us much understanding of the Periodic Table of the Elements butwe have not had as much success with the Particle Table. One major difference is that onenumber, the atomic number, fixes the position of a chemical element in the Periodic Table,and determines its main properties in a systematic way, while it takes several numbers to fixthe properties of a particle systematically. This seems to be because one group dominateschemistry, the rotation group; one statistics, Fermi; and one mass, the electron. The Lenzgroup expressing the conservation of the perihelion is an approximate symmetry group,valid for the non-relativistic Coulomb central potential, and also influences the PeriodicTable, but less than the rotation group. Several groups and two statistics and severalmasses enter into the Particle Table, according to the bstandard modelc. GUT provides aPeriodic Table of the Particles based on so(10).

These approaches do not analyze the space-time continuum but hang additional di-mensions onto it, quantum or continuous. The fully quantum event space is built up ofsimple quantum cells. Particle theories like the standard model and GUT can suggeststructures for these cells.

Relativity already provides particles with a deeper and still unresolved internal struc-ture. Relativistic mechanics represents an elementary particle as a curve in space-time, itspath or history. This can be composed out of a linear sequence of differential elements dx,by integration. These differential elements are its internal structure. A compound particlelike the Solar System is represented classically by a braid of such space-time curves. Ifspace-time is quantum then under high resolution, quantum particles resolve not into suchcontinuous braids but into cell networks of complex quantum space-time processes.


5.2 Earlier quantum space-times

In canonical theories, energy i~ d/dt generates a change in time t and conversely, in virtueof the relation

[E, t] = i~1, [1, E] = [1, t] = 0, (5.2)

although t and E are not observables. This defines the singular bcanonical Lie algebrach(1) := a(t, d/dt, 1). One structurally stable variant is Segal’s so(3;σ): [t, E] = r, [r, t] = E,etc., with extra variable r much like t and E, and signature σ. This time is a quantumvariable, one of the three generators of so(3;σ), and has discrete, bounded, and uniformlyspaced spactrum, like any angular momentum. The passage of this bquantum timec is noteffected by a one-parameter Lie group. Energy is another such variable, and so is r, thecommutator of time and energy, which is supposed frozen to the immediate vicinity of i~in the vacuum queue.

Such a theory can still represent the dynamics of the system through a generic history-amplitude instead of a time-translation generator. A history is typically a queue of a greatmany spinlike input and output actions, in which the observed system is an excitation.

In canonical physics we analyze the bfieldc into couples (x, f) of space-time events andfield values, coupled by a compound group. ϑo replaces the field by a queue with still morecoordinates, now coupled by a simple group. The queue events have momentum-energyand angular momentum as well as space-time coordinates, and in general each referenceframe in the vector space resolves these variables differently into space-time and energy-momentum.

5.2.1 Event energy

Simple quantum relativity continues Einstein’s physicalization of geometry. It supplementsformerly geometrical event coordinates, like position in space-time, with apparently dy-namical event coordinates, like momentum-energy. Quantum events like those of bSnyderc,bVilela-Mendesc, or Baugh have, in each admissible frame, not only space-time coordinatesbut also momentum-energy and other coordinates, mixed by the simple invariance group.

This is counterintuitive, for it relativizes the construct of absolute space-time pointthat has pervaded physics at least since Aristotle. Events are ordinarily supposed to havespace-time coordinates but no momentum-energy coordinates. Hume pointed out that wenever experience space or time by themselves. It is reasonable to ask how the notion ofan event as a purely spaciotemporal entity, with no other mechanical properties, enteredphysics and why it has lasted, when we never encounter such an entity in our experience.

The root of “line” means linen string and the root of “point” means puncture. Theserecall that geometry sprouted from annually flooded fields along the Nile, with strings ofEgyptian linen as its lines and sharp stakes piercing the ground as its points. A typicalBaconian idolization seems to have occurred: Surveyor’s stakes and strings have well-defined momenta, but their momenta are small in the terrestrial reference frame because

5.2. EARLIER QUANTUM SPACE-TIMES 191

they and the observer were both attached to one well-organized condensate, the Earth.These small momenta were first tacitly taken to be exactly zero, and then dropped fromthought completely, due to habituation. The idea of non-mechanical point emerged fromexperience with mechanical ones whose masses and momenta could often be ignored.

In space-time too one never encounters a massless event. To arrive at the space-timeconcept of today from actual space-time happenings, which all carry momentum-energy,we must have approximated the momentum-energy by 0 and then forgotten it.

One now corrects this idealized construct of empty space-time by attaching an energy-density tensor to each event. For other media, such tensors are surrogates for an underlyingatomism but space-time defies analysis into classical atoms in any relativistically invariantway. Here a quantum analysis is considered that accommodates the classical relativity ofspace-time within the quantum relativity of vector space.

As a first step, the simplicity principle, combined with Lorentz invariance, leads oneto restore the lost energy-momentum variables pµ — as Snyder already did [69] — and toposit a symmetry between xµ and pµ at the event stratum —- like Segal [66] — , broken inthe singular limit where pµ → 0, perhaps by an organization of events into space-time. Anorganization of space-time would now be expressed by a vector, fixed by the experimenter,not by the operator algebra, fixed by the system.

In canonical quantum physics, the dynamical phase-space of a theory is built on anallegedly deeper space-time, at least on a time axis; geometrical symmetries are regardedas fundamental symmetries of a hypothetical empty universe and are built into the dy-namical transformation group as a preferred subgroup; particles are characterized by therepresentations of the space-time group that they define.

In cellular q physics the macroscopically extended quantum space-time is a contingentorganization of the system queue. With no absolute space-time, there is no fundamentalspace-time group. The dynamical group is now simple and prior to the canonical singulargeometrical group. The system queue is a combination of event queues. Space-time sym-metries are at best phenomenological approximate statements about the ambient queue.

General covariance was indispensable for the creation of general relativity. It has anatural correspondent, fully quantum covariance (6.107), which implies covariance undera certain so(n;σ) for each stratum, with large n for the event stratum E. Full quantumcovariance is used in building a fully quantum theory of bgravityc in the way that generalcovariance was used to build general relativity. General covariance is treated here as not afundamental and exact law but as an approximate phenomenological statement about thecurrent ambient queue, that fails, for example, when the queue melts down.

Central space-time coordinates are built into the usual field theory, as part of thestrategy of absolute space-time. Consider the theory of bgravityc, for example. In hisgravitational action principle Hilbert varied bgravitational fieldc variables gµν(x) withoutvarying space-time coordinates x = (xκ). In the resulting Poisson Bracket Lie algebra, gµνcommutes with xκ, and the xµ are central in the Lie algebra of fields.

There are no central time coordinates in actuality, however. Our actual physical


coordinates xµ for a remote event are based on signals, usually electromagnetic, thatreach us through the intervening space-time and therefore inform us about the interveningbgravitational fieldc as well. The lattice of rods and clocks imagined by Einstein to coordi-natize events is not different in principle from optical coordinates since it too is connectedby electromagnetic interactions and modified by gravitational fields. Physical coordinatesare more relative than general relativity admits, being influenced by the ambient field aswell as the choice of reference frame. They are but field variables under another name, andnon-local as well. Since fields are non-central, so are coordinates.

5.2.2 Indefinite probability form

Fermi bfull quantizationc with real coefficients and a cut-off at a finite stratum leads to areal finite-dimensional vector space with no statistical metric. If the Lorentz group (forexample) is to be represented in this space, it cannot be by a unitary representation. Onemust learn to do fully quantum theory either with no absolute statistical metric or withan indefinite one.

The interpretation of indefinite probability forms is not yet fixed.One interpretation favored by bDiracc is that nature has good credit: its counters can

run into the negative and therefore so can probabilities.Another interpretation used in bGupta-Bleulerc electrodynamics, which also has a

vector space V of indefinite metric, is that negative-probability transitions do not oc-cur for other reasons. Each experimental frame can use a maximal subspace V+ ⊂ V withpositive-definite metric to represent feasible operations on the system. The complementarynegative-definite subspace V− is reserved to represent transformations that are unfeasible asexperimental operations but permissible as relations with other experimental frames; justas timelike vectors represent feasible translations of a mechanical system while space-likevectors represent unfeasible translations, which are nevertheless necessary to relate somespatially-separated observers.

The canonical theory of bgravityc has at least three strata D, E, F. The fully quantumtheory of gravity in ϑo imbeds these classical strata, with due regard for their cardinalities,in the strata of a fully quantum theory.

Here are some earlier quantum spaces that influenced this work.

5.2.3 Feynman space

bFeynmanc (ca. 1941) investigated quantum space-time before he developed his quantumelectrodynamics. He hoped that by eliminating arbitrarily small distances and times hecould avoid all ultraviolet divergences. The trouble with a classical lattice spacetime isthat it destroys the continuous symmetries of spacetime under translation, rotation, andLorentz transformation, leading to violations of the conservation of momentum and angularmomentum.


He considered a form of quantum space-time whose coordinates were sums of manyindependent commuting replicas of the Dirac operator-vector γµ [33]:

xµ = X[γµ(1) + . . .+ γµ(N)]. (5.3)

X is a fundamental subnuclear scale size to provide the physical dimensions of time. Thenthe four coordinates of a point do not commute with each other. This permits them allto have discrete spectra even though the theory is Lorentz invariant, forming a kind ofquantum lattice; just as all the components of angular momentum have discrete spectrumeven though their theory is rotationally symmetric. Nor are the coordinates xµ Hermitianin any definite metric; therefore they are not observable quantities in the strict sense.

In the queue models considered below there are many vector-operators γµ(n) but theymutually anticommute, generating a large Clifford algebra, instead of commuting.

The bFeynman spacec model shows that quantum space-time, regularity, and Lorentzinvariance are compatible if one sacrifices unitarity. Its time coordinate has a discretebounded uniformly spaced spectrum with a time quantum X.

Some propose on dimensional grounds that the time quantum X be identified with thebPlanck timec

TP =√

~G. (5.4)

Organized media typically have several scales of length and time, so the dimensional ar-gument is weak. Even an electron carries at least three very different empirical scalesof length, its Compton wavelength, its classical radius, and the range of its form-factor.We should allow the physical space-time queue as much. Indeed, all the lengths in theparticle spectrum are presumably characteristics of the vacuum. Dimensional arguments,moreover, cannot tell us how Newton’s G or Feynman’s X depend on the large number N .Therefore X and G will initially be kept independent, hopefully to be related later.

bFeynmanc and bHibbsc formulated a model of a quantum particle moving in a classicaldiscrete space-time similar in spirit to the Feynman quantum space [35]. It is usually calledthe checkerboard model and has also been studied by Jacobson [48] for example. Thecheckerboard is the infinite square lattice in which a piece moves like an uncrowned piecein the game of checkers: forward to the right or forward to the left. A rank of cells isinterpreted as a space line, a file of cells as a time line, and a diagonal of cells as a lightline, The cells accessible to a piece from a given cell form a discrete future light cone withapex at that cell.

On the one hand these models have no useful continuous symmetries and are thereforequite unphysical. On the other hand there is a surefire way of giving them the continuousorthogonal group symmetries they lack: One writes them in finite set theory and readsthem in queue theory. It is therefore worth examining the Feynman model further.

A path of a checkerpiece is represented by a sequence of position vectors

x(n) =(t(n)x(n)

)(5.5)


These define displacement vectors δ(n) := x(n+ 1)− x(n). By the rules of checkers, eachdisplacement δ(n) has one of the two constant forms

δ(n) =(

+1±1

)=: c± , (5.6)

corresponding to motion at light speed to the right or left.A motion-reversing operator R, with

Rc± := c∓, (5.7)

is useful.A Feynman path probability amplitude A is assigned to a path of such a piece by

summing contributions δA from each of the vertices (squares) x(n) in the path: A =∑n δA(n). The contribution δA(n) of vertex n to the sum is taken to be δA(n) = +1 if

there is no reversal during that segment, δA(n) = iM if there is reversal:

δA(n) = 1, for δ(n+ 1) = δ(n),= iM, for δ(n+ 1) = Rδ(n). (5.8)

iM is thus the probability amplitude for motion reversal relative to non-reversal. It followsthat a vector for one time can be expressed by an amplitude function of the coordinatesx, δ of a square and a displacement vector at that square, giving the location and motionof a piece. This vector will be written as a 2-component column vector

〈ψ =(ψ1

ψ2

), (5.9)

the 1- component giving the amplitude for a move to the right, and the 2-component tothe left. The amplitude function representing ψ is

ψ(x, δ) =: [x, δ〈ψ]. (5.10)

〈ψ propagates according to

[x(n+ 1), δ(n+ 1)〈ψ] = [x(n+ 1)− δ(n+ 1), δ(n+ 1)〈ψ]++ [x(n+ 1)−Rδ(n+ 1), Rδ(n+ 1)〈ψ : iM ]. (5.11)

A still more cellular version can give further insight into the Dirac equation. Considerthe two displacements c± of the checkerpiece as two abstract objects; call them bmovesc.Then a path on the checkerboard is a classical Maxwell-Boltzmann catenation — or se-quence — of moves. From this Maxwell catenation one forms a classical Bose catenation byignoring order, or symmetrizing over all permutations. This is the resultant displacementof the end of the path from the beginning. The two null coordinates n± of the end of the


path relative to the beginning are occupation numbers of this Bose catenation. The timeduration of the path is the total occupation number t = n∗+n−. The spatial extent of thepath is the difference x = n+ − n−.

The Feynman model lacks momentum variables and several strata, supplied in whatfollows.

5.2.4 Snyder space

bSnyderc space renounced geometric space-time in that its elementary events have energy-momentum and angular momentum variables as well as time and position. They mighttherefore be characterized as “particle-events”. bHeisenbergc had suggested that there isa bfundamental lengthc in nature, but attempted to incorporate it into non-linear bfieldcequations without modifying classical MInkowski space-time. Apparently following up anidea of Heisenberg communicated to him by bOppenheimerc, perhaps merely that of afundamental length, bSnyderc introduced new commutation relations for the space-timeposition coordinates xµ involving a fundamental length X, which became a quantum ofthe positional coordinates. This led him to a Poincare- invariant theory with a discretespectrum for each spacelike coordinate and a continuous one for each timelike and energy-momentum coordinate [69]. No working bfieldc theory was erected upon Snyder space,perhaps for lack of a guiding quantization principle. A theory like Snyder’s had beendiscussed by Pascual bJordanc and bvon Neumannc in 1938 without publication [79].

Later von Neumann expressed skepticism about such approaches and instead proposedto eliminate infinities with a bcontinuous geometryc for quantum physics. Continuousgeometries are more infinite than classical space-times, not less, in that even the dimensionsof their flats have a continuous range. Von Neumann supposed that continuous geometriesmight permit a finite physics because they lack “points”, projectors of a minimum non-zerodimension; but physics blows up at points of space-time, not at points of the vector space.The existence of sharp vectors is part of the solution, not the problem.

5.2.5 Segal space

bSegalc (1951) suggested a useful heurism for the evolution of physics [66]. He pointedout that quantum mechanics and special relativity both result from previous theories byhomotopies that carry singular compound Lie algebras toward simple ones. He showedthat simple – and even semisimple — Lie algebras are structurally stable against infinites-imal homotopies of their Lie product. He implied that canonical quantization is only thebeginning of a process of stabilization by simplification, and that the process should becompleted with simple groups. He gave two examples of simplification, h(1)⇐ su(2) andiso(3, 1) ⇐ so(6;σ); generalizations are clear. This implies that the quantum spaces ofphysics based on singular Lie algebras are singular limits of more physical quantum spacesbased on regular Lie algebras [66]. This includes all the quantum phase spaces resulting


from canonical quantization or Bose statistics, for example.In one illustration, here called the three-dimensional Segal space, Segal regularized the

canonical commutation relations for one coordinate variable q and momentum p [66]. Inthe canonical theory i is a central element of the real three-dimensional bcanonical Liealgebrac

h(1) : qp− pq = i, iq − qi = 0, pi− ip = 0. (5.12)

An irreducible representation of this Lie algebra in a complex Hilbert space defines whatmay be called a bcanonical quantum spacec. Segal simplified the canonical Lie algebra h(1)to a special orthogonal Lie algebra

so(3) : qp− pq = r, rq − qr = αp, pr − rp = βq (5.13)

with structure coefficients α, β. Any faithful irreducible representation of this Lie algebradefines a Segal space. The extra variable in (5.13) is r, which must freeze to a large valuein the canonical limit.

Similarly Segal suggested simplifying the Poincare-invariant bcanonicalc algebra h(4) =a(xµ, pµ, i) to an so(6;σ) algebra fixing a quadratic form of unspecified signature σ.

Let the six real basis vectors supporting this defining representation of so(6;σ) bedesignated by 〈α with α = 1, 2, 3, 4, 5, 6. When r freezes and breaks so(6;σ) symmetry,the first four variables will undergo the Lorentz group, the last two the SO(2) group of thecomplex plane.

The usual canonically dual variables xµ, pµ relate to the dimensionless generators ofso(6;σ) through a quantum of time X and a quantum of energy E:

xµ = XωEµ5,pµ = EωEµ6; (5.14)

and obey commutation relations

[xν , pµ] = 2XE δνµ ωE 65. (5.15)

This suggests that i, the fully quantized variant of i, is a component of angular momentum,suitably normalized:

i =1lωE 65 = ΣE−CωC 65 (5.16)

where −l2 is the maximum eigenvalue of [ωE65]2 .Presumably the real quantum time t is found by factoring the imaginary operator i

into the skew-symmetric operator t, the skew-time. This results in the anti-commutator

t =X

2l[ω65, ω45]−. (5.17)

Segal’s proposal influenced the classic retrospective studies of Inonu and bWignerc oncontraction and the Galilean limit [47], and Gerstenhaber’s cohomological theory of Lie


algebra stability [44]. These in turn had numerous consequences [36]. It seems that Segalpublished no further thoughts along this line.

Others subsequently found the bSegalc simplification h(1) ← so(3) independently ofSegal [5], [50, 52, 51],[59], [74].

It should still be verified that the chosen representation of so(6; ?) is a physically usefulapproximation to the canonical representation of the canonical Lie algebra h(4; 2). This istaken up in §5.3.2.

This left open such questions as how to choose a representation of the simple Liealgebras that arise in this way, how to cope with instabilities on other strata, how to expressquantum fields over such a quantum space-time, and how to express the dynamical relationsin the resulting kinematics. These questions are answered here by full quantization.

5.2.6 Penrose space

Following a separate path, Roger bPenrosec pioneered the quantum combinatorial approachto space-time by analyzing the sphere S2, with its infinitude of infinite tangent planes, intoa singular limit of an assembly of spins 1/2 with Bose statistics, representing the rotationgroup SO(3) in a double-valued way [60]. bPenrose spacec can be regarded as a non-relativistic variant of Feynman space, in that its point is a Bose catenation of Pauli spins1/2 instead of Dirac spins 1/2, and its coordinates are direct sums of Pauli spin matricesinstead of Dirac spin matrices. It too is quantum on only one stratum.

But there is no experimental distinction at any one time between an infinite planeand a sufficiently large sphere, and of the two it is the sphere that has the simple group.From the viewpoint presented here, therefore, Penrose quantized not only S2 but also itssingular limit, the plane R2. The conversion from dimensionless Pauli spin operators tolengths introduces a small length X. The expression of the coordinate variables

xk =N∑1

Xσk (5.18)

as sums of N spin variables, fixing the representation, introduces the large integer N . Thena Penrose space too has a fundamental quantum length and a large integer. NX sets thescale of the radius of the S2.

This work catalyzed many others. One effort reconstructed Minkowski space-time assingular limit of a Fermi assembly. It had the strata needed for field theory but lackedstructural stability and a quantization process connecting to the standard model [38]. Theyare provided here.

5.2.7 Palev statistics

bPalevc (1977) took a major step toward a generic post-quantum physics by simplifyingthe compound algebra of Bose statistics [59]. A bPalev statisticsc is one defined by the


commutation relations of a classical Lie algebra in a representation that approximates theBose statistics Lie algebra; or more formally put, in a sequence of representations that hasthe Bose commutation relations as a singular limit.

A bPalev statisticsc can be substituted for Bose statistics everywhere with only smallexperimental consequences in the present experimental regime. The most conspicuouseffect of this substitution is to cap the occupation numbers of one-quantum vectors. Thesize of this upper bound is a parameter that can be adjusted to fit the data. The Liealgebra of bPalev statisticsc defines a quantum space on which it acts, a bPalev spacec.

The Palev statistics in the orthogonal or D series looks like the real quantum theoryof a high-dimensional angular momentum in an irreducible representation. The imaginaryi must arise from one antisymmetric component of angular momentum, say ω56 to con-form with (5.16). ω56 cannot be diagonalized in the real theory. Define the non-negativesymmetric operator

|m| := +√−(ω56)2 .= 0, 1, . . . , l. (5.19)

(No operator m is defined.) (ω56)2 has a spectrum of the form

(ω56)2 := −m2 .= 0,−1, . . . ,−l2 (5.20)

with multiplicitiesd0 = 1, dm = 2 for m = 1, . . . , l (5.21)

The operator i must have two-dimensional invariant subspaces in the real theory. For realPalev statistics to approximate a canonical quantum theory, which is complex, l must bevery large, and the canonical quantum theory must work well enough in a subspace withm ≈ l. Call this subspace the bcanonical subspacec Vcan. Its dimension must be largeenough to pass for infinite and small enough for m ≈ l to hold throughout it. Let P [. . .]designate the projector on the eigenspace of the operator relation [. . .]. Then, for example,the subspace

Vcan = P [l2 − l < −(ω56)2 < l2]V ⊂ V , (5.22)

meets these criteria.This raises the question of what physical processes make m ≈ l. A reasonable hypoth-

esis, based on experience with organized matter, especially ferromagnets, is that organiza-tion of many small elements has occurred. This requires, first, that the angular momentumω56 be composed of a great many much smaller spins, and that these are aligned in thevacuum.

The quantum event of bVilela-Mendes spacec (§5.2.8) has a bPalevc coordinate algebra.The bPalevc Lie algebra is the second-grade part of a Fermi or Clifford algebra, so bPalevcevents can be formed out of pairs of Fermi events. It is diffidently assumed here that thisis the origin of all bosons. (§3.4.3, §6.4.1).


5.2.8 Vilela-Mendes space

Inspired by the Gerstenhaber theory of rigid (structurally stable) Lie algebras, bVilela-Mendesc constructed a simple quantum space based on an so(6;σ) Lie algebra with gen-erating variables ωβα ∈ so(6;σ). He used multiples of the ω’s to approximate Minkowskispace-time coordinates xµ, the canonically conjugate momentum vector pµ, the infinites-imal Lorentz transformation Lνµ, and the imaginary i, and discussed dynamics in such aspace [75]. This simplifying group was also proposed briefly by Segal [66]. The resultingbSegal-Vilela spacec is thus a regularization of the singular Poincare-Heisenberg quantumspace. It may have one, two, or three timelike dimensions among its six xα (or pα).

To construct a quantum space of the Segal-Vilela family, one must fix a high-dimensionalrepresentation of so(6;σ) that approximates the infinite-dimensional SPH representationclosely enough.

To represent a physical space-time a Segal-Vilela space also needs both a probabilityform on its vectors and a causality form on its differentials. These are provided in §5.3.

This then provides a single-stratum quantum space for a single event, not a space-time.A higher-stratum quantum space-time is constructed in Chapter 6.

We index the usual four axes of Minkowski space time with µ = 1, 2, 3, 4 and thetwo complexual axe with indices α = X,Y . The index α ranges over 1, 2, 3, 4, X, Y . Aconvenient complete set of Segal-Vilela-Mendes admissible coordinates is

xµ = XωµX ,pµ = EωµY ,

Lνµ = xν pµ − xµpν = XEωνµ,i = N−1ωXY (5.23)

with scale factors X and E having the units of time and energy. The quantum numberN is the square root of the maximum eigenvalue of −(ωXY )2 in a matrix representationRJ of so(V ), not fully pinned down by either Segal or Vilela-Mendes. The contraction toclassical space-time includes the limits

X,E→ 0, N→∞ (5.24)

and the freezing(ωXY )2 ≈ −N2, (5.25)

which can be treated as a subsidiary condition, restricting the system to a relatively smallsector of S. The physical origin of such a condition is taken up in $5.3.

The use of so(6;σ) in Segal-Vilela-Mendes space should not be confused with other usesof so(6) that have been suggested, such as the E numbers of Eddington [28]. Specifically,the so(6) of Segal-Vilela-Mendes space mixes x and p as discussed in §6.8.

To attain structural stability, bSegalc and bVilela-Mendesc combine and unify homo-topies separately introduced by bEinsteinc, bHeisenbergc, bde Sitterc, and bSnyderc. The


bSegal-Vilela spacec is a Matrix Geometry [27] without its connections and bgravityc. Itis more matrix than the Banks Matrix Model [6] in that its time variable too is a matrix.It has a fundamental time, the chron X, that fixes the quantum scale of space-time, andsuitably large integers that fix the representation of so(6R;σ). Other classical Lie algebrasbesides so(6;σ) also have ph(4) as a singular limit.

The events of a Segal-Vilela space could be realized by Palev collections of simplerspinlike quantum elements, generalizing the Penrose sphere, whose point is a Bose seriesof spins 1/2.

In general, if a configuration space has n dimensions then the classical phase space andits tangent bundle have the dimensionality 2n, but this phase space is the singular limit ofa regular quantum space with Lie algebra so(n + 2). One need not double the dimensionin this case, it suffices to add 2. The two added dimensions provide an axis of symplecticrotation that couples coordinates into momenta and conversely.

5.2.9 Baugh, Shiri-Garakani spaces

More recently, Baugh (2004) simplified the Poincare group to a special unitary group SU(n)instead of an orthogonal group. The quantum event of Baugh space can be represented asa pair of Palev sub-events with vector spaces 6C [8].

Structural stability forbids the Newton commutation relation [d/dt, t] = 1 of the usualdynamics. Shiri-Garakani (2005) simplified the linear dynamics of a harmonic oscillator[68].

At the extremes of system time t, when t ∼ ±max |t|, the multiplicities of the eigen-values of |t| typically vary rapidly, namely linearly in t ∓ max |t|, and unitarity is a badapproximation, but in the middle times, when |t| max ‖t|, unitarity can still be a usefulapproximation. The usual singular limit keeps only the middle times as the extreme timesapproach infinity, and so can be unitary.

5.3 Fully quantum event spaces

The primary quantum variable considered in this section is the event, a queue of stratumE corresponding to a space-time point of the canonical theory.

Under sufficient resolution the “field”, the system of stratum F, is seen to be a queueof stratum E events. Each event of E in turn becomes a queue of elements of stratum D,“differentials”. The differential stratum has dyadics generating the group of stratum C.These include the space-time position operators, energy-momentum operators, and angularmomentum operators for the particle associated with the event, as well as charges of thestandard model.

16 stratum-D dyadics belong to stratum C as well. These include charge/spin operatorsgenerating the group of stratum C.

5.3. FULLY QUANTUM EVENT SPACES 201

The bself-Grassmann algebrac S makes it natural to specify the representation of Segal-Vilela-Mendes quantum event space that Segal and Vilela-Mendes left open. The vectorspace of the event E is taken to be a Clifford algebra V E = S[E] as high in S as necessaryfor bulk.

Segal-Vilela-Mendes space can be accommodated within a fully quantum Clifford the-ory by first imbedding the Lie algebra so(6R;σ) as a Lie subalgebra within the real linearLie algebra so(S[3]) = so(10, 6) of the 16-dimensional stratum 3 subspace of S.

so[S[3]] is isomorphically represented as Lie algebra by the second-grade subspace ofthe Clifford algebra S[4] with the commutator as Lie product. This choice means that weidentify stratum C with stratum 3 of S. The reduction 16R = 4R⊕2R⊕10R then reducesso(16) = so(6) ⊕ so(10) into the Segal-Vilela-Mendez combination of external quantumspace-time and the complex plane, and an internal quantum space like that of the GrandUnified Theory.

Charge stratum C = 3 is the first rank able to support the so(6) of a fully quantumspace-time. Event stratum E = 5 is the first able to represent these operators quasi-continuously.

If one adopts tentative stratum assignments C = 3 and E = 5 then there are 120independent Lie algebra elements

ωEβα = ΣE−Cω[C]βα (5.26)

representing so[S[C]] on S[E]. Four of the 120 must approach continuous variables in thecanonical limit, one must approach the imaginary constant i, and the remaining 115 mustapproach 0:

xµ = XωEµ5 → ixµ,

i =1

NEXωE56 → i,

pµ = EωEµ6 → 0,...

Lνµ = EXωEνµ → 0. (5.27)

The normalization factor N is needed to compensate for the many terms in i.The Minkowski causal metric form 〉g〈 is then isomorphic to a singular limit of the

bKilling formc 〉k〈 of the Segal-Vilela-Mendes Lie algebra so(6;σ), suitably restricted (As-sertion 6). This agreement in form can be used to represent the physical Lorentz groupwithin stratum C.

It is tempting to use this agreement in form to represent the Minkowski space causalityform g as well, but first a chasm between the meanings of 〉g〈 and the Minkowski gνµwould have to be crossed. 〉g〈 is a quadratic form on quantum 4-dimensional vectors; if itis used as a probability form, it is usually normalized to unity. gνµ is a quadratic form onclassical 4-dimensional space-time tangent vectors, defining the square of a proper time, aclock-reading in a frame where the space-components of the differential dxµ are 0.


The canonical Lie algebra h(N ) has essentially one faithful representation R~ by dif-ferential operators, and it is infinite-dimensional. The linear Lie algebra so(6;σ) resultingfrom Segal’s algebraic simplification has an infinite number of different faithful represen-tations RJ so(6) labeled by a triad of quantum numbers J = (J1, J2, J3). In what follows,a circumflexed variable designates the RJ representative of the un-circumflexed variable.

Physics requires a representation with dimension much greater than that of the definingrepresentation. Here this results from exponentiation, passing to a higher stratum bybracing and catenation.

5.3.1 Fully quantum spaces

Here it is hypothesized that a physical fully quantum space is a queue, whose stratum-Lvector algebra is a subalgebra of the Grassmann power space over its stratum-L− 1 vectoralgebra. It follows that the vector algebra of each stratum L is a subalgebra of S[L].

Classical space-time physics distinguishes between the Minkowski coordinates on anyone tangent space to space-time and the more general coordinates on the gravitationalmanifold. For brevity we call these special and general coordinates. They seem to havenatural fully quantum correspondents:

A bspecial coordinatec on the differential stratum DS D is one that is induced by acoordinate of stratum C, and so is of the form

ΣD−Cω for some ω ∈ so(CS). (5.28)

Special coordinates of the event E are defined analogously, in the form

ΣE−Cω for some ω ∈ so(CS). (5.29)

A bgeneral coordinatec of the event E is of the form

ω = 4s, s ∈ so(ES). (5.30)

Canonical commutation relations among these coordinates arise as singular limits ofthe so(W ) commutation relations.

In the terms of bsupermanifoldsc [23] and bsupersymmetryc theory, the first-gradeelements Lsα ∈ S[C] define odd coordinates Lsα and the second-grade sβα define evencoordinates ωβα = 4sβα. A Clifford space has less symmetry between odd and evencoordinates than a supermanifold: The even are polynomials in the odd. There is still asymmetry group that mixes them: the isometry group SO(W ).

The usual space-time and energy-momentum coordinates for Minkowski space-time aresingular limits of special coordinates ωβα = 4sβα ∈ so[C], much as in §5.2.8.

The fully quantum theory ϑo provides the odd-grade generators sα as well as the even.Such odd variables can be useful for the construction of fermion field variables, so we


accept them as vectors. The 16 sα in S[C] are coupled into each other by the Lie algebraso(S[C]) = Grade2 S[C]2. The earliest stratum that will accommodate both the Poincaregroup and the standard model group is C = 4, with Dim S[C] = 16. For simplicity wearbitrarily suppose that all the dimensions of S[C] have physical meaning.

The symmetry Lie algebra so(S[C]) couples the 16 monadic generators sα ∈ S[D] andseems to be broken in higher strata. The 16 sα divide into 10 spacelike and 6 timelikegenerators. Four second-grade generators sβα give rise to long dimensions, the runners inthe space-time truss, and others give rise to the short dimensions, the spreaders in thecosmic truss. One s65 gives rise to the frozen constant coordinate i.

The four vectors sα used to construct generators of long variable (Minkowskian) di-mensions are designated by sµ. The two generators of the two (Argand-Euler) dimen-sions of the complex plane of the quantum imaginary i = sY X are designated by sζ ,ζ = 5, 6; these being the first two with negative signatures beyond the sµ. The 10sα used to construct the short variable (Kaluza-like) dimensions are designated by sκ,with κ = 5+, 8+, 9−, 10+, 11+, 12+, 13−, 14+, 15+, 16−, the subscripts giving the signatures(Table 1.1).

In a model of the present vacuum, the dimensions X and Y associated with space-timeand momentum-energy are long (external; longitudinal relative to the cosmic btrussc dome).When the corresponding sums ΣF−CωCµ6 include enough spins, they become effectivelycentral. This may not be true for the dimension XY associated with i, since i is effectivelythe same for all events. To be sure, to be approximately central, ΣF−CωC56 too mustinclude a macroscopic number of spins, isomorphs of ωC56. But these could arise from asum over a space-time region of a great many transverse spins in parallel, as well as froma sum of a great many longitudinal spins in sequence.

5.3.2 A fully quantum space-time

A fully quantum theory must first account for the great success of the canonical quantumtheory and then do better, just as the canonical quantum theory accounted for the greatsuccess of the classical theory and did better.

In the best case, all the elements of the low strata of S would have physical meaning;let us examine them with this possibility in mind.

A natural basis B for S is that fully generated without the use of addition. Then everybasis element sα ∈ B ⊂ S is a product of monads sβ,α, whose number is the bgradec ofsα, and each monad sβ,α belongs to some stratum Lβ,α, whose supremum over β is thestratum Lα of sα.

A typical physical field-stratum vector sf must be a product of a vast number ofmonads se, one for every quantum event or cell in space-time. Rank 5 contributes 264K

polyadic events, which suffice for now. The vast number must belong to rank 6 or higher.Let us stop provisionally at F ≈ 6 for economy.

If we open one of the monads of stratum F, we find a polyad of stratum 5 that we


can analyze again. Its monads are braces of polyads of stratum 4 proper. They generatea typically high-dimensional representation of SO(216) and provide a candidate for thequantum coordinates of event space.

One may now form quantum space-time events. The canonical Lie algebra a(x, p, i) ofspace-time and momentum-energy is generated by nine elements xµ, ∂µ, 1 (µ = 1, 2, 3, 4)subject to the canonical relations, with 1 central. One full quantization approximates thisLie algebra within a higher-stratum representation of the isometry Lie algebra so(10, 6) =so[C] of stratum C=3, the earliest possibility.

This Lie algebra is represented by generators of the form

ωDβα = Σ4sβα (5.31)

in the second-grade subspace of the next stratum S[D], with D = C + 1 = 4. The Cliffordalgebra S[C] of stratum C has 4 first-grade generators 〈sS[C]α, 24 dimensions, and a meansquare form 〉H〈 of signature 22 used both as causality form and probability form. TheClifford algebra S[D] has 16 first-grade generators 〈sDα, 216 dimensions, and a mean-square probability form 〉‡〈 of signature 28. The elements s1, s2, s3, s4, s5, s6 have signatures+,+,+,−,−,− (Table 1.1). We associate the basis element s4 ∈ S[C] ⊂ S[D] with timeand energy, and the basis elements s5, s6 with real and imaginary axes in a symplectic-complex plane. The dimensions of

X s4,5 = t, E s4,6 ∼ E, sXY ∼ i (5.32)

are assumed to be longitudinal in the cosmic dome relative to the vector 〈Ω of the vacuumqueue. A symbol like t[C] designates the predecessor in stratum C of a higher-stratumvariable, here the skew-symmetric time variable tE of stratum E.

The defining representation of so(3, 3) has skew-symmetric generators

ωβα =144[sβ, sα], where α, β = 1, 2, 3, 4, 5, 6. (5.33)

Rank C = 4 supports the adjoint (commutator) representation of so(10, 6). We maytherefore take the elements of the Lie algebra so[C] as the origins of the space-time andmomentum-energy variables of stratum E, setting

xµ = X ΣE−Cω[C]5µ, pµ = E ΣE−Cω[C]6µ, i = N−1 ΣE−CωC56. (5.34)

We now examine the conditions that must be met for these variables to be quasi-continuous.

For definiteness, consider the time operator ωE54 ∈ so(10, 6). The time dyadic t[C] :=iωC54 = 4(is5s4) has spectra (−1, 0, 1). By the exclusion principle the polyadics of stratumD have at most one monadic factor γ5 and at most one γ6, so the spectrum of the cumulativerepresentation

iωD 54 := iΣωC54 (5.35)


is still −1, 0, 1, but the multiplicities of these values are much greater than on stratumC. Of the 24 basic monomials in S[5], those that contain just one of the generators s4 ands5 transform according to a spin-1 representation of ω54; the other monadics transformaccording to spin 0. There are thus (24)/4 isomorphic replicas of spin 1 in stratum 5 andnone of higher spin.

TO DO: Check and finish. 080905Rank E = 5 has basic monomials with from 0 to (24)/4 spin-1 factors, and therefore

has a spectrum of ∼ 24 = 64K eigenvalues, still not adequate for the current resolutions ofspace-time-energy-momentum measurements.

TO DO: Complete this estimate.These variables of stratum C are represented in stratum E by

ωEβα := ΣE−CωCβα. (5.36)

The generator s4 is the first in S of negative square, and so is natural for the time-energy axis. Reserving the generators s1, s2, s3 for the space-momentum axes, and s5, s6

for the complex plane axes, we set

i = N−1ωE65, xµ = XωE54, pµ = EωEµ,6, µ = 1, 2, 3, 4. (5.37)

This uses the time quantum X, the energy quantum E, and the maximum N of |ωEY X |.Present experience, where the canonical relations work, is with a part of the spectrum

of |i| near the maximum value |i| .= N, and yet this narrow band must have a multiplicitythat passes today for infinite. It is possible to meet both requirements. For example only,the band

N−√

N < |ωEY X | ≤ N (5.38)

is both narrow and populous: |i| departs from 1 by about one part in√N ≈ 0, and the

multiplicity of this band is about√N ≈ ∞.

In the singular limit i → i, E → 0, and (supposedly) (ωY X)2 → −N 2, its extremeeigenvalue, so that the classical time axis emerges. This and the approximate validity ofthe canonical commutation relations imply

XEN = ~, N 1, NE ≈ 0,~X≈ 0. (5.39)

The conditions NE, ~/X ≈ 0 mean that these energies are below the present stratum ofdetectability in the particle laboratory.

To represent the physical theory of the time axis of which the singular limit is anapproximation, one must freeze i, eliminate E as a dynamical variable, and leave a large,finely spaced, quasi-continuous spectrum for t. Following the precedent set by classicalEuclidean geometry according to the hypothesis of §5.2.1, E is eliminated by making itclose enough to 0 to be ignorable.


Chapter 6

Fully quantum kinematics

which describes the queue.

6.1 History space

At this point we are like molecular geneticists of an earlier period in hypothesizing withoutconvincing evidence that beneath the great complexity of macroscopic nature lies a mi-croscopic stratum of great simplicity in which are encoded some of the order observed inhigher strata. The codes of Table 1.1 are one proposal for this structure, now of sub-queuesrather than genes. We also have a limited correspondence between some hypothetical lower-stratum structure — “genotypes” — and higher-stratum data — “phenotypes”. We havethe enormous algebra S of microscopic queue descriptors, but know the meanings of only(say) 120 second-grade elements representing position, momentum, angular momentum,and charge coordinates xµ, pµ, Lνµ, Qα, mostly frozen in the vacuum.

The semantics for canonical instant-based quantum kinematics is known. The problemis to extend the semantics to Q history-based descriptions of quantum scattering andproduction experiments. The predicted amplitudes for such experiments are then traces ofthe descriptors.

We must therefore cross two significant conceptual crevasses by reversing two singularlimits

CI←CH←QH. (6.1)

We pass from the canonical instant-based kinematics (CI) to the canonical quantum history-based kinematics (CH) in the next section §6.1.1; and thence to a fully quantum history-based kinematics (QH), in the following section §6.1.2.

207

208 CHAPTER 6. FULLY QUANTUM KINEMATICS

6.1.1 Canonical quantum histories

The Dirac concept of system history is first illustrated for a quantum mechanical systemwith a one-dimensional space-time with discrete time coordinate t, to which is attachedone coordinate q. The history probability-amplitude vectors of the system form a lin-ear associative algebra A, whose sum is quantum superposition, the addition of relativetransition-probability amplitudes, and whose product is catenation of experimental pro-cesses. H[A] stands for the quantum system with history-vector algebra A. It is assumedthat the generators of this algebra carry their own time-stamps, so that when they arecatenated they define their own chronological order in the product, which is not necessar-ily the order of writing. For example, if 〈I is a fermionic input, 〈I〈I is not an input of twoseparate consecutive fermions, but 0. The corresponding identity for Clifford statistics is〈I〈I = ‖〈I‖. The written order may still be the order of occurrence of control processesrelative to the metasystem.

A chronological structure of history is defined for each frame by a time-shifting energyoperator ∆E = ∆p0 = ∆E† and an energy-shifting time operator ∆t = ∆x0 = ∆t†,canonically conjugate in the sense that

[∆E,∆t] = i~. (6.2)

∆E is the departure from energy conservation for the history and ∆t is the duration of thehistory; hence the ∆’s.

The instant construct of binary observable or projector gives rise to three historyconstructs, representing the input, filtration, and output of a kind of quantum system.

[Conjecture:] They are associated with history vectors E, t in algebra A, which maybe regarded as brief histories, through

∆E = 4E, ∆t = 4t (6.3)

One can express the dynamics of quantum mechanics equivalently in terms of twoinstant vectors or in terms of one history vector with two instants. The development of astatistical distribution looks like the motion of a material distribution but the meaningsare distinct.

In instant terms, one writes a dynamical development as ψ2 = Uψ1, meaning that aninput at time t1 with vector ψ1 followed by the dynamical development U is equivalent,for final measurements, to an input at t2 with vector ψ2.

In history terms, one writes the amplitude for the briefest possible experiment historyas

A = TrD〈E = ψ2〈U〈ψ1, D = U, E = ψ2 ⊗Ψ1. (6.4)

This is as if a history is input with dynamics vector D = U and output with experimentvector E = ψ2〈 ⊗ 〈ψ1.

6.1. HISTORY SPACE 209

For a slightly longer history, with four equally spaced instants t1, t2, t3, t4, a typicaldynamical history vector has the form 〈D〈 = 〈U〈 ⊗ 〈U〈 , and an experimental historyvector might have the form

E = ψ4〈 ⊗ 〈3 = 2〈 ⊗ 〈ψ1. (6.5)

The factor ψ1 represents an initial input at time t1. The factor 〈3 = 2〈 is an identityoperator representing a direct connection from time t2 to t3 that means “do nothing” tothe system in the specified time interval. 〈3 = 2〈 is a primitive representation of a vacuum.The factor ψ4 represents a final output at the time t4.

The Schrodinger equation assigns a vector ψ(t) to each time t. It should not beconcluded that ψ(N) ⊗ . . . ⊗ ψ(2) ⊗ ψ(1) is the resulting history. This would representperforming an input operation ψ(t) at each time t, while in the actual experiment there isjust one input operation, followed by a sequence of waits, followed by the output operation.

It will be assumed here that each monadic factor in a polyadic history carries its owntime information within itself. Then the order of factors in a catenation need not be theorder of occurrence of the corresponding operations on the system; and the square of amonad in a queue does not indicate consecutive reiteration but is just the statistical normof the monad. And linear operators, assembled from monads and dual monads, likewisecarry time information.

080619 . . . . . . [Define E in general. Prove CI ≡ CH.]

6.1.2 Fully quantum histories

The queue vector space used in the following is SF. An experiment vector is a history vector〈E ∈ SF describing the experimenter’s actions during the experiment. The dynamicsvector 〈D ∈ SF is another history vector analogous to the operator 〈U〈 , representingthe dynamical law that operates between experimental actions. The relative transitionprobability amplitude is then

A = TrD〈E. (6.6)

The vacuum vector 〈Ω is a special case of 〈E representing a grand do-nothing for the entireexperiment: nothing in, nothing out. Usually all transition amplitudes are expressed asvacuum expectation values. The construct of the vacuum is dependent on the dynamics,so it will be taken up later.

6.1.3 Physics without functions

The classical construct of a relation can be fully quantized; but that of a functional relation,bfunctionc, or mapping cannot without breaking quantum invariance. This is expressed inmore detail as follows.

A classical relation is a set of pairs. If X and Y are state spaces of classical variables,the state space of the generic relation R between X and Y is

R = Rel(X,Y ) := P(X × Y ) (6.7)


To form random variables one forms the linear closures of the three state spaces X, Y , R.The quantization is immediate: A quantum relation is a quantum set of quantum pairs, thequantum entity with vector space 2X⊗Y If X and Y are Grassmann algebras it is naturalto use a Grassmann product X ∨ Y ⊂ X ⊗ Y instead of the entire tensor product X ⊗ Y .

The classical construct of a mapping or function, however, understood to be single-valued, has no such natural quantization. The key obstacle is the concept of the diagonal.Recall that for any state space X, the bdiagonalc of X × X is the graph of the equalityrelation:

[=]X := (x× x′) ∈ X ×X : x = x′, (6.8)

where × is the Cartesian product. The diagonal in X × X is a subset of X × X thatis naturally isomorphic to X. Further, a brelationc between I [X] and I[Y ] is an elementr ∈ 2X×Y , and induces a relation r × r between I [X ×X] and I [Y × Y ]. Finally, abfunctionc f : X → Y is a relation f ∈ 2X×Y with the property that the induced relationf × f of pairs preserves the equality relation [=], mapping diagonal in X to diagonal in Y :

f × f : [=]X → [=]Y . (6.9)

These concepts do not quantize because the quantum correspondent of the classical Carte-sian product X × X of sets is the tensor product X ⊗ X of vector spaces, which hasno diagonal, no subspace that is naturally isomorphic to either factor X. There is nomeaningful concept of equality between quanta, no useful concept of quantum clone.

This means that any construct of function or field in quantum theory has to be basis-dependent. In a simple quantum theory with a simple quantum event space, field theoriesmay still use the concept of relation, but not the concept of function. The theories usingthis concept must be singular limits of ones that do not.

These facts of quantum life were expressed by Wigner as the impossibility of quantumreproduction, and later, in quantum computation, as the impossibility of quantum cloning.Living systems are not produced by processes represented by vectors but by processes thatgenerate entropy at the same time. The impossibility of quantum reproduction has littlerelevance to biology.

6.1.4 Field variables

Central space-time coordinates are built into the foundations of canonical quantum bfieldctheory, as part of the strategy of absolute space-time. Consider the theory of bgravityc, forexample. In his gravitational action principle bHilbertc varied bgravitational fieldc variablesgµν(x) without varying space-time coordinates x = (xκ). In the resulting bPoisson BracketcLie algebra, gµν commutes with xκ, and the xµ are central in the Lie algebra of fields.

There are no such central time coordinates in actuality, however. Our actual physicalcoordinates for a remote event are based on signals, usually electromagnetic, that reachus through the intervening space-time. These signals inform us about the intervening


bgravitational fieldcs as well as about the remote event. The material lattice of rods andclocks imagined by Einstein to coordinatize events is not different in principle from op-tical or radar coordinates, since it too is connected by electromagnetic and gravitationalinteractions. It seems that physical coordinates are more relative than general relativityadmits. They are influenced by the ambient fields as well as by the choice of referenceframe. Physical fields are non-central and non-local, and so are physical coordinates.

In generalized bKaluzac models like the present one, bfieldc variables are bshortc coordi-nates of an event, short also in that what are usually described as two different field-valuesat one event are actually two different events too close together to be resolved by currentchronometry. In metrical terms, the ranges of the bfieldc variables we encounter in nature,measured in just noticeable differences, are much smaller than the ranges of external co-ordinates, in the current vacuum. It is assumed in the rest of this work that this metricaldifference is the main distinction between a field variable and a coordinate variable. Theusual distinction — that one is the value of the field function, the other the argument —must then arise in a singular limit of the organization of the vacuum. We no longer postu-late two different constructs, coordinate variables and field variables, just one construct ofevent coordinate variables. The field is replaced by the queue. The vacuum is a predicate〈Ω〈 of the queue.

The bosonic field variables we must fully quantize are bgravityc and the bgaugec vectorpotentials of the bstandard modelc. The fermionic fields are the four leptoquarks in theirthree generations. The bHiggs fieldc we tentatively identify with i, after an earlier model[73].

In a fully quantum theory, events are powers (braced catenations) of differentials. Thatis, the event vector space is a Clifford algebra over a lower-stratum differential Cliffordalgebra. Similarly, it is proposed to replace the bfieldc by a queue that is a power ofevents. The queue vector space is a Clifford algebra, supposedly of stratum F, based onthe canonical quantum theory, where a field polyadic of stratum F is a set of pairs of eventsand field-values of stratum E.

The dimensionality of the bstandard modelc gauge group is small enough to be thekinematic group of a stratum C with C < D < E. C = 3 permits the vector space V [C]

of stratum C to have 16 dimensions. Since so(5, 1) suffices for the position-momentumvariables and i, as in the theories of bSegalc and bVilela-Mendesc [66, 75], it seems possiblethat so(10, 6), the kinematical Lie algebra of stratum 3, is large enough to support thebgaugec vector potentials of the bstandard modelc as well.

Events of stratum E = 5 then have an event vector space S[5] of dimension 2216,

sufficient events for a queue of them to approximate a continuous field.When the simple field is replaced by a queue, namely of quantum events, iterated

Clifford statistics produces as many bstratumcs as needed. This results in a generic fullyquantum theory with vector space S produced from the empty set 1 by iterating bbracingcI and Clifford bcatenationc operations. The entire simplified physical system under study is


the bqueuec, and the vacuum is represented by a queue eigenprojector, of minimum energyin some sense, presumably of several dimensions.

The Clifford algebra on SE is generated by first-grade Clifford elements gamma thatflip particle-events into and out of existence in accordance with the Clifford law γ2 = ‖γ‖.Call the sign of ‖γ‖ the bsignaturec of the element γ. Fermionic creators and annihilators,obeying γ2 = 0, are linear combinations of anti-commuting Clifford elements of oppositesignature.

Each generic event carries space-time-energy-momentum variables as well as extravariables for charges. These are all spins of an appropriate orthogonal group. Taken alltogether, according to the regularity hypothesis, they generate a semisimple Lie algebra,like the so(6) of Segal-Vilela-Mendes space. The enormous difference between long andshort dimensions we must encode in the projector 〈Ω〈 of the vacuum.

S[E] is the vector space for the event, stratum E of S. The vector space for the queuestratum F is another subspace of the Clifford algebra,

SF = PF−E SE. (6.10)

This allows for F − E iterations of association, the number F − E being adjustable toprovide the necessary large numbers. In the following F − E = 1 suffices. Both bfieldcvariables and space-time coordinates of the canonical quantum singular limit derive fromqueue operators on SE.

The classical event of E in turn is an integral combination of classical differentials (ordifferential events) of stratum D:

SE = PE−DSD. (6.11)

The state space of a typical classical field is a set of pairs of field values and events,

F (x) ⊂ F × x. (6.12)

6.1.5 The cosmic crystal

To illustrate the concept of the quantum crystal film, here is a simple cellular lattice thathas both internal (short) and bexternal (long) dimensionsc:

(6.13)

It is sometimes convenient to call monomials in the Clifford algebra S “cells”, their subsets“faces”, their grade-2 subsets “chords”, and their grade-1 subsets “nodes”.

Like the cells of a classical mechanical phase space, these cells combine by superpositionto form the entire one-monad vector space and by catenation to form many-quantumvectors for the quantum field. Cells may also have common elements and be entangled


by superposition. This one-dimensional btrussc composed of 2-cells stands in for a four-di-mensional truss composed of 16-cells

:= 〈 int]⊗ 〈ext]. (6.14)

Each cell here is the bbracingc of a tensor product of an internal 1-cell, the vertical edgeof each square, symbolizing an internal 10-cell, and an external 1-cell, the horizontal edgeof each square, symbolizing an external 6-cell. Thus the model event cell is the bracedproduct of 16 monads: 10 internal ones forming an internal 10-cell 〈 int] and 6 externalones forming an external 6-cell 〈ext]. The lower edge of each cell in (6.13) symbolizes all itsexternal or longitudinal dimensions 〈ext]; the left-hand edge, all its internal or transversedimensions 〈 int]. The other two edges of each square are drawn only to make the cells lookcellular. The four vertices of each square stand for the 216 vertices of the 16-cell.

Turn now from this toy model to one that is slightly closer to physics. The vacuumtruss is a vector Ω supposedly in S[7].

There is no pressing reason to suspend either Lorentz SO(3, 1) or Standard U(2 ×3) symmetry at the cell stratum. The Segal-Vilela-Mendes SO(6) variant of Poincaresymmetry will also be assumed for each cell of a first model, although this symmetry isalready broken by the vacuum environment of the cell, and therefore is quite an approximatesymmetry of the cell at best.

One crude model of a quantum space-time is a bhypercubicc model, a four-dimensionalcheckerboard, with time axis along a principle diagonal. Each cell represents a Cliff(4R),and adjacent cells share some of their four generators.

A more physical model, allowing for internal dimensions and subsequent organization,is composed of 16-dimensional hypercubical cells, still connected into a 4-dimensional array,now forming a thin truss or dome. This 16-hypercube four-dimensional model is an ana-logue in higher dimensions of the primitive truss of (6.13). Its cell has the group SO(16R),omitting signatures for the moment. This is broken to SO(4,R)× SO(2R)× SO(10,R) bythe dome structure, representing the De Sitter variant of the Poincare group, the complexphase group, and internal groups. The SO(4,R) × SO(2R) factor is represented by thetruss with a spectrum of macroscopic extent, to describe the long dimensions and i. Theinternal SO(10,R) must be broken to the standard model S(U(2)× U(3)) as in GUT.

In a 16-hypercubic model, the prototype cell may result from a vector sC>, the topClifford element of stratum S[C], representing the input of 16 independent generating unitsas chords of the truss cell. All the coordinates ωEβα vanish for this cell:

ωEβα sC> = 0. (6.15)

Therefore it can be regarded as an borigin cellc of the vacuum truss. The group of its longdimensions might be SO(5, 1), making the imaginary axes X and Y spacelike, and leavingSO(5, 5) for the short dimensions transverse to the dome. The bKaluzac-Klein-Yang-Mills-bDeWittc middle bgauge groupc is the orthogonal group SO(5, 5) of a transverse face.


6.1.6 The organization of the imaginary unit

The organization i→ i must occur by the event stratum E, whose singular limit has onlyfour dimensions.

In the model ϑo, E = 6, and i is the infinitesimal transformation l−1iΣE−CωC65 : SE →SE induced by : S[C] → S[C], and normalized to unit maximum magnitude by a scale factorof its maximal eigenvalue l:

i := l−1ΣE−CγC65, γC65 = γC6γC5. (6.16)

MORE TO DO: Estimate i.The maximum eigenvalue of i2, occurs midway between the top and bottom of at the

grade ME = 26 = 2216.

The quantized imaginary has the form

i = l−1ΣF−CωCXY , (6.17)

normalized to have the extremum eigenvalue −1. The vacuum 〈Ω is to be an eigenvectorof

(i)2 .= −1. (6.18)

The hypercharge-isospin-color invariance Lie algebra u(2× 3) is a direct sum; relativequantum phases between isospin vectors of su(2) and color vectors of su(3) have physicalmeaning. Therefore u(2× 3) is an invariance Lie algebra of a composite system composedof one system with su(2) invariance and one with su(3) invariance. If we were workingwithin complex quantum theories we could regard the two parts as a 2-cell and a 3-cell.Since a real quantum theory has been posited, u(2× 3) is represented in the commutant ino(3× 6) of a generator i that provides a complex imaginary i in the singular limit, and wemust assume an organization that permits us to identify the is of color and isospin. Thecomposite system would then be a 9-cell composed of a 3-cell and a 6-cell with no mutualcoupling.

080510 . . . . . . [ Define the vacuum fine structure further.]Since the momentum coordinates are dual to the space-time coordinates relative to

i, one expects that large masses are associated with the small internal differentials andmuch smaller masses with the external differentials. The extended btrussc of large massesconnected by small ones begins to feel like an extended molecule of nuclei bound to eachother by electrons.

6.1.7 Statistical form

The probability form H : S → SD of the self-Grassmann vector algebra S must be fixed,especially its signature. Then the available signatures must be allocated to the various Liealgebras of the fully quantum theory.


The construction of the Grassmann algebra 2V makes no mention of a probability formon V . It is an affine construction. It builds on the defining contractions of vectors in Vwith dual vectors in V D. However if there is a probability form H on V , it can be usedto define one on 2V , still designated by H. And there is a natural one on C, the absolutenorm ‖z‖ = |z|2, to begin the recursive definition. One can proceed as follows:

One posits the canonical commutation relation (4.16) between I and ID; and the prod-uct relation

H(u ∨ v) = (Hv) ∨ (Hu). (6.19)

Note that this is not about the hermitian adjoints of operators. It relates the hermitianadjoints of input vectors, which are output vectors.

If u, v ∈ V are anticommuting vectors of norm +1, then their Clifford product hasnegative norm:

H(u t v)(u t v) = (Hv t Hu)(u t v) = −Hv t Huu t v = −1, (6.20)

then‖I(u t v)‖ = (H(Iu t Iv) (Iu t Iv) = ((HIv) t (Hu)) (Iu t Iv) (6.21)

This leads to the signatures indicated by tildes in Table 1.1. The signature of stratum 3 is√16 = 4.

If 6 dimensions of S[4] are used for the long dimensions of the cosmic dome, of neutralsignature, then the internal groups must be accommodated within the remaining 10 dimen-sions of signature 4, defining the algebra su(7, 3). The reduction su(7, 3)→ su(7)⊗ su(3)easily leaves room for color su(3) in one factor and the remaining internal groups withinthe other.

The dramatic change is in the enlargement of the Lorentz group to quantize the canon-ical i1. One may look to the transition from the Galilean group to the Lorentz groupfor some guidance. There were in principle two possible signatures, the definite and theMinkowskian. Lightspeed settled the question in favor of Minkowskian. This answer wasconfirmed by the fact that Newtonian kinetic energy p2/2m is positive rather than negative;an orthogonal rotation would reduce the energy — the timelike component of momentum-energy — below its rest value, not increase it. The law of transformation of the electro-magnetic field provides further confirmation: the invariant is the indefinite E2 − B2, notthe definite E2 + B2.

The variation from the canonical to the fully quantum is more involved than thevariation from the Galilean to the Lorentz, in that it not only adds dimensions to thespace but also adds an index to the canonical coordinates: xµ ∼ ωEC ∈ so[E], with C = µ5in an appropriately adapted frame.

The canonical theory has a Minkowskian metric on 4R; the fully quantum theory hascorrespondingly an invariant metric on so[C]. In the singular limit the square of one skew-symmetric element ω[C]

56 has a frozen value of cosmological magnitude and all other basic


elements ω[C]βα are small by comparison. Such vectors represent unitary rotations in a plane

close to the 56 plane. What is regarded in the canonical theory as an infinitesimal unittranslation pµ becomes a small unitary rotation in a distant µ6 plane:

pµ ⇐ (ξ/X)ω[C]µ6 (6.22)

Translations xµ of momentum, similarly, are singular limits of rotations in a remote µ5plane:

xµ ⇐ (ξ/X)ω[C]µ6 (6.23)

In order that i be skew-symmetric, γ5 and γ6 must have the same signature. The unitaryrotation in (say) the µ6 plane will respect the value of

Hω65H−1ω65 + Hωµ6H−1ωµ6 + Hωµ5H−1ωµ5 = const. (6.24)

It can significantly increase the last two terms relative to their initial values while decreasing‖ω56‖ imperceptibly.

The usual four differentials dxµ5 ∼ ωµ5 are the ones ordinarily perceived as eventcoordinate differentials, inherit the Minkowski signature, and give rise to long dimensionsof the cosmic dome. The remaining transformations ωρσ (ρ > σ ) whose signatures are inquestion ordinarily not seen as event differentials.

The probability form on the Fermi algebra S[L] of stratum L is designated by H[L].The Killing form on su[L] is designated by k[L]. Relative to any basis sα for S[L], a basisfor su[L] can consist of operators

ω[β,α], ωβ][α‖, (6.25)

in which [. . .] indicates skewsymmetry and real matrix elements, and . . . indicates sym-metry and imaginary matrix elements.

The hypothesis that the canonical unit 1⇐ r is the singular limit of ω65 implies thatthe 5 and 6 axes of the space S[C] have the same signature. Suppose C = 4, so that S[C]

has 16 dimensinsions. The definite nature of the metrics underlying the standard modelgroups implies that the remaining axes 7, 8, . . . , 16 of S[C] have the same signature. Thereare thus three signatures to be combined: ±2 for the axes 1, 2, 3, 4, ±2 for the axes 5,6;and ±10 for the remaining 10 axes. The intrinsic signature of S[C] is 4 if C = 4. It doesnot accommodate the signatures inferred from the canonical theory.

6.2 Fully quantum scattering

In a scattering experiment, quanta of known properties come together and interact, andparticles of changed properties are produced and are measured, for comparison with theo-retical prediction.

6.2. FULLY QUANTUM SCATTERING 217

In canonical quantum theory it is assumed that the transition amplitude for a scatteringexperiment can be expressed as

A = [D〈E] (6.26)

in terms of a dynamics vector 〈D〈 representing effectively rigid external influences on thesystem, and an experiment vector 〈E] representing irreversible influences on the system,such as creation or annihilation with registration. Both [D〈 and 〈E] are poorly definedsingular expressions because the underlying events have continuous spectra.〈E] is often assumed to have the form

〈E] = ss′ . . . s′′〈Ω] (6.27)

where 〈Ω] represents a vacuum and s, s′, . . . , s′′ represent input and output quanta ofdeterminate properties. The vacuum 〈Ω] is supposed to be determined by the dynamics〈D]. When 〈D] has the Hamiltonian form

〈D] = . . . e−iHδt . . . e−iHδt . . . , (6.28)

where each exponential connects two adjacent times, and H has a unique eigenvector 〈ω]of minimum eigenvalue, with projector 〈ω〈 , then

〈Ω] = . . . 〈ω〈 . . . 〈ω〈 . . . (6.29)

Due to spontaneous symmetry breaking by organization, the vacuum may be describedby a a projector 〈Ω〈 of dimension greater than 1 instead of a vector 〈Ω]. Then none ofthe vectors in the subspace ΩS need have the full symmetry of [D〈 . This descriptionassumes an experimenter who can resolve the degrees of freedom that distinguish thevarious eigenvectors of the vacuum projector 〈E].

MORE TO DOThe Q amplitude has the same form A = [D〈E], but its quanta have discrete spectra

and the vectors [D〈 and 〈E] are regular.In the example of Segal-Vilela space, the vacuum 〈Ω〈 is assumed invariant under the

skew-momenta QµY but not under the skew-coordinates QµX or the symplectic spin QY X .The dynamics [D〈 is supposed invariant under all the QC .

A two-event propagator (or correlator) is the amplitude for experiment projectors ofthe form

〈E〈 = 〈sE〈s′E〈Ω〈s′E〈sE〈 , (6.30)

in which each of the operators 〈sE〈 , 〈s′E〈 represents a terminal operation (a superpositionof input and output) and 〈Ω] represents a vacuum.

In the c and q theories, the one-quantum terminal processes are conveniently chosento be eigenvectors of the momentum-energy pµ. Since pµ is conserved for each quantum inthe absence of the other, and so can be readily prepared by the experimenter at a greatdistance from the scattering interaction.


It is natural to do the same in the Q theory, but the components of the Q momentumenergy do not commute. The be joint eigenvectors of the momentum-energy, the scatteringvectors

MORE TO COMECanonical quantum theories compute transition amplitudes between momentum-energy

eigenvectors to predict experimental scattering cross-sections. Here a fully quantum cor-respondent is constructed for a momentum-energy eigenvector.

[Do 080808 To be continued.]

6.2.1 Experiment time

The experiment vector must specify the duration ∆T of the experiment, from input tooutput. In the canonical quantum theory ∆T is allowed to approach infinity, on the groundsthat the input and output take place so far from the target, relative to the scale of thesystem, that their exact times are unimportant. The assumption ∆T → ∞ incorporatesthe assumption that the spectrum of time is homogeneous and unbounded, and neitherassumption holds in a fully quantum theory, nor presumably in nature. If this limit ∆T →∞ were taken too literally, in nature the experiment would run afoul of the Big Bang.The vector eigenspaces of the end times t = ±T have few dimensions, even if those for themiddle times |t| T have enough dimensions for a scattering experiment. What passes asan infinite time lapse ∆T in the canonical quantum theory is an infinitesimal one ∆T Tfrom the point of view of the fully quantum theory.

To deal with this inhomogeneity of time in the fully quantum theory ϑo, a fixed ∆T isassumed to exist that is large compared to relevant scattering times and small comparedto relevant metasystem times, and experiments are supposed to have proper duration ∆Tas a working approximation. In the canonical limit ∆T, T → ∞. A plausible first guessis ∆T =

√XT , the mean proportional between the quantum of time X and the maximum

time T .

6.2.2 Momentum vectors

A canonical quantum momentum(-energy) vector 〈k has the factored form

〈k = 〈eikµxµ〈0p. (6.31)

The operator 〈eikµxµ〈 translates momentum by ~kµ.Its operand 〈0p is a fiduciary eigenvector with the properties of

1. zero momentum,

2. zero angular momentum: Lorentz invariance,

3. totally uncertain position, and


4. projective invariance under i (implicit).

Such a vector is called a bmomentum originc here. One such vector is the bprinciple vectorc〈1x of a space-time xµ frame [24], whose components in that frame are all 1: 〈1x = 〈0p.

In the Dirac theory it is automatic that the operator i commutes with the translationoperator and fixes the ray of the momentum origin vector. In a real version of a fullyquantum theory, i invariance is contingent. It will be arranged ad hoc until a theory of theorganization of i can be worked out.

Both the momentum-translation operator and the momentum-origin vector are singularin the canonical quantum theory. Regular variants are required in the fully quantum theory.

The theory ϑo assigns the role of the fully quantum variant of i to the spin componenti := ωE

65/l acting on SE. For such spin components to act as infinitesimal isometries ofGrade1 SE they must act as the bcommutator representationc

ωEβα = 4sE

βα (6.32)

This i fails to commute with either the momentum components ∼ ωEµ6 or the position

components ∼ ωEµ5.

The momentum components ωEµ6 span a linear subspace of 4Grade2 SE that is not

a Lie subalgebra but generates a Lie subalgebra soEp∼= so(3, 2) called the bmomentum

subalgebrac of SE. There is a canonically conjugate position subspace that generatesanother Lie subalgebra soE

x∼= so(3, 2), the position subalgebra. The two together generate

an so(3, 3) ⊂ soE.The integer l > 0 is determined so that the maximum eigenvalue of −i2 is 1; and in

the canonical limit only vectors with −i2 ≈ 1 are used, and only operators that respectthis condition.

A probability operator rho of the fully quantum theory that commutes with i has theform

ρ = ρ− + ρ−i, [ρ±, i]± = 0. ρ±† = ±ρ±. (6.33)

(6.31) is an instant vector, not a history vector, and so need have no natural fullquantization. But a history vector of the form

E(k, k′) = k′〈 ⊗ 〈1〈 ⊗ . . .⊗ 〈1〈 ⊗ 〈k (6.34)

describes an experiment with an input of momentum k and an output of momentum k′

and so should have a fully quantum correspondent E(k, k′).A fully quantum variant 〈0p ∈ SE would have at least the properties

〈 pµ〈0p = 0, i2〈0p = −〈0p, (6.35)

useful for a fully quantum scattering theory, with large uncertainties ∆xµ and zero uncer-tainties ∆pµ.


(6.35) implies that all four ωEµ6 ∈ soE, for µ = 1, . . . , 4, annul 〈 0p. This implies that

their six commutators, the ωEνµ, also annul 〈 0p.

The Schrodinger generalization of the uncertainty inequality is

(∆A)2(∆B)2 ≥ 14|〈 [A,B]〉|2 +

14|〈A− 〈A〉, B − 〈B 〉〉|2, (6.36)

in which A,B can be any two operators, and their expectation values 〈A〉, 〈B 〉 and disper-sions ∆A,∆B refer to a given vector s. This assumes a positive-definite probability form.It is therefore still valid if the probability form is indefinite and the orbit of the vector sunder the algebra generated by A and B lies in a positive-signature subspace of the vectorspace. The uncertainty inequality (6.36) clearly excludes simultaneous eigenstate for thecanonical quantum position and momentum, which support a definite probability form,but not for the fully quantum ones, since their commutator is another spin component andcan have eigenvalue 0.

The top element sE> ∈ SE is invariant under all of soE and so has exactly zeromomentum. It also has exactly zero position, and it does not maximize but annulls −i2,so it is no candidate for a momentum origin but a kind of vacuum, which can be useful formaking one.

The eigenvalue of −i2 can be raised from 0 with a ladder operator. For example, thecomponent Lz of angular momentum is raised by the ladder operator Lx + iLy. To make aladder operator, however, an imaginary operator i that can play the role of i is necessary.It should have the properties

1. i = −i†,

2. i2 ≈ −1 for enough vectors,

3. i commutes with all the other operators among the ωEβα that enter into the ladderconstruction, and

4. i→ i in the canonical limit.

By “enough” vectors is meant, enough to simulate the low-energy sector of the Hilbertspace of the canonical limit.

A product of regular monadics will be imaginary if and only if the number of monadicfactors with positive square is even and the number with negative square is odd. This isreadily proven case by case.

To raise ωE65, a ladder built of ωE45 and ωE64 is useful.An imaginary that commutes with these three is required for such a ladder. Such an

imaginary is readily available on stratum C, where the spins in question originate. Thepolyadic

i[C] := s[C]2 s

[C]1 = s

[C]21 (6.37)


commutes with ωC65, ωC54, and ωC46.Such an imaginary i can be used to construct the four ladder operators and their

adjointsΛµ := iωE

µ5 + ωEµ6, Λµ† := −iωE

µ5 + ωEµ6 (6.38)

that increment ωE65 by i or annihilates its operand:

ωE65 [ΛµΨ] = Λµ

[ωE

65 + i]

Ψ. (6.39)

080621 . . . . . . [Construct a momentum origin!]The cumulative spins sE

βα = ΣE−Cs[C]βα generate a representation on stratum E of the

relativity group of stratum C Among the left multiplications ωEβα = LsE

βα are both themomentum and coordinate operators of the fully quantum theory.

The catenations

[C]4 s

[C]3 s

[C]2 s

[C]1 (6.40)

is annulled by each momentum component pµ = 4sEµ6, µ = 1, . . . , 4.

The four momentum operators γEµ6 relative to one frame generate a Lie subalgebra

so[E|p] ∼= so(3, 2) ⊂ so(3, 3) that contains approximate correspondents pµ ∼ γEµ6 ∈ so[E|p]

of the canonical momentum variables, and angular momentum operators LEµν coupling

them to one another. Call this the (fully quantum) momentum subalgebra for stratumE, relative to its frame. Its infinitesimals include cumulative spin angular momenta onstratum E of individual spin angular momenta γ[C]

µ6 = Ls[C]µ6 on stratum C, four of the 15

spin components that generate so[C].There is no monadic vector s of stratum C which the spin angular momenta γ[C]

αβ allannul and which could therefore provide a bmomentum originc. The top polyadic vector

sC> ∈ S[C] is simultaneously annulled by all 15 γ[C]αβ

.= 0. In particular the eight meancoordinates and momenta all vanish exactly on this vector, in consequence of the Clifford-algebraic commutation relations. This is not the metaphoric North Pole of Sphereland butthe center, so central that it cannot be shifted by any relativity transformation in SO[C]and so s useless as fiduciary vector.

The more restricted catenation

γ[C]6 γ

[C]4 γ

[C]3 γ

[C]2 γ

[C]1 (6.41)

is invariant under all the fully quantum momentum operators γ[C]µ6 and not under the

fully quantum space-time coordinate operators γ[C]µ5 . Therefore its brace is adopted as

momentum origin for stratum D:

〈 0p := I(γ[C]6 γ

[C]4 γ

[C]3 γ

[C]2 γ

[C]1 ) ∈ Grade1 SD. (6.42)


This monadic of SD also belongs afortiori to SE, and the E stratum coordinate and mo-mentum operators act upon it.

080616 . . . . . . [What is the preferred E-stratum isomorph of the C-stratum first-gradegenerators γ[C]

α ? They themselves? Or their double-bars? Or their Σ’s? And what is thepreferred E-stratum isomorph of the grade-16 top C Clifford polyadic? There are evenmore possibilities for the top polyadic than for the generating monadics.]

The experiment description is ignorant about the remote past and the distant futureof the system, the times outside the interval ∆T . In a canonical quantum theory thisignorance is consigned to infinity and oblivion. In the fully quantum theory, however, ourignorance concerns almost the full range of time, and cannot be left out of the picture.One way to express this ignorance is to assume a sharp description confined to the interval∆T . A full quantization of (6.34) that is restricted to ∆T will suffice. This is formulatedfactor by factor.

One fully quantum correspondent of the translation operator factor replaces the trans-lation of canonical quantum momentum-energy space through k by a rotation of its corre-sponding angular momentum quantum space through an angle θ:

〈θ = 〈eθµ5ωEµ5 ⇒ 〈k = 〈eikµxµ〈0k. (6.43)

As the four angles θµ5 vary, 〈θ ranges within the subgroup SO(3, 2) ⊂ SO(3, 3) that fixesthe 6 axis. This group has 10 parameters instead of the 4 of the translation group or ofthe four θµ5. However the four parameters θµ5 are related to the momentum energy kµ inthe singular limit by

θµ5 → lXkµ, ωEµ5 → 1lXxµ5 (6.44)

where l is the maximum eigenvalue of : iωE65 :. The other 6 parameters of SO(3, 2) enteras commutators of the operators θµ5ω

Eµ5 with one another, in the higher-order terms ofthe exponential in (6.43). These higher-order terms are small for ordinary values of kµ dueto the large denominator lX. It is necessary that the large number l overwhelm the smallHeisenbergchron X, making the product lX large on the laboratory scale.

080612 . . . . . .

6.2.3 Quantum topology

One does not use the heuristic cells of phase space to infer the topology of the canoni-cal quantum phase space with coordinates (q, p); nor can one use the heuristic cells of aquantum event space to infer its topology. A topology can be defined by the algebra ofits admissible coordinates. The algebras of continuous, differentiable, or analytic functionson a manifold define corresponding topologies. There are then at least two quite differentsenses in which a topology can be quantum or non-commutative. A topology whose coor-dinate algebra is the operator algebra of a vector space, here an event Clifford algebra, is


quantum in a weak sense. A topology is quantum in the strong sense if it itself is a quantumvariable, necessarily of a higher stratum than the event, defined, say, by a Clifford algebraover an event Clifford algebra.

Evidently a fully quantum theory can describe quantum topologies in the strong sense.

6.2.4 Origins of the coordinates

The coordinates on canonical space-time are supposed here to be classical limits of quan-tum operators of a fully quantum stratum E. One must distinguish between space-timecoordinate operators of one elementary particle and those of astronomical bodies. Parti-cle coordinates belong to flat Minkowski space-time, supporting the Poincare group, andthey and their canonically conjugate momentum-energy variables might have origins in theso(16) of stratum C, with representations on all higher strata. Astronomical voordinates oncurved Einstein space-time support the diffeomorphism group, which is first approximatedon stratum E.

One maximal commuting set of event variables in so(16;σ) is

(ω12, ω34;ω56;ω78, . . . , ω15 16). (6.45)

In the 120 spins ωβα one finds the

ωµ5, µ = 1, . . . , 4 (6.46)

which provide four Heisenbergexternal coordinate generators, a quantum imaginary gen-erator ω65 = li whose representation in stratum E is a large but frozen pure imaginary li,and all the remaining spins, whose representations in stratum E become 0 in the classicallimit.

080612 . . . . . . [Check whether the signatures matter.]Coherent, perhaps superconducting, propagation occurs along the Heisenbergexternal

dimensions. The great differences between internal and external masses and couplingconstants are differences between propagations in the long and short dimensions.

The quantum event cell I[X] has a vector space X. A natural candidate for a quantumfield is a quantum variable I[Y ] with vector space

Y = 2X , (6.47)

the real Clifford algebra over X. This formula can be read classically too: if X is thesample space for an event, 2X is the sample space for space-time as a queue of events. IfX is simple, the random object 2I [X] is a random set.

6.2.5 Fully quantum fermions

Here the fully quantum correspondent of a quantum field of leptons or quarks is formulated.In the quantum kinematics neither field values nor coordinates commute, and the concept


of functional relation among non-commuting variables is not well-defined in any invariantsense.

A spinor field of the standard model can have the general structure ψτ (x), acceptingexternal event coordinates x and an internal basic vector sτ specifying Lorentz spin, hy-perspin, isospin, and chromospin, to deliver a fermionic bgeneratorc ψ for a fermion withthe specified properties.

The corresponding fully quantum operator ψ should be a fermionic annihilator ψ foran event with specified coordinates and spins. It is assumed that sτ belongs to stratumC, sξ belongs to stratum E, and the operator ψ belongs to stratum F. It is convenient forthis unification of internal spin and external coordinates that S[C] ⊂ SE; the internal spinoperators of stratum C are among the spin operators of stratum E, and so are the eventcoordinate operators.

080522 . . . . . .If bfield variablesc and coordinate operators merge into one operator algebra, one must

explain the canonical quantum limit, where some coordinates xµ commute and others yα

are functions of them determined by the dynamics.Organization can do this. For example, consider a gas of particles with coordinates

xn, yn, zn at one time t. The class of all these triples defines a ternary relation among thethree variables x, y, z, but not a functional dependence of two coordinates on the third. Tobe sure, all three are functions of n, but the label n is arbitrary, without physical meaning,so this function has no physical meaning either.

Suppose however the gas condenses into a droplet, thread, or bubble, organizationswith 0, 1, or 2 long dimensions and 3, 2, or 1 short dimensions, respectively. If enoughparticles are so organized, 0, 1, or 2 manifold coordinates emerge that are approximatelyclassical variables, even though the particles are quantum, and the atom variables maybe smooth functions of these manifold coordinates. The condensation then creates newcollective physical variables, makes them effectively classical, and establishes a functionalrelation of the old quantum coordinates on these new collective variables. Space-time andfields seems to originate when cells organize into a quantum crystalline structure with somelong dimensions for space-time and some short ones for the field.

One can formulate how this process might take place in a quantum cell system. Cliffordbfull quantizationc converts the bgaugec potential into a cumulative coordinate Σnωαµ, arepresentation in GE of the Lie algebra ωαµ ∈ so[D], converts the coordinate xµ → Σnωµ5,and converts the momentum pµ into ωµ6, up to a sign. The stratum difference F−D is atleast 2. The action contains the commutator

[Σnωµ6,Σnων6] ∼ Σnωµ,ν ], (6.48)

which becomes a derivative with respect to xµ in the classical limit. One can expectthat owing to such derivatives in the action, discontinuities in the classical limit result inunbounded momentum and kinetic energy, and so are energetically suppressed.

6.3. FULLY QUANTUM GRAVITY 225

Therefore in the canonical quantum limit a bfieldc F arises that is effectively differen-tiable with respect to the classical space-time point of X.

In general the effective field will be multivalued. If the bfieldc is double-valued at apoint, it will therefore be double-valued in a neighborhood for the same energetic reason.Hopefully, then, an n-valued field is effectively the same as n single-valued fields on thesame region of space-time. If single-valued fields have a significant zero-point energy inthe canonical quantum limit, there will be energy-gaps between the single-valued, double-valued, . . ., fields. The multivalued field will require more energy than the single-valued.Energetic considerations then favor single-valuedness much as they favor differentiabilityin the classical limit.

6.3 Fully quantum gravity

To fully quantize gravity one may analyze the global field into events and local fields atevents, fully quantize both the event and the local field, and then assemble them into theglobal gravitational field. These steps are taken in §6.3.1– 6.3.3.

6.3.1 Fully quantum events

It has been posited that space-time coordinates, momenta, and i emerge in a stratum Eas cumulative representations of swaps ωβα ∈ so(10, 6) of stratum C. This assumption istested here.

For it to hold, a vector subspace ΨE ⊂ SE must exist with a nearly constant valuei2 ≈ −1 where i = N−1ωE56, macroscopically large and variable values for the space-time coordinates xµ = XωEµ5, negligible values for the momentum-energy coordinatespµ = EωEµ6 ≈ 0 and the other generators ωEβα and for the uncertainties in all theseoperators.

The ωβα act on the 16-dimensional vector subspace SD1 ⊂ SD as commutators with

elements sβα ∈ SD2 :

ωβα :=124sβα. (6.49)

Here

s1 := I1 = 1, s2 := Is1 = 1, s3 = Is2 = 1, s4 = I(s1s2) = 1 1, . . . , (6.50)

with s5, . . . , s16 given in Table 1.1. This is the defining representation of so(10, 6).These operators induce operators on the next stratum by the Lie homomorphism (3.22).

Each ωβα of stratum C acts on Iψ ∈ SE2 as

ωEβα := ΣE−CωCβαID (6.51)

Since the vector space SD is real, antisymmetric operators like ω21 on this space can-not be diagonalized. But looking back to the condensation already required by a real


Segal-Vilela-Mendes space, we can set aside s56, which commutes with s12, s23, s31, as an“incipient i” and use it to construct “incipient eigenvectors” of (say) ω12, anticipating thatthese will give rise to the physical i and proper eigenvectors after organization on a higherstratum F centralizes i = N−1ΣF−Ds56/l→ i. Then the Clifford element s31 + s56s23 is anincipient eigenvector of ω12 with incipient eigenvalue s56:

ω12 (s31 + s56s23) :=12

[s12, s31 + s56s23] = s56 (s31 + s56s23) . (6.52)

These are the seeds of the event coordinates of the next stratum D:

xβα ∼ ωEβα = ΣE−DωDβα = ΣE−D4sDβα (6.53)

up to dimensional constant coefficients. This is a concrete algebraic realization of theSegal-Vilela-Mendez commutation relations for so(6; sigma).

In fully quantum physics the system interface is shifted to move some of the basic space-time variables x from the metasystem to the system, where high-resolution measurementexposes their non-commutativity and discrete spectra. In the fully quantum theory, thecomponents of x do not commute, and there is no basis of simultaneous eigenvectors 〈x. Itis supposed here that the fully quantum correspondent of x is, up to dimensional constantfactors like X and ~, the many-quantum operator

ωFβα = ΣF−EωE

βα = ΣF−Cω[C]βα (6.54)

on SF induced by the operator ω[C]βα ∈ so(10, 6).

6.3.2 Time form

The spin form γµ of the Dirac equation defines the causality form by its self-anticommutatorsand defines the Lorentz group by its self-commutators. To quantize gravity it seems enoughto quantize the spin form field γµ(x).

The classical spin form maps each vector in a certain vector space, originally a 4-dimensional Minkowski tangent space R4, into a first-grade Clifford element. The brace Itoo maps vectors into first-grade Clifford elements. The restriction of I to any stratum L isdesignated by I[L]. Then I[3] is isomorphic to the spin form of a 4-dimensional Minkowskispace-time R4 and I[4], which includes I[3], is the spin form on a 16-dimensional space ofsignature 4. These spin forms are native to a fully quantum theory and do not have to beimported. In the following constructions, the Clifford algebra S4 of stratum C is used asvector space for a lower-stratum seed of a space-time cell, I[3] is the seed of the gravitationalfield, and I[4] is the seed of all the standard model fields as well.

6.3. FULLY QUANTUM GRAVITY 227

6.3.3 Fully quantum gravitational potentials

It is reasonable within the conceptual framework of full quantum theory to seek the rootsof both the causality form and the probability form in lower strata of the system queue. Itseems doubtful, however, that a linear space of low dimensions can support two differentmetrical forms without leading to a clash of symmetry groups.

The physical interpretations of the forms, moreover, make it implausible that theyare independent. Their functions overlap when the probability form is indefinite like thetime one. The causality form distinguishes timelike translation of negative norm fromspacelike translations of positive norm. Operators can be used reflexively, to transformfrom the initial experimenter to another, or transitively, to transform the system; sometimesthese are called passive and active interpretations. Timelike translations are feasible bothtransitively and reflexively; spacelike ones are feasible only reflexively.

An indefinite probability form also distinguishes by its sign between transitively feasibleor unfeasible actions, namely the input-output actions represented by vectors. This meansthat these two forms must be related, although one concerns classical elements — space-time events — and the other quantum events — quantum input-output generation. It isnatural to explore the assumption that

Assumption 8 (Chronometrical hypothesis) On some stratum C < E, the causalityform is the probability form:

〉g[C]〈 = 〉H[C]〈 (6.55)

080211 . . . . . . [Segue]Classical gravitation theory has a sequence of increasing groups with correspondents

in fully quantum theory proposed in §4.4.1.To represent the classical gravitational field, schematically speaking, one combines

Dirac spin variables γµ and classical coordinate variables x into a variable field functionγµ(x) that defines the chronometrical tensor field

gνµ(x) =12

[γν , γν ]−. (6.56)

It is supposed that in the fully quantum theory corresponding operators x and γµ are tobe combined in a corresponding way.

The fully quantum field is a descendant of stratum C:

SF = ΠF−CS[C]. (6.57)

It therefore has the natural spin structure

γalpha := L ΣF−Csα, sα ∈[C] S. (6.58)

By the chronometrical hypothesis (Assumption 8) the probability form of stratum C definesthe causality form of that stratum and all higher ones, and none needs to be attached. Like


the Cartan repere mobile, the quantum cell provides a local metrical structure; and thenthe connection between cells defines the global one.

080319 . . . . . .Canonical quantum bfield theoriesc use a complex vector space, sometimes with an

indefinite Hermitian form, as in the theories of Gupta and Bleuler. To generate the complexLie algebra of such a vector space from an underlying real Lie algebra, however, we mustinclude an bimaginaryc i among the generators. This creates a radical and makes thecanonical Lie algebra structurally unstable in principle, though it seems to result in noinfinities. For the sake of structural stability a real finite-dimensional vector space istherefore assumed. To create an approximately central i near a singular limit, a globalorganization that freezes a generator i suffices. to convert a real quantum theory into acomplex one [72]:

If A is (at least) an algebra and e ∈ A then

A\∆e := a ∈ A : ∆e · a = 0 (6.59)

is called the bcentralizerc (or bcommutantc) of e (in A). If A is a real algebra and i ∈ A hassquare −1 then A\∆i (often designated by i′) is a complex algebra with imaginary uniti. A can be factored as A ∼= A0 ⊗ Alg 2, where Alg 2 consists of 2 × 2 real matrices, ande = 10 ⊗ ε, where 10 is the unit operator in A0 and

ε = 0,−11, 0 . (6.60)

Any two-dimensional projector ρ ∈ A\∆e can be written in the form

ρ = ψψ†, ψ = ψ0 ⊗ 12 + ψ1 ⊗ e, ψk ∈ A0. (6.61)

Organization is the most plausible physical origin of such centralization in the presentcontext.

We have provisionally assumed that every bstratumc has the same real Clifford statis-tics. Therefore every bstratumc has an invariance Lie algebra so(N+.N−) belonging to theD sequence of classical groups, until broken by organization or a singular limit.

Conveniently, the signature of the bstratumc of S with dimension n+ + n− is

n+ − n− =√n+ + n− for n+ + n− > 2. (6.62)

Therefore the so(4;σ) of stratum 3 is the Lorentz Lie algebra; while so(n, σn) for n, σn →∞is asymptotically neutral as required for bFermi statisticsc, in that signature σn =

√n =

o(n); and a Pavel variant of Bose statistics is defined by the Lie algebra of the second-gradeelements of each stratum.

6.4. CONSTRUCTION OF THE VACUUM 229

6.4 Construction of the vacuum

In canonical quantum theories like the standard model, the space-time continuum is builtinto the algebra of the theory and the vacuum is the instant vector with minimum energy.In the canonical theory one can specify a time to fix an instant and then specify threespace coordinates to determine an event. A history is a succession of instants; an instantvector therefore says something about the history of which that instant is a part, and cantherefore be expressed in terms of history vectors.

Formally speaking, from the historical perspective an instant vector seems local intime and grossly submaximal in its description, and so might be expected to correspondto a projector of high dimension in the space of history vectors. In fact the canonicaltheory usually supplements the explicit local information in an instant vector with enoughtacit global information to define a unique history vector. It is always understood that thespecified instant occurs between the input and output phases of a experiment, and that thesystem is adiabatically shielded during the experiment. Then a fixed dynamical historyvector

〈D = 〈UT 〈 ⊗ . . .⊗ 〈Ut〈 ⊗ . . .⊗ 〈U0〈 (6.63)

governs the development of the system through a series of unitary operators here labeledby time. The background experiment vector 〈E has the form

〈E = φ〈 ⊗ 〈 lT−1〈 ⊗ . . .⊗ 〈1t〈 ⊗ . . .⊗ 〈11〈 ⊗ 〈ψ (6.64)

where each 1t is an identity operator associated with a specific intermediate instant t.When the canonical theory tells us that the vector at the instant t is ψt, the meaning isthat for outputs after time t, the transition amplitude from 〈ψ0 at time 0 is the same asthe transition amplitude from ψt at time t. In the history theory, the transition amplitudefor D〈E is also given by D′〈E′ with truncated history vectors with input delayed to timet:

〈D′ = 〈UT 〈 ⊗ . . . ox < Ut <, 〈E′ = φ〈 ⊗ 〈 l〈 ⊗ . . .⊗ 〈ψt. (6.65)

Little of this works in a fully quantum theory. In each frame it is still possible to definea skewsymmetric time coordinate operator t in each frame, but different values of t are notconnected by unitary or orthogonal transformations. There is a plausible skewsymmetricenergy operator E in each frame but it does not increment time by a fixed amount but bya third operator, essentially the commutator of time and energy.

In a Poincare-invariant theory the vacuum energy is adjusted to 0 and the vacuumvector is independent of time. In a fully quantum theory, Poincare invariance is replaced,for example by an so(6;σ) invariance in the model ϑo, and time is replaced by a skew-symmetric operator like

t = TΣF−Cγ45 (6.66)

a suitable normalization factor T providing the units of time.


Here we consider how to relate the fully quantum vacuum to the dynamics.The bvacuumc is what is left in a target chamber when the air is pumped out and

no quanta are injected. Its symmetries are determined by experiment, not definition. Itis the ambience of the experiment, including in principle long-range influences of the theexperimenter, apparatus, and cosmic environment, which break all symmetries. Clearly thephysical vacuum is only approximately Poincare invariant. so(6;σ) invariance or so(10, 6)invariance may be a better approximation on a cellular scale.

The vacuum is represented here by a projector 〈Ω〈 . In the main application thevacuum has a non-trivial multiplicity or “degeneracy” Tr 〈Ω〈 1.

It is convenient to reach the vacuum from the dynamics vector in several stages in acanonical quantum theory.

First, to pass from the history mode to the instant mode of representation, one factorsthe dynamics vector into a time-ordered product of infinitesimal unitary transformationse−iHdt/~ and defines a category of unitary operators U(t′, t).

This recasts the canonical quantum action principle in Hamiltonian form, describingthe temporal development of any instantaneous variable Q = Q(t) with no explicit timedependence:

[E −H,Q] = 0 (6.67)

with Hamiltonian and energy operators

H = −L+d

dt

∂L

∂t, E := i~d/dt. (6.68)

Then to define the canonical quantum vacuum one seeks instantaneous vectors ofminimal energy eigenvalue ε:

〈H〈Ψ = ε〈Ψ for minimum ε. (6.69)

If Ψ(t) is a time-dependent Schrodinger vector, it would be a mistake to think of thesuccession of its values as a history vector. It is a history of a vector but not a vector ofa history. To be sure, a classical time-dependent state q(t) defines a history. The functionΨ(t), however, describes a single input process, carried out at any one of the times t, not asequence of input processes at all the times. After the input process Ψ(t), all that happensat later times t′ > t, before the final output process, is a passive transport, represented bya sequence of unitary transformations. Ψ(t′) is not another input process besides Ψ(t) butan alternative input process that could be performed instead of Ψ(t) and have statisticallyindistinguishable consequences for future times t” > t′.

When the canonical quantum theory is based on a dynamics vector of the singularform (1.39), the classical action density L acquires a quantum interpretation. In a finite-element approximation, the space-time can be divided into finite hypercubes or 4-intervals∼ I4 of 4-volume ∆x. Then each factor ei∆xL/~ in the dynamics vector 〈D representsa quantum transition probability amplitude for a process represented by a vertex with


8 lines or generation processes, one through each 3-face of the I4. The Galilean space-time dissection parallel to the coordinate axes results in 6 spacelike lines representinginfinite velocity and two timelike lines representing zero velocity, a relic of pre-relativisticthought. In a less unphysical dissection appropriate to special relativity, the hypercubes,diagrammatically speaking, stand on one corner instead of one 3-face. The normal to each3-face is a null vector, representing generation at light speed rather than infinite speed orzero speed. Now the eight lines are clearly divided into four inputs and four outputs foreach vertex. The four input vectors form a basis of null vectors for the entire space. Inthis null basis the Minkowski metric form is a matrix with 0 on the diagonal and (say) 1everywhere else.

A man in the game of checkers or draughts moves along such null diagonals in theplane. The proposed dissection of space-time defines a 4-dimensional checker-board, withcheckers in the center of hypercubes and moves across 3-faces. Projected on a 3-space ofconstant time, the four null directions appear as four unit vectors from the center of aregular tetrahedron to its vertices.

This classical cellularization is highly artificial, a computational expedient. A fullyquantum dynamics vector 〈Ψ does not have to be artificially cellularized, however, sinceit is composed of finite quantum elements in stratum E from the start: 〈Ψ ∈ SF = 2SE

.Each monadic factor ψ in Ψ counts as a cell or vertex. Each cell ψ of the vacuum in turnhas several monadic factors of stratum D within it. These represent generation processesconnecting the cell to other cells.

The canonical dynamics is a development in time, one variable, generated by a canon-ically conjugate variable, the energy E = −i~∂t. Both variables have fully quantum coun-terparts. One must not proceed too formalistically, however. The time at which the energyof the system is determined is not determined by the system but by a clock in the meta-system. One determines system energy at a metasystem time.

First one forms the fully quantum correspondents of the system energy and time.In a fully quantum theory, canonical conjugacy is not a binary relation but a ternary

one, involving a quantum imaginary. A canonical conjugate is not unique and or absolutebut is relative to a choice of i. For the choices of (5.23), the canonical conjugate ofimaginary time XωE45 is EωE46, which is therefore the imaginary energy. The imaginaryunit is i := N−1ωE65. It must be used to convert imaginary time to real time, a symmetricoperator, that can be used to define instants by its eigenvectors. The imaginary energysuffices to generate the time development.

The fully quantum imaginary unit i and imaginary time t do not commute and theirproduct is not a symmetric operator; the easiest symmetric operator that approaches theproduct in the canonical limit is the anticommutator. Therefore the fully quantum realtime on stratum E is defined by the anti-commutator

E =X

2ωE45, ωE65. (6.70)


Now the general history must be partitioned into instants, slices of constant time, anda unitary development U(tn+1, tn) from one instant to the next must be formed from thedynamics vector. The fully quantum development is closest to unitary for the two centraltime intervals.

Then the vacuum energy condition can be expressed as

E 〈Ω =E

2ωF

56, ωF46〈Ω = ε 〈Ω, for minimum ε. (6.71)

080603 . . . . . . [Segue.]The quantum imaginary is the i that appears in both the Heisenberg equation of motion

and the quantum action principle. It is used to convert elements a of the Lie algebra toobservable variables ia that are conserved when the dynamics is invariant under i, one ofthe main principles of quantum kinematics. It has three defining characteristics: Its squareis −1; it is central; and it is a bmeta-operationc, in the folllowing sense.

When we compose two systems into a product system, important quantities composein one of three ways. We add corresponding infinitesimal operators (like translation gener-ators), multiply corresponding finite operators (such as parity), and equate correspondingmeta-operators (those which act on the metasystem) . The main meta-operator is time tin elementary quantum mechanics. When we combine two particle systems, each with atime variable t, the composite system still has only one time variable, inherited from bothsubsystems. This is because the composite system has but the one metasystem for bothof its parts, and time is read from a clock in the metasystem, not from the system. Theimaginaries of the two subsystems are also equated. Moreover under time reversal, i isreplaced by −i. It seems clear that i too is a meta-operator.

In stratified quantum theory we quantize part of what was previously metasystem.This part must have a non-central operator i, the quantized i, that becomes the centrali in the singular limit of canonical quantum theory. It is necessary to give a concreteexpression for i to make the theory definite, and for i to approach a unique central i in thecanonical quantum limit.

It is common in canonical quantum theories for observables to be bilinear in variablesof the system and the metasystem. For example, when we speak of the energy of the systemin special relativity we mean the bilinear expression

E = V µpµ (6.72)

where V µ is the experimenter’s time axis and pµ is the momentum-energy of the system.We cannot make maximal quantum determinations on the experimenter without disruptingthe experiment, but we can measure quantities like the world velocity V µ and position xof one experimenter with respect to another without harming either, if we do not exceedmacroscopic precision. This respect for the experimenter introduces errors that are usuallynegligible compared to those we make anyway. They permit us to regard V µ as central,


even though the laws of quantum theory tell us that V µ = Pµ/M is a multiple of themomentum energy of the experimenter and fails to commute with the centroid coordinatesXµ of the experimenter. Evidently the central i is an operator like V µ, determined by themetasystem.

In the classical limit all variables commute and the energy of a mass m is positive andbounded away from 0:

E := +√−∂0

2 > mc2 > 0 (6.73)

where +√ is the positive square root. Then we may take i to be the phase of the antisym-metric time-translation generator ∂o:

i ∼ ∂oE, (6.74)

which is trivially central.Simple quantization replaces this i by a generator of a simple Lie algebra a on a deeper

stratum, for example the generator ω56 of Segal and Vilela-Mendes. A candidate is neededwithin fully quantum theory for i, the quantum i that is used to form observables fromgenerators on stratum F. i is required to transform like ω56 under the simple Lie algebraa. ω56 acts first on stratum C, and induces a transformation ΣF−Cω56 on stratum F of themetasystem, that, suitably scaled by physical constants, is the proposed i.

It seems permissible that i belong to the metasystem, since it is used to form observ-ables, and observations are interactions between system and metasystem. The i of theexperimenter also has the important property of near centrality relative to the algebra ofthe system, and so seems to be the natural choice for forming observables from systemgenerators.

6.4.1 Bosonization

It is especially easy to construct excitation quanta with bPalevc statistics and even spinfrom a Fermi aggregate of quantum entities with spin 1/2:

Assertion 4 The algebra of a Fermi aggregate of spinorial quanta includes the algebra ofa bPalevc aggregate as subalgebra.

Argument The basis vectors ψα ∈ V define creators ψα = Lψα ∈ FermiV in the Fermialgebra over V annihilators

ψβ := ψα†. (6.75)

If Λ = (Λβα) ∈ Alg V is any linear operator on V then its many-quantum representativeon PV is

ψβΛβαψα (6.76)

By (3.22) these obey bPalevc commutation relations for sl(V ) generators and have bosonicgenerators as singular limits.


To actually combine bPalevc quanta represented by such second-grade operators, onecannot merely brace the pairs and then multiply; bracing produces grade-1 monads, notPalev quanta, which have grade 2. One cannot simply multiply the second-grade operators;that would scramble the pairs. However if the two first-grade elements in each Palevquantum have a coordinate in common that distinguishes them from the other pairs, theproduct can always be unscrambled. In that case one may combine Palev quanta by Cliffordmultiplication.

One may go from one such quantum to a catenation of many by applying the Lie algebrahomomorphism Σ, which preserves the commutation relations and bPalevc statistics. Σmay be iterated to produce larger assemblies.

6.4.2 Antiparticles

6.4.3 Flavor

[Do: Since the one-quantum Dirac equation includes transitions from positive to negativeenergy, can it be a good approximation for an electron moving over the vacuum Dirac sea,which fills the negative energy strata? Does this question persist in the normally orderedtheory? Or must something like the Foldy-Wouthuysen equation, without such transitions,replace the Dirac equation at a deeper stratum than usual?]

6.5 Fully quantum gauge theories

A canonical quantum bgaugec theory has a variable covariant differentiator DµM (x) on agiven bundle with a given lower-stratum gauge group in the bundle group and a givenc space-time manifold for base. Its classical space-time destabilizes it. Here it is fullyquantized stratum by stratum.

The gauging of a canonical quantum field theory is a heuristic process that converts aquantum bfieldc theory with a lower-stratum gauge group G(C) represented on stratum Fto a richer quantum field theory with a much larger stratum-F gauge group GF consistingof group-valued fields: sections of the principle bundle with fiber G over the base E .

Gauging has developed along three quite different lines. All set out from Einstein’stheory of gravity, both have a boson vector field Bµ, the bgauge bosonc vector field, as abasic variable, and both have several descendants along the line.

In the Weyl line, which includes theories of Yang-Mills and the standard model, thegauge boson vector Bµ is a connection analogous to the Christoffel vector-connection Γµof gravity. One introduces a covariant differentiator Dµ = ∂µ − Bµ as basic field variable,making the replacement ∂µ ⇐ Dµ throughout the ungauged field theory. This is a bgaugeconnection theoryc.

In the Kaluza line, which includes theories of Klein, DeWitt, and others, the basemanifold is a Cartesian product EM ⊗ G, where EM is a pre-gauge event manifold and G

6.5. FULLY QUANTUM GAUGE THEORIES 235

is the lower gauge group. Vectors in E ⊗ G have a composite index (µ, α) of a Minkowskiindex µ and an index for a basis in the gauge Lie algebra dG. Therefore a causality form〉g〈 on E ⊗ G has a Minkowskian external term gνµ, an internal gauge term gβα, and achronometrical term gµα coupling internal and external differentials. On this line

Bµα = gµα. (6.77)

This is a bgauge metric theoryc.In the BRST line, the usual differential operators representing translation are sup-

plemented by nilpotent operators that can be interpreted as monadics of never-observedquanta, the ghosts. The higher gauge transformations become translations of the polyadicsof a new field, the BRST field.

Under full quantization, neither the lower nor the higher gauge transformations remainlocal. In general relativity the stratum F gauge group can be taken to be the diffeomorphismgroup G[E ] of the space-time event manifold E , with a Lie algebra a(E) defined by singularcanonical relations like

[∂µ, xν ] = δνµ, [δνµ, xλ] = 0, [δνµ, ∂µ] = 0. (6.78)

There are several choices for the lower bgaugec group whose gauging gives the diffeomor-phism group. For now we adopt Feynman’s choice, the translation group R4.

Every bgaugec theory has an equally singular relation of much the form (6.78), and areplacement

∂µ ⇐ Dµ. (6.79)

The coordinate xµ can be treated as a trivial scalar field. Therefore the simple bquantizationcof bgravityc can be a guide for that of the other bgaugec fields, especially if the other bgaugecfields are simply aspects of bgravityc in higher dimensions, as bKaluzac suggested.

6.5.1 Fully quantum gauging

Einstein created general relativity to provide a bfieldc theory of gravitational interactionsthat could replace the theory of action at a distance of Newton, who had explicitly declaredhis theory of gravity to be unphysical on grounds of its non-locality. To quantize his theoryis to synthesize Einstein’s bfieldc theory with Newton’s particle theory.

By gauging we mean a heuristic process for converting a special bgaugec theory to ageneral one. The gauging process of general relativity is a prototype for gauging in gen-eral. It represents the Minkowski space-time differentiator as singular limit of a covariantdifferentiator:

∂µ ⇐ Dµ = ∂µ + Γµ (6.80)

We construct a simple quantum counterpart, having the usual gauging as a singular limit;call it fully quantum gauging .


Gauging begins by enlarging the group of allowed coordinate transformations from aLie group to a function group. The allowed coordinates of special relativity form a vectorspace and a commutative Lie algebra a(xµ). The allowed coordinates of general relativityform instead a commutative algebra Alg(xµ), with the same elements as a(xµ) in a certainrepresentation. To fit this into a simple framework one drops commutativity and providesa fully quantum interpretation for the gauging process, as follows.

First the commutative Lie algebra of the four basic coordinates, a(xµ), is simplifiedto so(4, 2);σ) ⊂ so(10, 6), whose defining representation consists of mappings 6R → 6R.This 6R with its metric of signature σ is interpreted as a subspace of the vector space S[C]

of the hypothetical quantum element of stratum C. The Lie algebra so(6;σ) is interpretedas defining the statistics of this quantum. The defining 6 × 6 representation consists ofcertain spin variables of the q differential. Aggregating these variables by means of theiterated bcumulatorc ΣE−C results in an isomorphic Lie subalgebra of a much larger Liealgebra, that of the event stratum E:

ΣE−D’ so(6;σ) ⊂ so(V E). (6.81)

The associative algebra Alg V E algebra is interpreted as the kinematical algebra of theevent. The commutative algebra Alg(xµ) of general relativity is a singular limit of Alg V E.

According to simple quantum relativity, then, space-time is a catenation, probablymultiple, of q differentials, self-organized to freeze out one variable ω56 that serves as i,and approximated in a singular limit where Eωµ6 is ignorable [75].

The transition from special to general relativity is a singular limit of a transition fromthe kinematics of a q individual to that of a q catenation. Gauging is catenation seen in asingular limit.

In what follows hνµ is again the Minkowski metric, gνmu(x) is the Einstein metric,Dµ is the covariant differentiator, G is the dimensionless form of the Newton gravitationalconstant, and Lg is the Hilbert gravitational action. Gauging replaces Poincare-invariantstructures that break general covariance by local generally covariant structures, replacinghνµ by gνµ(x), and replaces the differentiator with respect to special (inertial) coordinatesby the covariant differentiator with respect to general coordinates. If one simply viewsspecial relativity in general coordinates, the Minkowski constant metric becomes a variablemetric field, hνµ ⇒ gνµ(x), and one that is flat, obeying Rνµλκ = 0. Einstein replaced thefixed metric by a dynamically variable metric field, and replaced flatness by the Einsteinequation Rνµ = κT νµ, based largely on analogy with electromagnetism.

At the same time, the constant differentiator ∂µ keeps its form while that for vectorfields is replaced by the covariant derivative, ∂µ ⇐ Dµ = ∂µ − Γµ(x). Γµ(x) is a Lie-algebra-valued vector field Γµ(x) = (Γµλκ(x)), and transforms so that Dµ transforms asa vector. The condition of metricity, Dµgλκ = 0, was assumed mainly for convenience inthe absence of experimental evidence to the contrary. In the present context, the absenceof experimental evidence is reason to assume non-metricity, since non-metricity is genericand metricity is not.

6.6. FULLY QUANTUM METRICS 237

The quantum vectors of stratum D are quantum correspondents of classical space-time tangent vectors, with extra components for binternal dimensionsc. The invariance Liealgebra of SD as bilinear space is, however,

so[D] := so(216; 64)n, Dim so[D] = 215(216 − 1) (6.82)

The simple theory unifies the space-time and a differentiator (or momentum) in the Liealgebra so(6) of of dimension 6, imbedded here in S[C] = 16R. The covariant differentiatoris an extension of the differentiator from scalars to vectors. These are singular limits ofoperators already included in the representation algebra of so(6).

In the canonical quantum theory the variable differentiator Dµ(x) is a bfieldc of localdifferentiators. In the fully quantum theory the field is replaced by a queue. Associationtakes us from the representation of so(6) on stratum C to that on stratum F.

Concretely, let the c space-time coordinates of the event be xµ and the differentiatorbe ∂µ. For any operator X : S[T ]→ S[T ], ΣX : S[T +1]→ S[T +1] designates the inducedrepresentation of X one stratum higher. Then the full quantization proposed is

so[C] ⇐ so[F],σDβα ⇐ σF

ba, (6.83)

The indices a, b label basic monadic generators of SF, and have enormously greater rangethan the indices α, β, which label basic monadics of SD. Indeed, the matrix of variablesσDβα is a small corner block in the matrix of variables σFaba.

This implies the correspondences

xµ ⇐ ΣF−Cω[C]µ5 ,

Dµ ⇐ ΣF−Cω[C]µ6 . (6.84)

6.6 Fully quantum metrics

This chapter fully quantizes the Einstein kinematics of bgravityc; the the Einstein-Hilbertdynamics is fully quantized in Chapter 7.

In his famous inaugural dissertation, bRiemannc noted that discrete multiplicities havenatural metrical structures determined by counting, while continuous multiplicities have nonatural metrics but must import them from outside themselves[61]. As though obedient tothe continuum clause of Riemann’s Principle, bEinsteinc and bCartanc imported a metric“from outside”, a real quadratic form g(v) := vµ

′(x)gµ′µ(x)vν(x) on space-time vectors v,

interpreted as the square of the proper time differential. The language Cartan chose fordifferential geometry was the exterior calculus, an instance of Grassmann algebra. Theexterior algebra, however, does not describe a metric or linear operators on itself. Cartanresorted to a foreign language, the tensor calculus, to represent the metrical form. This


could also be used to express linear operators on the exterior algebra. A more economicalextension that provides a metric is a Clifford algebra.

If one asks Riemann’s Principle whether metrical structures of quantum multiplicitiesare to be imported or domestic, doubt arises because quantum multiplicities are neitherpurely discrete nor purely continuous, but quantum, having aspects of both: Their Hasse(lattice) diagrams are horizontally continuous due to superposition and vertically discretedue to dimensionality.

Riemann himself seemed to prefer the discrete path and the domestic metric, due to itsconceptual unity, but the conflict between the known continuous symmetries of nature andthe discrete symmetries of any hypothetical discrete spaces closed this path to physicistsbefore the quantum theory.

The Clifford algebra language brings in a metric form, and so is not as totally inade-quate as the exterior algebra language, but its metric form is constant for a given algebra,appropriate only for a flat space. The self-Grassmann algebra S, however, constant metricforms, enough to approach the variable form of a Riemannian manifold as a singular limit.

Riemann’s principle works for topology too. Ordinarily a topology is defined by a setof sets of points; for example, by the set of all closed sets. Thus topology bridges threestrata. A continuous manifold must import a topology, for example that of its importedmetrical structure, and a discrete set can have a natural discrete topology like that of acheckerboard.

In a fully quantum theory, a continuum topology emerges as a singular limit of infinitelymany points.

Finally, Riemann’s dichotomy applies to dynamics too, which may be imported ordomestic.

Classical dynamics too has a kind of metric, the action integral of a history. A classicaldynamics must be imported for continua but may be domestic for discrete multiplicities.In quantum theory the dynamics of the isolated quantum system can be domestic. Thepresent fully quantum dynamics employs natural topological and metrical constructs forthe isolated quantum system. On the other hand the experimenter must be allowed toinfluence the dynamics from outside the system, or measurement would be impossible.

Since the probability form seems conceptually prior to a chronometric, in that it op-erates on a lower stratum, it is constructed next.

6.6.1 Fully quantum probability forms

Higher strata provide reducible representations of the isometry groups SO(V ) of lowerstrata. There are physical variables besides the symmetry generators, describing internalstructure. But such a reducible representation of SO(V ) can have several quadratic formsinvariant under SO(V ), so more physics must be injected to fix a probability form.

For example, a Clifford algebra C has a commutator Lie algebra, and therefore abKilling normc v 〉k〈v := Tr(∆v)2, as well as a bmean-square normc v 〉H〈v = Tr v2/Tr 1.


The physics of Fermi statistics is useful at this point. There a many-quantum vector〈Ψ ∈ W0 is a Grassmann polynomial in one-quantum vectors 〈ψ ∈ V . Its Fermi dualHΨ〈 ∈ W0

D is the transposed polynomial in the dual vectors Hψ〈 ∈ V D. The probabilityform defines and is defined by the quadratic form

‖Ψ‖ := Ψ〉H〈Ψ = Ψ〉H〈Ψ〈 . (6.85)

A corresponding procedure works word-for-word for Clifford algebras as well and pro-duces a probability form that reduces to the Fermi form in the singular limit.

In the Clifford case a probability form is built into the algebra 2V by Clifford’s Clause.Taking λ = 1, we have

∀ψ, φ ∈ V : ψHψ = ψ2. (6.86)

It follows that Hψ = Grade0 Lψ: to act with the breversec of ψ, multiply by ψ from theleft and then take the scalar part. The probability form H has a natural extension from Vto 2V :

∀Ψ ∈ 2V : HΨ = Grade0 LΨ (6.87)

This defines H in terms of left multiplication L and taking the scalar part Grade0. Moreover,if H is positive definite on V then H is positive definite on W = 2V . However an algebra A,Lie or associative, generally has many bnatural quadratic formsc. The H just constructedis one of many that have the proper specialization to Fermi statistics, but is the mostplausible and is tentatively adopted. Briefly,

H = Grade0 L. (6.88)

In other words, the probability form of a queue is assumed to be the mean-square form(3.10).

6.6.2 Fully quantum causality form

Now we turn to a construction of the fully quantum variable corresponding to Einstein’sgνµ(x).

In physical and mathematical practice, the system always has an underlying probabilityform, which enters into the construction of the chronometric. One such two-metric spaceis M4, the 4-dimensional Minkowski tangent space, consisting of tangent vectors vµ atsome arbitrary space-time origin O. The more familiar metric of the two is Minkowski’sbcausality formc

v 〉g〈v = vνgνµvµ (6.89)

giving the squared proper time of the displacement vµ. Implicit, however, is a classicalprobability form defining the Boolean logic of predicates about differentials. This is thesingular bprobability formc on M4 given by

v′ 〉H〈v = δ4(v′ − v); (6.90)


the transition probability amplitude from one classical tangent vector 〈v to another 〈v′is 0 unless they are identical. This is singular because these tangent vectors are classicalobjects with commuting coordinates and can be observed without changing them. Theprobability norm defined by 〉H〈 is then the ill-formed expression

v 〉H〈v = δ4(v − v) = δ4(0) (6.91)

telling us only that all the unnormalizable orthogonal vectors 〈v have the same infinitenorm.

The charge stratum C is surely at or below the differential stratum D. These formsdiffer so much on higher strata that they must migrate upward quite differently. In classicalgeneral relativity the causality form gν,µ(x) is the main dynamical variable. In the classicaltheory each value of gν,µ(x) defines a Clifford algebra C(x) over the tangent space at eachevent x. The C(x) are all isomorphic to each other. For each value of µ = 1, 2, 3, 4 thetangent vector ∂µ at x is written as γµ(x) when it is used as a Clifford algebra element.

The remaining classical variable is a connection (form) Cµ(x) used to construct anisomorphism from the Clifford algebra at one event to that at any nearby event,

Cµdxµ : C(x)→ C(x+ dx). (6.92)

It is important to fix a physical interpretation for these quantities. The relation of thecausality form to chronometry is not a useful clue for the microscopic interpretation, sincewe have no microscopic chronometers, but must emerge in the macroscopic limit. This isan important condition on the interpretation.

The components gνµ(x) are variables whose values are numerical functions of x in theclassical theory but operators in the canonical quantum theory, creating and annihilatinggravitons in the linearized approximation. This means that in classical gravity the γµ arenot regarded as quantum objects but as classical tangent vectors, provided with a productthat encodes both the Grassmann product and chronometrical information; or as Cliffordput it, both topological and metrical structure.

But it is also true that if a quantum electron is place in the classical gravitational fieldthe γµ serve as operators in the kinematical algebra of the quantum spin. It is inferredthat the γµ(x) Clifford algebra of the classical space-time manifold is like the Poissonbracket of classical mechanics: a tip of the buried quantum structure that protrudes intothe classical surface structure. It is not merely a variable to be quantized, like the space-time coordinates, but a piece of the quantum theory to be respected in the quantizationprocess. One did not quantize the Lie product of classical mechanics (the Poisson Bracket)but kept it intact as the quantum theory beneath it was excavated; one can do the samewith the Clifford product of special and general relativity.

The bchronometricc form 〉g〈 on the field stratum F, the gravitational potential, istaken to be the form in stratum F induced by the probability form 〉H〈 of stratum C.


080601 . . . . . . [Express the classical gravitational potential field gµν(x) is a singularlimit of 〉g〈 . Show that the classical limit of the fully quantum graviton annihilator definesa causality form like Einstein’s. ]

This relation between the time and probability forms is another balgebra unificationc,echoing the one in §3.3.3 that occurs in canonical quantum theories. This one provides adeep bunificationc of general relativity and quantum theory. It uses the uncertainty on adeeper quantum stratum C to define a clock and a meter stick for classical dynamics onthe field stratum F.

For a more symmetric choice of constants, one may replace the pair (~,X) by a pair(E,X) of natural energy and time quanta related by

EX = ~. (6.93)

ϑo has two adjustable quantum parameters: the quantum of action ~ and the quantumof space-time X. In addition it has a large pure number N that characterizes the specificrepresentation of its Lie algebra. In the singular limit, X→ 0 and NX→∞.

Existing theory also contributes relevant parameters c and G of special and generalrelativity. Since a non-relativistic limit is not taken here, one sets c = 1. G enters intothe Hamiltonian and thus into the dynamical Lie algebra of the canonically quantizedbgravitycof bArnowittc,bDeser, and cbMisnerc [2], but not into the kinematic Lie algebra ofthe functional quantum theory, which is based on history vectors rather than instantaneousones. Since the functional precedent is followed here, the constant G first appears here inthe dynamical history tensor.

Therefore a quantum event space SE need not be metrized from outside like Einstein’ssingular space-time. Its special coordinates form a simple Lie algebra, for example so(DS)in one model. In any case the Lie algebra of special coordinates has a natural causalityform derived from the probability form of stratum C.

In general relativity the proper time between two points 1 and 2 is a “continuous sum”

τ =∫ 2

1|dx|, |dx| = [gνµ(x)dxµdxν ]1/2 (6.94)

of contributions from each Minkowski form gνµ(x) along the path from 1 to 2. All theseforms are isomorphic but there is no natural isomorphism among them, before the pathdefines one. This is expressed by general bgaugec invariance, invariance under a translationx 7→ x + a(x) that varies from event to event. Similarly the Klein-Gordon action term∫

(dx)4 gνµpν(x)pµ(x) is a continuous “sum” of bilinear catenations of many momentumvectors at different events x.

A catenation of quantum differentials with Clifford statistics, and therefore with eventvector space ES = PDS, was already assumed in ( 4.33 ) as quantum correspondent tothe catenation of Minkowskian tangent spaces in a manifold. Now its metric tensor isconstructed.


Einstein replaced the Minkowski hνµ ⇐ gµµ(x) to make a theory with general covari-ance. Now we have one with quantum covariance constructed in the following strata:

1. The causality form 〉g〈 and the probability form 〉H〈 on stratum C are both thebmean square formc 〉H〈 .

2. The bKilling metric formc 〉k〈 defines an operator in any faithful irreducible repre-sentation of so(n), namely the quadratic Casimir operator k, which now representsa one-quantum observable.

3. Since so(n) acts on SD, so does its Casimir operator k.

4. The classical metric field arises from a many-quantum catenation whose vector spaceis SF.

5. Then the bquantum chronometricc operator is the many-quantum operator

g = IF−C k ID(F−C): SF → SF (6.95)

induced in stratum F by k in stratum C.

In words, to apply the quantum chronometric to a vector in stratum F, one unbraces andfactors as many times as necessary to reach stratum C, applies the Killing metric, andremultiplies and rebraces to return to stratum F.

Currently the exponent F− C is tentatively assumed to be 3.Thus if special relativity is regarded as having one quadratic form, that of Minkowski,

then general relativity has an infinity of quadratic forms, one on each tangent space, all un-naturally isomorphic; and simple quantum relativity has a large finite number of quadraticforms, one for each quantum event.

Schematically speaking, we can pass from the quantum back to the classical as follows.The quantum metric form Σk(L) is again a polynomial in a great many momenta pµ ∼ ωµ6.Each momentum is accompanied by coordinates xµ ∼ ωmu5. In a singular limit these p, xvariables can be treated as commutative and assigned simultaneous eigenvalues. If thedensity of values in the spectrum is low, most values of xµ do not occur, and most of thosethat occur do not occur twice. Then the collection of pµ, xν pairs nearly defines a partialfunction pµ(x), a co-vector field. If the dynamics prevents large changes in pµ for smallchanges in xµ, a smooth function can be formed from this partial function by interpolatingbetween the values already defined. Then the quantum metrical form approaches theclassical general relativistic one.

When two events in a Clifford vector are exchanged, the vector merely changes sign.The same values of the coordinates are paired with the same values of the momentumvector, so that the field pµ(x) is unaffected. The surface appearance of Bose statistics in

6.7. COVARIANT DIFFERENTIATOR 243

bgravitational fieldc theory is an artifact of the singular limit. It has nothing to do withthe deep statistics, which is Clifford.

Let 1µ(x) be a tangent vector to the µ coordinate axis at the event x in classicalrelativity. The c metricity and atorsional conditions are

Dµgλκ = 0, Dµ1ν −Dν1µ = 0 (6.96)

respectively. They reduce all the degrees of freedom of the c differentiator Dµ to thoseof the c metric gλκ. They are, however, unstable and so are not expressed in the genericLie algebra FG of stratum F. We must see what variants of metricity and atorsionality theaction principle implies.

Moreover in the canonical quantum theory Dµ and gνµ are Bose fields. Therefore theirfully quantum correspondents are assumed to be bPalevc catenations of Clifford pairs.

6.7 Covariant differentiator

In the simplified quantum theory, as in bSegal-Vilela-Mendes spacec, the events have mo-mentum coordinates which are unified with their space-time coordinates as generators ofthe same Lie algebra

so(6) ⊂ so(S[C]) P⇒ so(SD) P⇒ so(SE) ⊂ Alg SE. (6.97)

The distinction between space-time coordinates and momenta returns in the singular limitof physics.

The extension of the classical differentiator from scalar to multivector fields is theexterior differentiator. This extension is purely algebraic, introducing no new element ofgeometric structure. The same is assumed here for the quantum theory.

A covariant differentiatior Dµ(x) is an extension of the ordinary differentiator ∂µ fromscalar to the most general tensor fields, preserving the Leibniz product law. It is notunique but a dynamical variable. It defines a tensor connection ΓK

ΛM, where K,Λ,M, . . .

are collective tensor indices, and

Dµ(x) = ∂µ1 + Γµ; (6.98)

here ∂µ is the Lie differentiator with respect to the vector field 1µ tangent to the µ coordi-nate axis, 1 is the unit operator on the local tensor space, with elements δN

M, and Γµ(x) isgenerally a non-trivial Lie algebra element acting on the local tensor space, with elementsΓµΛ

K(x).The gauge variable Dµ(x) is usually subject to the conditions

1. Dµ(x) is a tensor differentiator: that is, it is a linear operator on the space of tensorfields, transforms as a co-vector, and obeys the Leibniz Relation.


2. Equivalence principle: At any event x0 there is a suitable general coordinatetransformation that transforms Dµ(x0)⇒ ∂µ at that point.

Here a factor of a unit operator is implicit. 1. implies that Dµ(x) restricted to scalar fieldsis indeed ∂µ. 2. implies that Dµ(x) restricted to vector fields differs from ∂µ by a metricaland atorsional Levi-Civita connection.

As Einstein set out for general relativity from the Minkowski space of special relativity,we start out for simple quantum relativity from Segal-Vilela-Mendes space.

6.8 Reciprocity

Let us call the x–p symmetry reciprocity, though bBornc’s breciprocityc acted only on ahigher stratum, that of dynamical law [15]. Since xµ is local and pµ, being off-diagonal inxµ, is slightly non-local, breciprocityc breaks blocalityc in the quantum theory.

To describe the field organization that breaks reciprocity and restores blocalityc wemay define a blocality algebrac homomorphic to 2R⊗ 2RD, a real version of the Pauli spinalgebra. It contains the seeds of both locality and reciprocity as anticommuting operators.

The blocality algebrac is a 2 × 2 real matrix subalgebra of the algebra with basismatrices

L0 =[

10

01

], L1 =

[01

10

], L2 =

[01−10

], L3 =

[10

0−1

](6.99)

regarded as defining a quantum space. In a basis of eigenvectors of blocalityc like t and E,the reciprocity operator L2 generates the transformation

L2 : δxµ = pµ, δpµ = −xµ. (6.100)

and the locality operator L3 generates the transformation

L3 : δxµ = xµ, δpµ = −pµ. (6.101)

The symmetric operator

L1 = [L2, L3]/2 (6.102)

generates blocalityc and breaks breciprocityc. We can call (6.99) the locality basis of thelocality algebra, since the locality operator L3 is diagonal in this basis.

In the Segal-Vilela-Mendes so(3, 3; R) Lie algebra, we may reserve the first four basisvectors 1A (A = 1, 2, 3, 4) for the usual Lorentz so(3; 1; R) Lie algebra and the last two(A = X,Y ) for the blocality algebrac. Since the anti-symmetric operator LXY interchangesxµ and pµ, it represents breciprocityc L2; the Segal-Vilela-Mendes basis is a blocalityc basis.

6.9. FULLY QUANTUM COVARIANCE 245

6.9 Fully quantum covariance

A fully quantum general relativization rule will now be developed that has the familiarclassical general relativization rule as a classical limit.

The generic event vector space SE has high dimensionality

Dim SE = PE−DDim SD Dim SD. (6.103)

Like general relativity, simple quantum relativity has special coordinates at the differ-ential stratum and general coordinates at the event stratum. The general coordinates forman algebra. The special coordinates do not form an algebra but merely the Lie algebraso(SD).

Since the Clifford event has vector space SE, the quantum field history F has vectorspace SF = PF−E SE = PF−D SD and coordinate algebra Alg SF.

A significant lapse of correspondence now appears. The vector space SE is a directsum of Grassmann products of SD’s. There is no natural isomorphism among tangentspaces; but the factor spaces in a Grassmann algebra are identical to each other. Generalcovariance blocks such an identity between tangent spaces at different events.

To be sure, Maxwell(-Boltzmann) statistics has no such antisymmetrization and re-quires no such identity between factor spaces. Distinguishable particles can form Maxwellcatenations, which are merely sequences. Maxwell statistics alone is invariant under inde-pendent automorphisms of all its factor spaces.

In the classical limit, all forms of statistics approach the Maxwell statistics. Thereforeeither the proper statistics for quantum differential events is Maxwellian, or general co-variance is an artifact of a singular limit. Since no quantum object is known with Maxwellstatistics, I conservatively suppose that it again arises in a singular limit.

If the proper statistics is Clifford, then to approach Maxwell statistics the differentialevent should have a vector space of high dimension, approaching infinity in the classicallimit, and the number of occupied vectors in that space should be low, so that identity oftwo occupied vectors becomes an event of low probability that can be ignored.

We already have been forced to the high state-vector-space dimension by other require-ments.

As for the low occupation, the classical limit of a bosonic field like gνµ(x) requiresorganizing a large number of effective bosons with high occupation numbers of bosonicvectors, although our bosons are pairs of fermions, which exclude each other. These re-quirements are compatible provided the bosonic pairs of fermions have internal degrees offreedom that are sufficiently excited, like Cooper pairs in superconductivity, so that theexclusion principle rarely applies.

Therefore if Clifford statistics holds, there are violations of general covariance far fromthe classical limit, at short distances and high energies. We already expect such violationsdue to non-locality, so this is not a serious problem.


To put this assumption on a more secure foundation we recapitulate the formulationof simple quantum general covariance for Clifford events.

Let a(v; 0, 0) be the Lie algebra of the diffeomorphism group of the c/c event manifoldE = R4. An element of a(v; 0, 0) is a vector field v = vµ(x)∂µ on E . The product in a(v; 0, 0)is the Lie Bracket [u, v]L. Let a(v; ~,X) be the simple variant of this Lie algebra that weseek. It is not the diffeomorphism Lie algebra of any space; call it the generic covariance Liealgebra, and rename the usual diffeomorphism algebra a(v; 0, 0) the “general covariance”Lie algebra to indicate correspondence.

Let ωβα be the 16× 16 matrices generating so(C). Let

ωβα = ΣE−Cωβα (6.104)

be the operators on stratum E induced by the ωβα, generating a Lie algebra so(E) :=ΣE−C so(C).

Assertion 5a(v; ~,X) = ∆ Alg (SD), (6.105)

the commutator (adjoint) Lie algebra of stratum D.

Argument The generic form of the vector field vµ(x)pµ is Poly(ωµ5)ων6. These polynomi-als generate the matrix Lie algebra Poly(ωβα) ⊃ so(SE) with the commutator Lie product.

[Do:Check.]Step 1 in bsimple quantizationc is to convert classical event coordinates and differen-

tiatiors into generalized Segal-Vilela-Mendes coordinates in the Lie algebra Σ so(SD) ⊂so(SE):

xµ ⇐ X ωµ5,∂µ ⇐ E ωµ6,

vµ(x) ⇐ V µ(X ωµ5),v = vµ(x)∂µ ⇐ v = E vµ(X ωµ5)ωµ6, (6.106)

where the vµ(. . .) are arbitrary ordered polynomials in the indicated operator arguments.Step 2 is to close the vµ into the Lie algebra that they generate, the quantum general

covariance Lie algebra. This is the Lie algebra of all ordered polynomials in the ω(βα)(SD),and must therefore be

a[E] = so(SE) ⊂ Alg[E]. (6.107)

Step 3 is to convert the variable hνµpνpµ of special relativity into the variable g(L) ofsimple quantum relativity.

[Do: Check.]The singular limit that converts simple quantum relativity to general relativity must

undo these steps in reverse order.

6.9. FULLY QUANTUM COVARIANCE 247

To invert Step 3[Do: Invert 3.]To invert Step 2 and recover the vector fields

v = E vµ(X ωµ5)ωµ6 (6.108)

evidently one must let E X → 0 with E ~/X, and keep only the leading term inE vµ(X ωµ5)ωµ6. This can be a Lie-algebra homotopy.

To invert Step 1 and recover the Lie Bracket, one must freeze the variable ω65 thatcontributes the right-hand side i of the commutation relation between xµ and pν , as in(5.23):

i := ω65/N, N =[max(ω65)2

]−1/2. (6.109)

In nature, only four dimensions of the hypothetical 15-dimensional so(6) of the dif-ferential stratum D organize into macroscopic space-time dimensions of the event stratumE, and the structure of so(6) does not determine which four, so a spontaneous break-down of so(6) invariance seems to make this determination, leaving an unbroken so(3, 1)Lorentz subgroup in the tangent space at each event. The kinematics permits a differentfour dimensional subspace of so(6) to be so favored at different events. There is no nat-ural isomorphism between these different subspaces, even of the same dimension, and anapproximate general covariance can emerge.


Chapter 7

Fully quantum dynamics

for gravity and the standard model.

7.1 The history vector

In classical mechanics the dynamical development is described by an equation of motionof the Lagrangian form

δS = δ

∫Ldt = 0. (7.1)

In the canonical quantum theory we learn that this is not the exact dynamical law, butresults from a deeper quantum dynamical law of a more statistical nature as an approx-imate condition for constructive interference. The deeper dynamical law merely assignsa quantum probability amplitude to each history. In the usual formulation, pioneered byDirac, a certain history vector 〈D~ (of the form (1.39) defines the quantum dynamics. Thisis less singular than the dynamics of the form (7.1), since it softens the commutativity ofthe coordinate with the momenta with the parameter ~.

But the form 〈D~ is still singular. It preserves the commutativity of the coordinatesand of the momenta separately and implies a measure on the space of histories that doesnot exist. It is a research program, not a mathematical statement. This leads to theexpectation that the history vector 〈D~ is a singular limit of a deeper regular dynamicallaw 〈D~X.

An exponential form〈D~X = 〈e−eS (7.2)

will be assumed for the dynamics vector in a fully quantum theory as in the canonicaltheory. A factor i has been absorbed into the skew-action S, which is an antisymmetricgenerator, not a symmetric observable. Usually a time-ordering operation T is applied tothe factors in 〈D; this can be omitted here.

249

250 CHAPTER 7. FULLY QUANTUM DYNAMICS

For well-known reasons Einstein chose a classical action density R bilinear in thecovariant differentiator D. What actually appears in the physics is the skew-action densityiR. The corresponding queue expression is designated by R. In R the canonical i must bereplaced by a suitable dyadic which in one frame is i = NωE56. Now the skew-action oflowest non-trivial order in the dyadics is trilinear in ω instead of bilinear.

The action trilinear in dyadics hopefully reduces near the vacuum to one that is bilinearin monadics and linear in dyadics, like the usual action for the fermions of the standardmodel.

Recall the Dirac relation between Hamiltonian and Lagrangian,

t〈e−iH∆t〈 t′ = Net〈 iL∆t〈 t′ (7.3)

with a normalization factor N that diverges as ∆t := t − t′ → 0. In (??) Ψ~X is writtenas an exponential of an operator, not of matrix elements, to be basis independent, so theexponent corresponds to a Hamiltonian rather than a Lagrangian.

In a fully quantum theory, the Lagrangian form of the history amplitude emerges onlyin the limit of classical time, which singles out a basis in which space-time variables are alldiagonal.

In the canonical theory the Lagrangian form is invariant under a larger group of space-time coordinate transformations than the Hamiltonian, because L is a scalar and H isone one component of a vector. Here the Hamiltonian form is more invariant than theLagrangian because it does not break the canonical group, as preferring a time variableover its canonical conjugate, the energy, does. This invariance is possible because the sumin (7.2) is over dynamical elements, which are invariant entities, not over instants of time,which are not.

Correspondence requires an approximation, admittedly singular, relating the fullyquantum dynamics vector 〈DX to the canonical one 〈D0:

〈DX = eΣeH → u(tn, tn−1 ⊗ . . .⊗ u(t1, t0) = 〈D0; (7.4)

u(t′, t) = e−iH(t)(t′−t) is a unitary operator representing a small dynamical developmentfrom time t to t′.

One still imposes a condition for constructive quantum interference on Ψ~X to recoverthe usual dynamics. Then Ψ~X can be regular because its space and time coordinatesbelong to a simple Lie algebra. This does not break Lorentz invariance, though it imbedsthe Lorentz group as a non-normal subgroup in a larger simple group.

If one tolerates singular formulas without clear meaning, then a classical dynamicstoo could be associated with a vector Ψ0, a functional of a classical history, giving infiniteprobability amplitude to histories obeying (7.1) and 0 amplitude to others. There is thena sequence of singular limits

Ψ~X → Ψ~ → Ψ0 (7.5)

that must be reversed to reconstruct the fully quantum theory.

7.2. HIGHER-ORDER TIME DERIVATIVES 251

7.2 Higher-order time derivatives

In the canonical theory it is assumed that the dynamical equations are of first order inderivation with respect to time, ∂t. To be sure, the classical canonical equation of motion,

dQ

dt= [H,Q]P, (7.6)

equating the time derivative to a Poisson bracket, is of first order in ∂t. Since the classicaltheory is merely an approximation to the quantum theory, all one can deduce from theclassical theory is that higher order time derivatives must be a properly quantum effect,vanishing in the limit of classical time.

Linearity in ∂t is equivalent to linearity in the energy operator E. Linearity in energy,however, would almost always be a good approximation for sufficiently small energies. Inthe fully quantum theory it is natural to expect that higher powers of the energy appearin the dynamical equations and can be neglected in the limit of classical time because thatis a low-energy limit.

7.3 Spinorial dynamics

A bFermi quantum theoryc is a fully quantum theory whose vector space is a subspace of Sof bounded stratum number. The operator algebra is then a Clifford algebra. The vectorspace is a spinor space for that Clifford algebra.

Let sα ∈ S[L] be homogeneous polyadics forming a basis for S[L]. It is convenient toindex them by their bBaugh numberc, since they can be naturally generated in that order.Then the first of them is s0 = 1 and the last is the top element sN = s>.

Their braces Isα form a basis for the first-grade elements of the next stratum S[L+1].Let γα := Lsα : S[L+1] → S[L+1] be left-multiplication by the sα.Let H be a symmetric form on the Grassmann algebra S[L+1] in the Isα basis:

[sβ, sα]− = 2hβα = Tr sβsα. (7.7)

sα := Hαβsβ.o : S[L] → S[L] is an infinitesimal linear transformation with matrix elements oβα in

the basis sα.Isα = sα ∈ S[L+1] are the monadics produced by bracing the sα.

ω = σ(o) =12γβ o

βα γ

α (7.8)

is a spinorial representation of o, with

∀Ψ ∈ S[L] : oΨ = [ω,Ψ], (7.9)

in virtue of the Fermi commutation relations.


7.4 Gravity action

In the following, we consider the Lie algebra so[S[C]], which includes the so[6R] underlyingbSegal-Vilela-Mendes spacec, and its representation R by induced transformations of somehigher stratum L, constructed by iterating the cumulator Σ the necessary L− C times.

Let γβα = −γαβ be the usual basis for the Lie algebra so[S[C]] represented on S[D].Let kβ′α′βα be the invariant Killing form of so[C]. Let ΣF−Cγβα be the representation onstratum L of γβα, induced by F−C applications of the cumulator Σ.

Then the usual Casimir operator of the representation R is, up to a conventionalconstant,

K = ΣF−CγβαΣF−Cγβα. (7.10)

The Casimir operator for an irreducible representation is a scalar. The representationΣF−C is highly reducible, and K is highly non-scalar and non-central.

This permits the following tentative

Assertion 6 The operator for fully quantum gravity corresponding to the Hilbert actionfor classical gravity is the Casimir operator K on S[F] up to a numerical factor. It isunacceptable as a queue action.

Argument In the limit where ωµ6 → ipµ and the other ωβα → 0,

K → K1g66gνµpνpµ. (7.11)

To relate this mathematics to the physics of moving bodies and clocks one first observesthat the bKilling formc has defining features of the Minkowski quadratic form: It is invariantunder the special covariance group, now simple, and it reduces to proper time dτ2 = dt2

in the rest-frame of the singular limit. Before the singular limit, to be sure, there is norest frame: the spacelike momentum variables do not commute and cannot all be 0 exceptwhen all the ωβα are 0, and in the vacuum the variable ωY X is not 0 but as large as itcan be. The primary interpretation of this form is as the bKilling formc, and its furtherinterpretation is fixed by how it appears in the action principle.

7.4.1 Cosmological constant

This raises the question of the contribution of event energy to bdark energy densityc, thebcosmological constantc Λ, which appears in the Einstein equations as a part

T νµ(Λ) =c4Λ8πG

gνµ (7.12)

of the source stress tensor T νµ. The observed baccelerating expansionc of the universeindicates a dark mass spatial density

ρ(Λ) =c4Λ8πG

≈ 6× 10−27kgm−3, (7.13)

7.4. GRAVITY ACTION 253

about 75% of the critical value, according to

http://www.astro.ucla.edu/~wright/cosmo_constant.html

[?]. In Planck units this is about 10−123. The edge of a cube containing one Planck massof dark energy is about 2137 Planck lengths, or 106 m. The size of a light diamond (doublecone) containing one quantum ~ of dark action is then about 1031 Planck units, 10−4 m,or 10−12 s.

This is to be compared with the contribution of quantum event energy to dark energydensity, which depends on the vacuum values of the spatial density of events and the meanenergy of each event:

e(Λ) = ρ(E) AvE (7.14)

A stress tensor calls for a conjunctive evaluation of the energy, momentum, position,and time of a volume element. These are, however, already complementary in canonicalquantum theory, and more so in a fully quantum theory. The Einstein Equation must beregarded as a singular limit of a fully quantum equation with a different meaning.

It is natural to wonder whether a dark energy density might emerge naturally fromevent energy in the singular limit. Vacuum events cannot all have energy eigenvalue 0.The fact that i ≈ i in the vacuum, and the commutation relation

[E[C], i] =E

NXt[C], (7.15)

imply a non-zero uncertainty product for energy and i unless t .= 0. This does not preventE from having expectation value 0 in the vacuum, however. Experimentally dark energydensity is locally small and globally large, dominating other sources on the cosmologicalscale. The question will be reopened when a vacuum vector is available.

080609 . . . . . . [Find the contribution to the cosmological constant from the vacuumexpectation of event energy-momentum flux.]


Chapter 8

Output

which summarizes our results.

Where canonical quantization greatly reduces the radical on one stratum, full quanti-zation eliminates it on every stratum, making a finite quantum theory possible. A bfullyquantumc theory is developed and applied to gravity and the bstandard modelc in a loop(2-form) formulation, imbedding both four-dimensional gravity and the standard modelbgaugec potentials within a higher-dimensional gravitational potential. It is constructed innested modular strata, with nested simple classical groups. In any application the stratumconstruction cuts off at a finite stratum, here the seventh. All fully quantum variables havediscrete bounded spectra. Symmetry groups of the canonical theory are preserved, eitherexactly or approximately.

In the theory ϑo under current study, Fermi quantification relates the strata, leadingto a family of nested so(nR;σ) relativity Lie algebras. Six strata A – F suffice. RanksD, E, and F have familiar classical correspondents, which are the strata of Differentials,Events, and Fields. This raises the question of a physical origin for the causality form g ofrelativity and the probability form H and imaginary i of quantum theory. The organiza-tion of i reduces the algebra so(4R) ⇒ sl(2C). The organization of g reduces the algebraso(4R)⇒ so(3, 1). But so(3, 1) ∼= sl(2C). This is taken to indicate that the organization ofi is also the organization of g, pointing to a loop (2-form) theory of gravity, where both areexpressed as a Hodge star. The space-time coordinates are generators of a representationof the sl(16R) of stratum C in the sl(2216R) of stratum E. In stratum C, g ∼= H. The g andH of stratum C organize into those of stratum F, of which the standard g and H are sin-gular limits. Events have momentum-like coordinates as well as position-loke coordinates,all generators of an sl(16R). The quantum non-commutativity of the momentum-energycomponents is the origin of space-time curvature.

One may think of the quantum structure of ϑ(20008) as a four-dimensional modular

255

256 CHAPTER 8. OUTPUT

btruss domec composed of a cosmologically large number of simplicial cells each with 16vertices. Four vertices of each cell fit into long stretchers, two are frozen into a globalcomplex plane, both organizations involving very many cells. The remaining 10 verticesare short internal stiffeners. The cosmic truss dome has four blong dimensionsc with vari-able coordinates, seen as space-time in a singular limit, two long dimensions with frozencoordinates, seen as the complex plane, and is about one strut thick in each of 10 bshortdimensionsc. A further organization must be invoked for the Higgs. There are no fields inthis theory in the usual sense, until a singular limit of classical space-time is taken. Theelements of stratum F are not functions on stratum E but quantum aggregates of quantumevents of stratum E, with Fermi statistics.

Chapter 9

ACKNOWLEDGMENT

in which the following are thanked for beneficial and pleasurable com-munications during this work

Yakir bAharanovc, James bBaughc, Walter bBloomc, Eric bCarlenc, Giuseppe bCastagnolic,Martin bDavisc, David bEdwardsc, Andrei bGaliautdinovc, Tenzin bGyatsoc, Werner bHeisenbergc,William bKallfelzc, Alex bKuzmichc, Garrett bLisic, Danny bLunsfordc, Dennis bMarksc,Tchavdar bPalevc, Aage bPetersenc, HeinrichbSallerc, Joseph bSamuelc Frank bSchroekc,Jack bSchwartzc, Sarang bShahc, Mohsen bShiri-Garakanic, Henry bStappc, Carl-Friedrichsvon bWeizsackerc, Eugene bWignerc, and Julius bWessc.

257

258 CHAPTER 9. ACKNOWLEDGMENT

Bibliography

[1] V. Ambarzumian and D. Iwanenko. Zur frage nach vermeidung des unendliche selb-struckwirkung des elektrons. Zeitschrift fur Physik, 64:563–567, 1930.

[2] R. Arnowitt, S. Deser, and C. W. Misner. Dynamics of general relativity. In L. Witten,editor, Gravitation: an introduction to current research, chapter 7, pages 227–265.Wiley, 1962. Reproduced as arXiv:gr-qc/0405109v1.

[3] R. T. W. Arthur. G. W. Leibniz, The Labyrinth of the Continuum: Writings of theContinuum Problem, 1672-1686. Yale University Press, New Haven, 2001. Translationand commentary. Page 205 of translation. I thank O. B. Bassler for this quotation.

[4] E. Artin. Geometric Algebra. Interscience, New York, 1957.

[5] N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf. Contraction of the finite one-dimensional oscillator. International Journal of Modern Physics, A18:317, 2003.

[6] T. Banks, W. Fischler, S. Shenker, and L. Susskind. M theory as a matrix model: Aconjecture. Physical Review, D55:5112, 1997.

[7] J. Baugh, 1995. Personal communication.

[8] J. Baugh. Regular Quantum Dynamics. PhD thesis, School of Physics, Georgia Insti-tute of Technology, 2004.

[9] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer. Quantummechanics as a deformation of classical mechanics. Letters in Mathematical Physics,1:521–530, 1977.

[10] P. G. Bergmann. The fading world point. In P. G. Bergmann and V. de Sabbata,editors, Cosmology and Gravitation. Spin, Torsion, Rotation, and Supergravity, pages173–176. Plenum Publishing Co., New York City, 1979.

[11] P. G. Bergmann and A. Komar. The coordinate group symmetries of general relativity.International Journal of Theoretical Physics, 5:15–28, 1972.

259

260 BIBLIOGRAPHY

[12] D. Bohm. A proposed topological formulation of the quantum theory. In I. J. Good,editor, The Scientist Speculates: An Anthology Of Partly-Baked Ideas, pages 302–314.Heinmann, London, 1962.

[13] N. Bohr. Causality and complementarity. Philosophy of Science, 4:293–4, 1936.

[14] G. Boole. The mathematical analysis of logic; being an essay towards a calculus ofdeductive reasoning. Cambridge University Press, Cambridge, 1847. Reprinted, Philo-sophical Library, New York, 1948.

[15] M. Born. Reciprocity theory of elementary particles. Reviews of Modern Physics,21:463, 1949.

[16] R. Brauer and H. Weyl. Spinors in n dimensions. American Journal of Mathematics,57:425, 1935.

[17] Paolo Budinich and Andrzej Trautman. The Spinorial Chessboard. Springer, Heidel-berg, 1988.

[18] E. Cartan. Lecons sur la theorie des spineur. Hermann, Paris, 1938.

[19] C. Chevalley. The algebraic theory of spinors. Columbia University Press, New York,1954.

[20] C. Chevalley. The construction and study of certain important algebras. The Mathe-matical Society of Japan, Tokyo, 1955.

[21] A. Connes. Non-commutative geometry. Academic Press, San Diego, 1994.

[22] B. S. DeWitt. Dynamical theory of groups and fields. Gordon and Breach, 1965.

[23] B. S. DeWitt. Supermanifolds. Cambridge University, 1992.

[24] P. A. M. Dirac. The Principles of Quantum Mechanics. Oxford, 1930. (First edition.Revised fourth edition 1967.).

[25] P. A. M. Dirac. Forms of relativistic dynamics. Reviews of Modern Physics, 21:392–399, 1949.

[26] P. A. M. Dirac. Spinors in Hilbert Space. Plenum, New York, 1974.

[27] M. Dubois-Violette, R. Kerner, and J. Madore. Gauge bosons in a non-commutativegeometry. Physics Letters, B217:485, 1989.

[28] A. S. Eddington. Fundamental Theory. Cambridge, 1948.

[29] A. Einstein. Physics and reality. Journal of the Franklin Institute, 221:313–347, 1936.

BIBLIOGRAPHY 261

[30] A. Einstein and W. Mayer. Einheitliche theorie von gravitation und elektrizitat.Sitzungsberichte der Preussische Akademie der Wissenschaft, 25:541–557, 1931.

[31] A. Einstein and W. Mayer. Einheitliche theorie von gravitation und elektrizitat. zweiteabhandlung. Sitzungsberichte der Preussische Akademie der Wissenschaft, 12:130–137,1931.

[32] L. D. Faddeev. How we understand “quantization” a hundred years after Max Planck.Physikalische Blatter, 52:689, 1996.

[33] R. P. Feynman, 1941. Personal communication ca 1961. Feynman began this line ofthought in about 1941, before his work on the Lamb shift, and may have published asimilar formula.

[34] R. P. Feynman. Nobel address. In L. M. Brown, editor, Selected Papers of RichardFeynman, with Commentary, page 3. World Scientific, Singapore, 2000.

[35] R. P. Feynman and A. R. Hibbs. Quantum Mechanics Via Path Integrals. McGraw-Hill, New York, 1965. Problem in Chapter 2.

[36] A. Fialowski, M. de Montigny, S. P. Novikov, and M. Schlichenmaier (organizers).Deformations and contractions in mathematics and physics. report 3/2006, Mathe-matisches Forschungsinstitut Oberwolfach, 2006.

[37] D. Finkelstein. On relations between commutators. Communications in Pure andApplied Mathematics, 8:245–250, 1955.

[38] D. Finkelstein. Space-time code. Physical Review, 184:1261–1271, 1969.

[39] D. Finkelstein, J.M. Jauch, and D. Speiser. Quaternion quantum mechanics I, II, III.Report CERN 59–7, 59–11, 59–17, CERN, Geneva, 1959.

[40] D. R. Finkelstein. Quantum Relativity. Springer, Heidelberg, 1996.

[41] M. Flato. Deformation view of physical theories. Czechoslovak Journal of Physics B,32:472–475, 1982.

[42] H. Georgi. In C. E. Carlson, editor, Particles and Fields-1974 (APS/DPF Williams-burg). American Institute of Physics, New York, 1975.

[43] H. Georgi and S.I. Glashow. Physical Review Letters, 32:438, 1974.

[44] M. Gerstenhhaber. Annals of Mathematics, 32:472, 1964.

[45] Robin Giles. Foundations for quantum mechanics. Journal of Mathematical Physics,11:2139, 1970.

262 BIBLIOGRAPHY

[46] E. Grgin and A. Petersen. Relation between classical and quantum mechanics. Inter-national Journal of Theoretical Physics, 6:325, 1972.

[47] E. Inonu and E.P. Wigner. Proceedings of the National Academy of Science, 39:510–524, 1953.

[48] Ted Jacobson. Feynman’s checkerboard and other games. In N. Sanchez, editor, Non-Linear Equations in Classical and Quantum Field Theory, pages 386–395. Springer,Berlin/Heidelberg, 1985.

[49] J. R. Klauder and B.-S. Skagerstom. Coherent States: Applications in Physics andMathematical Physics. World Scientific, Singapore, 1985.

[50] A. Kuzmich, N. P. Bigelow, and L. Mandel. Europhysics Letters, A 42:481, 1998.

[51] A. Kuzmich, L. Mandel, and N. P. Bigelow. Physical Review Letters, 85:1594, 2000.

[52] A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N. P. Bigelow.Physical Review A, 60:2346, 1999.

[53] R. A. Laughlin. A Different Universe: Reinventing Physics from the Bottom Down.Basic Books, New York, 2005.

[54] S. W. MacDowell and F. Mansouri. Unified geometric theory of gravity and super-gravity. Physical Review Letters, 38:739–742, 1977. Erratum: ibid. 38: 1376.

[55] S. Mandelstam. Loop quantization of gauge theories. Physical Review, 175:1580, 1968.

[56] Y. I. Manin. Quantum Groups and Non-Commutative Geometry. Les PublicationsCRM, Universite du Quebec a Montreal, Montreal, 1988.

[57] Y. I. Manin. Topics in Non-Commutative Geometry. Princeton University Press,Princeton New Jersey, 1991.

[58] N. Mankoc Borstnik. Unication of spins and charges enabels unification of all interac-tions. In N. Mankoc Borstnik, H. B. Nielsen, and C. Froggatt, editors, Proceedings tothe international workshop on what comes beyond the standard model, Bled, Slovenia29 June- 9 July1998. DFMA, Ljubljanna, Slovenia, 1999. arXiv:hep-ph/9905357v1.

[59] T. D. Palev. Lie algebraical aspects of the quantum statistics. Unitary quantization(A-quantization). Preprint JINR E17-10550, Joint Institute for Nuclear Research,Dubna, 1977. hep-th/9705032.

[60] R. Penrose. Angular momentum: an approach to combinatorial space-time. InT. Bastin, editor, Quantum Theory and Beyond, pages 151–180. Cambridge UniversityPress, Cambridge, 1971. Also available at http://math.ucr.edu/home/baez/penrose/.

BIBLIOGRAPHY 263

Penrose kindly shared some of this seminal work with me as his theory of ‘mops’, ca.1960.

[61] Bernhard Riemann. Uber die hypothesen, welche der geometrie zu grunde liegen. Ab-handlungen der Koniglichen Gesellschaft der Wissenschaften zu Gottingen, 30, 1854.Reprinted, ed. H. Weyl, Springer (1919); English trans. On the Hypotheses which lie atthe Bases of Geometry , Transl. W. K. Clifford, [Nature, Vol. 8. Nos. 183, 184] availableonline.

[62] H. Saller. Operational Quantum Theory I. Nonrelativistic Structures. Springer, NewYork, 2006.

[63] H. Saller. Operational Quantum Theory II. Relativistic Structures. Springer, NewYork, 2006.

[64] J. Samuel, 2008. I owe this apt formulation to Joseph Samuel. (Private communication2008).

[65] J. Samuel and S. Sinha. Surface tension and the cosmological constant. PhysicalReview Letters, pages 161302–4, 2006.

[66] I. E. Segal. Duke Mathematical Journal, 18:221–265, 1951.

[67] I. E. Segal. A non-commutative extension of abstract integration. Annals of Mathe-matics, 2nd Ser., 57:401–457, 1953.

[68] M. Shiri-Garakani and D. R. Finkelstein. Finite quantum theory of the harmonicoscillator. arXiv, 2004.

[69] H.P. Snyder. Quantized space-time. Physical Review, 71:38, 1947.

[70] A. Sommerfeld. Lectures on theoretical physics. Volume 1. Electrodynamics. AcademicPress, New York, 1954.

[71] R. D. Sorkin. Causal sets: discrete gravity. In A. Gomboroff and Marolf, editors,Lectures on Quantum Gravity. Springer, New York, 2005.

[72] E. C. G. Stuckelberg. Quantum theory in real Hilbert space. Helvetica Physica Acta,33:727–752, 1960.

[73] M. Tavel, D. Finkelstein, and S. Schiminovich. Weak and electromagnetic interactionsin quaternion quantum mechanics. Bulletin of the American Physical Society, 9:435,1965.

[74] G. ’tHooft. Determinism in free bosons. International Journal of Theoretical Physics,42:355, 2003.

264 BIBLIOGRAPHY

[75] R. Vilela-Mendes. Journal of Physics A, 27:8091–8104, 1994.

[76] L. von Mises. Human Action. Yale University, New Haven, 1949. 4th revised edition,Fox & Wilkes, San Francisco, 1996.

[77] J. von Neumann. Eine Axiometisierung der Mengenlehre. Zeitschrift fur Mathematik,154:219–240, 1925. Also in J. von Neumann, Collected Works, volume 1, page 35.

[78] J. von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer, Berlin,1932. Translation by R. T. Beyer, Mathematical Foundations of Quantum Mechanics,Princeton, 1955.

[79] J. von Neumann. John von Neumann: Selected Letters (A. H. Taub, ed.). AmericanMathematical Society, Providence, 2005.

[80] S. Weinberg. Quantum Theory of Fields I – III. Cambridge, Cambridge, 1996.

[81] C. F. v. Weizscker. Komplementaritt und logik i – ii. Naturwissenschaften, 42:521–529,545 – 555, 1955.

[82] E. P. Wigner, 1960. Personal communication.

[83] E. P. Wigner. The problem of measurement. American Journal of Physics, page 3,1962.

[84] E. P. Wigner. Symmetries and Reflections: Scientific Essays of Eugene P. Wigner.Indiana University, Bloomington, 1967.

[85] E. P. Wigner. Events, laws of nature, and invariance principles: 1963 Nobel lec-ture. In Unknown, editor, Nobel Lectures, Physics 1963–1970, pages 6–18. Elsevier,Amsterdam, 1972.

[86] F. Wilczek. Physical Review Letters, 48:1144, 1982.

[87] F. Wilczek. Fractional Statistics and Anyon Superconductivity. World Scientific, Sin-gapore, 1990.

[88] I. Wilkins. An Essay Towards a Real Character and a Philosophical Language. SamuelGellibrand & John Martyn, London, 1668.

Index

’ , 31H, 21ø , 21gentle determination, 50queue, 169reversal, 73Grade, 145SL[V ], 74SO[L], 74Semi, 159ϑl, 16ϑo, 16, 46, 62∨, 46, 175b-ary expansion, 168g-adics, 146m-tuplet, 141p-ad, 141〈H〈 , 118S, 72, 142S[L], 142X, 24, 73ID, 35Aharanov limit, 137Aharanov, 253Ambarzumian, 28Arnowitt, 238Bacon, 69Baugh expansion, 169Baugh number, 247Baugh, 168, 253Berkeley, 123Bleuler, 22Bloom, 253

Bohr atom, 100Bohr, 56, 101, 106, 119, 125, 128, 177Born, 125, 128, 241Bose quantification, 154Brauer, 158C gauge group, 94C-gauge, 91Carlen, 253Cartan, 84, 158, 234Castagnoli, 253Chevalley, 158Clifford algebra, 38, 146, 156, 160–163, 170Clifford form, 145Clifford product, 99Clifford statistics, 149, 162, 185Clifford, 77Complete, 62Compton, 108Connes, 28, 84Copenhagen quantum theory, 132Davis, 253De Witt, 92DeWitt, 150, 211Deser, and , 238Dirac algebra, 146Dirac, 22, 143, 190Dynkin diagrams, 149Edwards, 253Einstein Equivalence Principle, 89Einstein, 28, 108, 197, 234Everett, 136Fermi algebra, 60, 154, 155Fermi full quantization, 179

265

266 INDEX

Fermi graded Lie algebra, 155Fermi operator algebra, 41, 76Fermi quantification, 154, 158Fermi quantum theory, 247Fermi statistics, 49, 60, 77, 86, 149, 151, 153,

167, 179, 186, 225Fermi(-Dirac) statistics, 146Feynman space, 191Feynman, 125, 190, 191Fresnel, 97Full Fermi quantization, 85GUT, 96Galiautdinov, 253Galileo, 186Gell-Mann, 127Giles, 118Grade, 31Grand Unified Theory, 49Grassmann differentiator, 152Grassmann product, 99Grgin, 153Gupta-Bleuler, 190Gupta, 22Gyatso, 253Heisenberg, 56, 106, 120, 125, 127, 128, 178,

186, 193, 197, 253Hertz, 108Hessian determinant, 179Hibbs, 191Higgs field, 83, 209Hilbert, 208Inonu, 134Ivanenko, 28Jacobi identity, 147Jordan, 193Kallfelz, 253Kaluza strategy, 182Kaluza-Klein, 92Kaluza, 209, 211, 232Killing form, 29, 178, 199, 248Killing metric form, 238

Killing norm, 235Killing, 84Kleinian, 76Kuzmich, 253Large Number, 52, 53Law of Large Numbers, 135, 183Lisi, 253Lunsford, 253MacDowell-Mansouri, 92Mach, 52Malus Relation, 112Malus-Born, 23Malus, 108, 125Manin, 120Many World Theory, 136Marks, 253Misner, 238Newton, 97, 124Nicolas of Cusa, 124Nietzsche, 69Noether, 82Oppenheimer, 193Palev space, 196Palev statistics, 31, 178, 185, 195, 196Palev, 149, 195, 196, 230, 231, 240, 253Pauli, 125Peano, 99Peirce, 52Penrose space, 78, 195Penrose, 18, 28, 195Petersen, 153, 253Planck time, 191Planck, 108Poincare group, 69Poisson Brackets, 58Poisson Bracket, 208Rank, 31Regge, 19Riemann, 234Saller, 253Samuel, 253

INDEX 267

Schrodinger cat, 136Schrodinger’s cat, 138Schrodinger, 111, 125–128, 133, 136Schroek, 253Schwartz, 253Schwinger, 125Segal-Vilela space, 197Segal-Vilela-Mendes space, 150, 182, 240, 248Segal, 12, 28, 193, 195, 197, 209Shah, 253Shiri-Garakani, 253Snyder, 28, 188, 193, 197Spinorial Clock, 161Stuckelberg, 117Stapp, 253Vilela-Mendes space, 196Vilela-Mendes, 188, 197, 209Vilela-Mendez, 28Volterra-Burgers, 82Weizsacker, 253Wess, 253Weyl gauge strategy, 91Weyl, 158Wick product, 157Wick, 58Wigner’s friend, 138Wigner, 127, 130, 132, 134, 138, 156, 159,

194, 253Yang-Mills, 92accelerating expansion, 248acceleration, 88algebra unification, 99, 153, 237algebra, 27amonadic, 171ancestors, 168anti-canonical, 45anti-system, 110apostrophe, 31aspect ratio, 61association, 71assured transitions, 38

assured, 114, 144atomism, 15bilinear space, 38brace, 20, 33, 35, 46, 170bracing, 87, 98, 142, 209, 210c set, 16canonical Lie algebra, 188, 194canonical classical strategy, 57canonical codicil, 58canonical gauge theory, 79canonical group, 69canonical limit, 64canonical quantization, 14, 40, 45canonical quantum space, 194canonical quantum theory, 23canonical subspace, 196canonical, 45, 194catenation, 209causal structure, 25causality form, 21, 25, 73, 236cell, 19, 20, 42, 95center, 186centralizer, 225central, 186charge level, 77, 99charge, 17chronometric, 237chron, 33, 95classical logics, 108, 175, 183classical logic, 108, 125, 135classical set, 16classical system, 22class, 115, 141cliffordons, 162coherent superposition, 117collapse, 133commutant, 120, 225commutative reduction, 111commutator representation, 147, 216compactification problem, 92, 150complete, 12, 111, 120, 165

268 INDEX

complexual, 19compound, 12, 13, 69, 88, 115, 176–178, 185constraints, 81contact, 13contiguous action, 55continuous geometry, 193coordinate state, 120correspondence, 119cosmological constant, 75, 248cosmological leap, 44, 67creations, 26creators, 26crisp, 62, 118crystal film, 20cumulant, 68cumulator, 48, 68, 181, 233cut, 132dark energy density, 248de Sitter, 197de-brace, 99defects, 82definite, 22deformation quantization, 70, 177degravitization, 60density matrix, 117, 118derived, 69diagonal, 208diffeomorphism group, 69diffuse, 62, 118diff, 94doublet, 141dual space, 144dual system, 110duality, 118dual, 75ductor, 21duplex form, 34, 76duplex norm, 76duplex space, 34, 156duplex vector, 75duplex, 75

dyad, 141dynamics vector, 32, 37, 51dynamics, 37eductor, 21electromagnetic field, 81emergent, 25equivalence principle, 88, 94ether, 97event level, 35event number, 181even, 146exchange parity, 159experimental, 26experiment, 118exponential logics, 142exponential, 35exponentiation, 34external (long) dimensions, 210family property, 162field level, 35field theories, 224field variables, 221field, 39, 48, 49, 60, 85, 170, 181, 187, 188,

193, 208–210, 221, 231, 232, 234film, 19filtration, 57, 130forbidden, 144fragile, 13full quantization, 15, 24, 40, 45, 71, 72, 77,

83, 85, 179, 184, 190, 221fully quantize, 26fully quantum, 25, 26, 251function, 14, 207, 208fundamental length, 193gauge boson, 231gauge connection theory, 231gauge equivalence principle, 89, 92gauge groups, 43gauge group, 69, 211gauge metric theory, 231gauge potential, 73

INDEX 269

gauge theory, 179gauge, 49, 150, 183, 209, 221, 231, 232, 238,

251gauging, 91general coordinate, 94, 200general relativization, 85generating Lie algebra, 86generating vectors, 77generations, 99generation, 46generator, 220generic, 13geometric algebra, 77ghost particles, 80grade operator, 145grade, 35, 141, 146, 201grand, 33gravitational field, 81, 153, 189, 190, 208,

239gravitization, 64graviton, 97gravity, 77, 85, 87, 97, 103, 149, 186, 189,

190, 197, 208, 209, 232, 234, 238growth problem, 93, 150harmonic coordinates, 89high-temperature superconductor, 97history, 88hypercubic, 211hyperexponential, 45iconoelastic, 28idol, 69imaginary, 224incoherent superposition, 117, 118incoherent, 172incomplete, 119induced, 94, 180induce, 73inductor, 21input, 95, 109, 110, 112, 143internal dimensions, 233io, 144

irreducible, 12isometry, 23kinematical Lie algebra, 86kinematical algebra, 185kinematical group, 69, 185lattices, 107lattice, 106left-multiplication, 147levels, 28, 99level, 14, 49, 72, 142, 161, 162, 175, 176, 209,

225linear logics, 142locality algebra, 241locality, 13, 15, 25, 60, 241long dimensions, 252long, 95mean square form, 148, 238mean-square form, 77, 148, 151mean-square norm, 46, 68, 235meta-operation, 229metasystem, 50, 101, 112, 115, 118, 119, 160,

170, 183minimal coupling, 13, 85minimal differential order, 13, 85miss, 35mixing, 118mixture, 117momentum origin, 215, 218momentum subalgebra, 216monadics, 36monadic, 171monadization, 71monad, 15, 141moves, 192multilevel Clifford algebra, 172multiplicative norm, 147multiplicity, 110, 141natural quadratic forms, 236negative norm, 67neutral, 76, 158non-commutativity, 84, 107, 119

270 INDEX

normal ordering, 58normally ordered product, 157number operator, 152observable, 57occlude, 35odd, 146ontologism, 100, 127, 135, 136ontology, 56, 57, 100orbital part, 91orbital, 91origin cell, 211orthonormal, 45output, 112, 143palevons, 31phase space, 100polyad, 141polynomial logics, 142polynomial, 94power set, 34, 168praxics, 106praxic, 21praxiological, 56, 121praxiology, 27, 56, 120predicate, 115, 141principle vector, 215probability form, 21, 23, 25, 38, 48, 73, 112,

114, 118, 144, 153, 236probability function, 128probability operator, 50, 118probability tensor, 117, 118product unification, 153projective geometry, 114, 144projector, 57proper state, 120pseudo-bosons, 97pure sets, 40q bit, 38q set, 16quantification, 39, 149, 185quantifier, 152quantization, 63, 150, 232

quantized imaginary, 83quantum chronometric, 239quantum generators, 18quantum logics, 106, 107quantum non-commutativity, 84quantum praxics, 108quantum set theory, 115quantum set, 16quantum time, 188quantum variables proper, 63quantum, 27quasicontinuous, 73queue algebra, 72queue gauge group, 94queue strategy, 77queue, 16, 24, 27, 31, 60, 115, 141, 168, 170,

175, 209q, 116radical, 69random set, 40, 169, 172rank jump, 171rank, 14, 33, 34reciprocity, 241reduced, 181regular, 13, 29, 145relation, 208relativity, 26reverse, 236rigid, 13robust, 13root vector, 84scalar, 148sea, 16second quantization, 39, 158self-Fermi vector algebra, 76self-Grassmann algebra, 40, 49, 62, 198self-organization, 75semi-simple, 84semiquanta, 158semiquantum, 154semisimple, 12, 69

INDEX 271

semivectors, 76semivector, 158set, 95, 141, 146sharp, 37, 62, 118short dimensions, 252short, 95, 209signature, 210simple quantization, 243simple system, 116simplex, 19simple, 12, 84, 85simplicity principle, 16simplicity, 15simplification by quantization, 70, 185simplification strategy, 12singlet, 141singular Lie algebra, 29singularities, 149, 150, 179singular, 13, 29solvable, 69sortation, 130special coordinate, 200special, 13, 89spin 1/2, 158spin parity, 159spin part, 91spin vector, 47spin-statistics correlation, 159spin-statistics, 41, 77, 149, 158–160, 162, 163spinor, 158, 161spin, 17, 47, 91, 162standard basis, 149standard model, 49, 85, 87, 167, 176, 180,

187, 209, 251state space, 100state strategy, 57state vector, 113state, 119, 120statistical Lie algebra, 86statistics, 149strategy, 100

strongly continuous, 165structurally stable, 13structurally unstable, 13, 60, 186structure manifold, 13successor, 99supermanifolds, 200superposition, 35, 115, 118superquantification, 158supersymmetry, 200swap, 17symmetry, 26throughput, 144time level, 224time quantum, 24time reversal, 156torsion, 89transformation theory, 143transition probability amplitude, 118transition probability, 118truss dome, 95, 252truss, 96, 201, 210, 212unification, 49, 237unimodular diffeomorphism, 89unimodular, 89unimodulat covariance, 89unit function, 109unitarity, 60unitization, 71vacuum, 53, 226variable, 57vector strategy, 57vectors, 145vector, 16, 117, 124, 143, 162, 168vinculum, 171von Neumann, 106, 107, 130, 193wave function, 125–128zero-point action, 44

, 46

atomism, 15

272 INDEX

curvature, 96

monadic, 46

polyadic, 46

simplicity, 16

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