ISBN: xxx-xx-xxxx-xmy.ilstu.edu/~lmiones/DWTv1.pdfAppendix A. VIReQuEST: a V.I. Virtual Institute 219 1. Mathematical-physics and top-down design 219 2. DWT v.1 implementation goals

Printed in Romania by Olimp Press www.olimpress.roCopyright c©Lucian M. Ionescu, 2006Copyright c©2006 Olimp Press for the present edition.

ISBN: xxx-xx-xxxx-x

Keywords: Quantum physics, quantum mechanics, quantumcomputing, quantum field theory, Feynman process, quantumgravity, entropy, quantum dots, qubits, information dynamics,mind-matter.

Lucian Miti Ionescu

The

Digital World TheoryVersion 1.0 : An Invitation!

A Mathematical-Physics and Computer Science unifyingapproach:

V I ReQuEST

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Physics 2.0

((RRRRRRRRRRRRRComputer Science//oo

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Mathematics

The MPCS-Alliance

Olymp Press

I dedicate this book to my family.

Abstract

The Digital World Theory (DWT) is the trademark of anew approach in modeling interacting and interactive systems,from fundamental interactions including quantum gravity, tocomplex systems like communication networks, i.e. DWT mod-els natural and artificial reality alike!

It gathers various new trends consistent with the author’sown research and philosophy, into a “project description” play-ing the role of a grant proposal for “A New Kind of Science”([Wolfram]), and prone to “open science development” (seeAnnex A).

The major “upgrades” of DWT are:

1) There is no apriori space nor time, not even a “Space-Time”:

1800 1900 2000Space × Time Space-Time Causal structure

direct product ”warped” “multi-extension”

The causal structure is rather a “multi-dimensional timeextension of space”, and a “linear time” might not exist evenlocally. There is no “time flow”, but:

Quantum Information Flows !

2) The causal structure with “variable geometry” is a dis-crete model (q-digital!), not a continuum model (“the reals are

not real”): the Quantum Dot Resolution (graded of finite

type: “no functions, but generating functions!”)

1

2 ABSTRACT

3) A new Energy − Information Unifying Principle is

introduced at a fundamental level, binding Einstein’s energy-matter with information beyond the well known thermody-namic balance between entropy and energy. It provides a com-

mon framework for a unified description of Mind and Matter .

4) It is a concerted modeling effort unifying contribu-tions from Physics and Computer Science at the appli-cation level, under the auspices of Mathematics at the imple-mentation level: Computer-Physics 2.0 for [Human 2.0]! It ismandated by the duality:

Quantum Interaction ∼= Quantum Computing ,

at the hardware (matter - quantum computer) and software(physics model - computer science interpretation) levels. It pro-vides an “umbrella description” of interacting systems (System-System) and interactive systems (System-Observer, i.e. mea-surements and Observer-Observer, i.e. communications).

This last upgrade is probably the most valuable ideologicalunification, representing the GUT character of the DWT.

Contents

Abstract 1

Introduction 71. The DWT unifying ideas 92. What was “Space-Time”? 103. About the book 164. Acknowledgments 22

Part 1. User’s Manual 25

Chapter 1. The Search for a New Unifying Principle 271. Models, models, models! 272. Three ... revolutions: returning to principles! 303. A “New” Principle 384. Conclusions 42

Chapter 2. Miscellaneous: a “warm-up” 431. On the measurement “paradox” 432. Modeling a complex process 443. What is the Information Flow? 454. The “understanding” of QM 475. Is it all in the brain? 486. Classical or quantum mechanics?

... what’s the difference anyway! 497. To think or not to think classically? 518. Add a pinch of ... Einstein relativity! 53

3

4 CONTENTS

Part 2. Reference Manual 55

Chapter 3. The Principles of DWT 571. Why Path Models? 572. The Causal Resolution 583. The Fundamental Principles of DWT 59

Chapter 4. Causality: Information versus Time flow 61

Chapter 5. Modeling classical information flow 631. Measurement: a “paradox”, or what? 632. Shannon or Von Newmann? 653. Shannon’s measure of information 664. Probabilities and Partition Functions 745. The thermodynamics legacy: a hidden message? 826. Feynman path integrals and entropy: graphs

invariants! 947. Relative entropy and information channels 103

Chapter 6. Modeling quantum information flow 1111. Constructivism: the critique of the critique 1112. Quantum entropy 1133. Two good examples 1154. Quantum information 115

Chapter 7. Classical and Quantum logic 1171. Propositional calculus 1172. The lattice: creation and annihilation! 1183. The modular identity and ... entropy! 1194. Relation to projective geometry 1195. Conclusions 120

Chapter 8. Quantum dots and bits 1231. Quantum dots: theory and practice 1232. Reversibility: information flow and time reversal 1273. Categorifying classical mechanics 128

CONTENTS 5

Chapter 9. Quantum gravity and information 1331. Artificial Intelligent Geometry 1342. Comments on Loop Quantum Gravity 1353. Is “reality” 1-D? 1374. The Laws of Black Hole Radiation - revisited 141

Chapter 10. Conclusions and further developments 1491. Dimensions: 1,2,3 ... ∞! 1502. The Holographic Universe 1513. Time flow is Subjective 1524. Cosmological constant: nowadays Cinderella 1545. What next? 1556. Epilog 156

Part 3. A Research Diary 159

Chapter 11. Why is mass energy: E = mc2? 1611. Information flow revisited 1662. 4D Hyperdynamics 1673. Space-Time duality and Hypersymmetry 1694. QG as a Deformation Theory 1695. String Theory as a Deformation Theory 171

Chapter 12. Quantum Digital Gravity 1751. Energy-momentum tensors: external and internal 1762. More clues ... 1803. Quantizing I/E DOFs 1834. Currents in String Theory 1835. Gravity and electromagnetism: the revival of an old

story! 1856. The Quantum Temperature-Entropy Field Theory 1867. Bohmian mechanics: a “hidden” message 1908. Heat transfer and entropy flux 191

Chapter 13. The Bohmian mechanics interpretation 1951. The quantum dice 195

6 CONTENTS

2. Entropy and String action 1973. Information current revisited (II) 2004. Again, why is energy mass? 2015. Bohmian mechanics’ Schrodinger equation 2026. Interpreting the new “Klein-Gordon equation” 2047. Entropy entanglement and Riemann surfaces 2078. Probabilities and amplitudes 207

Bibliography 213

Appendix A. VIReQuEST: a V.I. Virtual Institute 2191. Mathematical-physics and top-down design 2192. DWT v.1 implementation goals 221

Introduction

The book is an invitation to a collective effort for a new kindof theoretical approach to understand reality, whether “natu-ral” or artificial.

It is not The Theory of Everything nor a Final Theory! Itis a proposal for a “Grand Unification”, not only as a unifieddescription of the fundamental forces and of complex systemsalike, but also of theoretical contributions from Mathematics,Physics and Computer Science.

The DWT expresses a personal point of view; it is a “BigPicture” getting its shape from lots of pieces of the reality puz-zle, put together with a lot of ... wishful thinking (Ahem ...“design specifications” I meant to say :-). Therefore “I conjec-ture/believe/etc.” are suppressed as ... the default!

One cannot understand any of the foundational concepts,e.g. space, time, matter, information, mind, conscience etc.,without finding a unified approach to understand all of them.

To understand Quantum Gravity for instance, we adoptthe view of Gauss, Euler, Grothendieck etc. to broaden thegoal, rather then trying to simplify our task and focus on aunification of quantum mechanics and general relativity only,without modeling the other interactions (e.g. Loop QuantumGravity [Rovelli-1]; see Section 2, p.135).

It is time for a concerted effort bringing together not thetraditional mathematical-physics “team” (with its glory but

7

8 INTRODUCTION

also deficiencies [Kirrilov], [Gosson]), but also Computer Sci-ence the “New World” of science, as it become more and moreobvious with the advent of quantum mechanics and quantumcomputing, starting perhaps with Feynman (see [Milburn]).

The DWT is also a (grant) proposal towards building theteam which will design a unified scientific model of mind andmatter, human and computer, life under its various guises. Thepresent “project description” sketches a few new ideas which, asusual, can be later recognized being implicit in various scatteredplaces ... It is not the time for “priorities”; the priority is for a“new kind of science” [Wolfram].

The DWT message is that “there is no Time”, universal ornot, global or local, meaning that in order to progress beyondthe current stage, science has to develop models which are notbased on a “linear time”, with its companion assumption of an“orthogonal space” (if it’s not parallel it must be sequential!?),i.e. we have to develop the Theory of Causal Structures, beyondMarkov and Feynman’s pictures, causal structures which arenot a (warped) product of space and time, not even locally.

The historic regret after the “lost paradise” of deterministicscience, should be compensated by the excitant of recoveringthe “freedom” of information ... flow, with the hope of tam-ing the “apparent chaos” which is the trademark of efficientcommunication protocols ...

The double parenting of the “new science” (Physics andComputer Science) 1 brings the dual interpretation of “change”as physical interaction and communication, e.g. particles andwaves or information sources and messages, timing of events orprocess scheduling and information flow. Obedient as always(or rather with hind site!) mathematics moved away from staticSpace-Time models to models with “variable geometry”.

1Therefore it needs a new name, reflecting a common scope under twodifferent disguises: (Quantum) Information Dynamics/Processing.

1. THE DWT UNIFYING IDEAS 9

And speaking of the “Science Drift Theory”, the ComputerScience - Mathematics alliance might be able to stop the currentdrift of the science’s focus/interest from physics to biology [ST].

1. The DWT unifying ideas

The main unifying and upgrading ideas fall into two cate-gories:

I) Methodology:1) a broad goal and theoretical technology used (unified

math/ phys/ CS approach): in order to design the theory tolast a few decades (not just another resonance / “theoreticalphysics cycle”).

2) Building theories for machines, for higher processingpower (e.g. lattice QCD for computability [Mackenzie]), withcare for the Human Interface, (with a conceptual high-levellanguage description for physicists). It is thought of as “Physics2.0” for [Human 2.0], i.e. new software for the new human-machine merger in the near future;

3) A top-down design approach (necessary for the strati-fied complexity of the future’s theoretical constructs), in orderto make it “upgradable” and to allow each specialist to designhis/her own layer, within his/her own level of expertize, andwith care for the interface between layers;

II) Content:1) The overall philosophy can be summaries as:

(Modeling)Reality IS Quantum Computing.

This reflects the dual interpretation of mathematical models:Quantum Physics and Computer Science. More specifically:

Quantum interaction = Quantum communication.

2) A New Unifying Principle, beyond Einstein’s E = mc2

unification of energy and matter, stating the balance of en-tropy/information and energy.

10 INTRODUCTION

If almost a century of efforts to incorporate gravity as anexchange interaction (force-field) failed, perhaps it is time totry an alternative approach. Gravity is a theory of space-time,i.e. of causality, and quantum mechanics emphasized the roleof measurement and that of the observer.

The idea that gravity is related to the measurement paradox[Penrose] and it is an organizational principle acting againstthe entropic principle [D1] is promoted to a goal:

Gravity and Mass are Entropic Effects.

Interpreting an interaction as a quantum communication re-quires a unifying point of view such that any interaction, system-system, system-observer and observer-observer, be treated onan equal footing, i.e. as a quantum communication.

Therefore a quantum theory of space-time must incorporateentropy and information at its foundations. Then space-timeis not a “fixed model”, independent of the observer’s modelinggoals; it is a device to keep track of degrees of freedom (DOFs /memory to store information). This is only half of the picture:the external DOFs; the other half, keeping track of internalDOFs, comes with a duality ... then energy and informationare the two faces of the same coin!

2. What was “Space-Time”?

The author’s search for a substitute of space-time as a wayto resolve the incompatibility between general relativity andquantum mechanics evolved through several “stages of enlight-enment”.

The categorical language as an “object-oriented language”was clearly the starting point, and a categorical analog for the“spectrum of the algebra of observables as a substitute for spaceseemed highly desirable and sketched in [Ion00] to expressFeynman’s ideology that one should think of quantum processesas “histories”, or paths. Of course, this (the Path Model) is a

2. WHAT WAS “SPACE-TIME”? 11

disguise of a states and transitions approach, i.e. the Automa-ton Picture (again quantum computing!). It became clear thatFeynman’s paths are in fact a substitute for “space-time”, andlater on [VIRequest-UP], that all one needs (and can hopefor) is a “causal structure” (there is no “Time”, really - in themodern models, that is: there should be no global “internaltime”).

While studying Kontsevich’s graphs as a tool in deforma-tion quantization [K97] and Kreimer’s use of graphs in renor-malization [Kreimer] (“IHES illumination period”), the au-thor realized the additional importance of graphs, not just as aperturbational device in QFT, but one which enables a ScalingPrinciple (“zoom in/out” on a system). Graphs were providingsomething like a resolution of a ... “point” (!?). The “missingwords” (of encouragement) were found in Gelfand and Manin’sbook [GM], p.6) (where else?); the message (for me) was “For-get the space(-time), all you need is a resolution”!

Now, classical “time”, representing our bold conviction thatwe can put a linear order on phenomena, was longtime for-gotten, in the search of a less strict requirement to representcausality (causal structures, e.g. Feynman graphs etc.), in ahomological algebra vain by “generators and relations”. So anew principle of modeling “space-time”, and in general causalstructures, thought of as the structure modeling the “externaldegrees of freedom” (EDOF for short), was adopted.

Definition 2.1. A Causal Structure is a resolution of a“point” (“The System” S), with duality between external andinternal degrees of freedom, called the Quantum Dot Resolution(QDR).

It provides a substitute for “Space-Time” (we avoid furthertechnical terms at this point). The duality refers to the ca-pability of “trading” internal (I) and external (E) DOFs, sincecollapsing subgraphs (“zoom out” on the system’s model) leads

12 INTRODUCTION

to “loosing” EDOFs (and therefore information!), which mustbe compensated by IDOFs.

This Scaling Process implemented by collapsing of graphsenables a generalized micro-macro observable correspondence(see Boltzmann’s approach to entropy), requiring/enabling theintroduction of a “balance between entropy and energy, andshowing that a “flexible model” of “space-time” (no more fixedconfiguration spaces) is both an “effective theory” and an “ex-act theory” at each scale level. This approach should allow, itis hoped, to connect with the black-hole radiation laws, whichhint to a “discrete structure” of space-time (which loop quan-tum gravity actually gets to [Rovelli-1] - with a lot of work,though!).

The differences between “deformations” (perturbation) andresolutions, both aspects present in the FPI perturbative ap-proach to QFT are discussed, hopefully providing a link be-tween the perturbative and non-perturbative approaches.

The more important lesson learned from the above, is thatentropy and information gain/loss enter into the picture earlyon, probably at the axiomatic level, and the CS-interpretationis crucial to model and understand the implications.

2.1. It’s all about “Time” ... Returning to the exis-tence of “Time”, the internal structure of a model of a quan-tum system reveals the presence of an information flow as asubstitute. The “global time” corresponds to a labeling of theexternal “symbols”/ states received/observed by another in-teracting system/observer, under the dual Physics-ComputerScience interpretation (PCS-interpretation for short).

At this point it is productive to think about the EDOFs,which incorporate the symmetries of “space-time” also as “in-ternal DOFs” by duality, as qubits, due to the “rich coinci-dences”: quaternions H viewed as C⊕C and SU(2) ∼= SL(1, 1)


etc. A “(anti)particle” traveling up and down in time (Feyn-man), or using the PCS-correspondence, the information flow-ing “up-and-down” relative to the observer’s global time, is aheavy parallel and sequancial (quantum) computing process,which corresponds in the physics language to s-correlation andt-correlation, the cousins of String Theory’s s and t channels.They are “coordinate dependent” concepts, allowing to splitthe quantum computation into a parallel and sequential com-putational blocks (“space” and “time”), subject to some gaugesymmetry related to conformal invariance (again some “coin-cidences” to keep in mind: Conf ∼= Diff(S1) ∼= Diff(S1) ∼V irasoro algebra and qubit symmetries).

2.2. ... and measurements. As an additional benefitof the PCS-alliance, we mention the possibility of resolvingthe quantum measurement paradox by modeling the measure-ment process as an “eavesdropping on a quantum communica-tion”. Recall that in our DWT, in principle there is no differ-ence between “observers” O and “systems” S (remember theHuman 2.0 merger?), since “quantum interactions” are “quan-tum communications”. Now modeling/interpreting a prepara-tion/observation measurement process as a “classical communi-cation”, one might expect to be able to “explain” (model) themeasurement process as the eavesdropping by Eve, the “Ob-server”, on the quantum communication between “Alice”, thepreparation apparatus, and “Bob”, the measurement appara-tus:

AliceQ−Interaction

QC// Bob

Eve

CC

ddHHHHHHHHH CC

;;xxxxxxxx

The abbreviationsQC and CC stand for quantum and classicalcommunications.

14 INTRODUCTION

So God/Nature does not play dice, but rather (quantum)talks to us out-laud, and the miracle is that we occasionally getan idea about what He/She is talking about :-)!

This brings us to the question regarding how the observer“thinks”. A “brainstorming start” consists in believing that ourconscience is a classical computer, since we need to interpretethings in our familiar classical terms (models), while proba-bly the inner subconscience mechanisms are quantum compu-tations. We get a glimpse upon them when we remember ourdreams (decoherence, as in any Output operation) ... It isa speculation only, but it’s hard to believe that God/Naturemissed such a remarkable implementation opportunity.

Of course, such “dangerous speculations” should be “con-fined” to the introduction, and mentioned only to disclose thepossible implications of a fully developed DWT, unifying thepowers of Math-Physics and Computer Science in order to “cre-ate” the unifying theory of Mind and Body. The potential ben-efits to the many other sciences studying life and mind etc. areobvious.

2.3. What is a Causal Structure? Causal structures ap-peared long time ago in QFT in the disguise of approximatingschemes: Feynman graphs (perturbative approach to QFT).The present author suggested in [Ion00], still in search of theappropriate mathematics model for Space-Time, that there ismore to it than it meets the eyes: Feynman graphs are a sub-stitute for the possible paths in a space-time. The idea thatthere should be a more general structure including cobordismscategories besides Feynman graphs emerged. The term general-ized cobordism categories, denoting the would be more generalstructure, was coined during graduate school before the authorbecame aware of the advent of operads and PROPs. Moreover,later on, it became clear that there should be no “fixed causalstructure”, allowing to account for the “scaling problem”, andalso a duality keeping a balance, allowing to trade “geometry”


for “physics, as suggested in [VIRequest-UP] (“packing andunpacking DOFs”).

2.3.1. Zooming in and out on a System. The appropriatetechnical tool seems to be the insertion and collapse of subsys-tems (zoom-in and out on a system: how natural!). The precisemathematical implementation is a generalization of Kreimer’sinsertion and deletion operations on Feynman graphs, opera-tions which allowed Kramer in his work with Connes [Kreimer,CK1, CK2] to rephrase and finally tame renormalization.

The present author provides the physical interpretation be-yond the mathematical tool itself: the Quantum Dot Resolution(QDR) 2.1 is believed to be the promising framework for devel-oping (classifying) QFTs, thought of as “Space-Time with vari-able geometry” (causal structures). In a way the interpretationcame post-factum, as sometimes happened in the past. Indeed,as relativity as a conceptual break through, appeared basedon the already existing physics and mathematics of Lorentz,Minkowski etc. causal structures have already appeared as op-erads and PROPs, related to the operator product expansions(OPE). OPEs establish a relation between causality and theexternal global time. In some sense, introducing a macroscopi-cal time consistent with the classical limit corresponds to OPEsand quantization.

2.3.2. A “new” algebraic-geometry principle. The conceptof “variable geometry” imposed by the modern homologicalalgebra-geometry principle “forget the space, use a resolution...” (see 2.1) had already found a “back-door entry” into dis-crete models (lattice models and Monte-Carlo simulations [DJ],p.254; see also [Ion01]).

The Feynman causal structure is not just a perturbativeapproach, it is rather a substitute, and in fact a generalizationof the concept of Space-Time; moving from (loop) degree todegree, as in Hilbert’s “syzygy theorem”, is an approximatingprocedure.

16 INTRODUCTION

Kreimer’s insertion-elimination operations should be inter-preted as “Space-Time bubble fluctuations” of the causal struc-ture: “Dirac’s vacuum”. At a more technical level, the generalframework is that of a 2-category of “generalized cobordisms”,where extensions (e.g. graph extensions) play the role of “ho-motopies”.

3. About the book

The book is organized into parts. The User’s guide targetsa larger audience, being an exposition of the main ideas andof the corresponding development projects. The correspond-ing implementation descriptions are provided in the ReferenceGuide. Annex A presents the Virtual Institute ReQuEST asan interface between sponsors and researcher teams developingthe proposed projects. It is also a permanent (non-standard)grant proposal.

The focus is on putting the ideas on the table (usually rep-resenting about 1% of a project), with care not to get swampedby technical digressions. It is not a question whether the envi-sioned theory is true or false, but rather, as the author believes,how hard it is to implement it at a level accounting for thepredictions of the present theories (99%). The benefits of theambitious unifications proposed should overwhelm the reasonsnot to depart from the traditional technology and current theo-retical products. On the other hand, since the past and presenttheories provide key ideological jewels which could be imported(general relativity, quantum loop gravity, lattice gauge theoryetc.), it is rather a matter of time to put together this bigger“puzzle” using already forged pieces.

The style of the book is adequate to an invitation to thinkand develop (or challenge) the ideas and statement made bythe author; quite often “...” ends a paragraph, to invite the

3. ABOUT THE BOOK 17

reader to ponder about what was said, if not asking the readerfor a most welcome feedback 2.

3.1. About the merger of QFT and GR. There arevarious “indications” that something is “wrong” with Gen-eral Relativity, beyond the well-known (implicit) incompati-bility with Quantum Theory (it is a “local theory”, not a “cor-relation theory” [Ion00]), in spite of its impressive age and“stature” among “Venerable Theories”. One such indicationis termed “dark energy”, as a wide carpet where many incon-sistency may be swiped under. Let’s not forget that there isalways an alternative when changing our models, for exam-ple Newton’s F = ma theory, where RHS sets the theoreticalframework and LHS allows for the experimental input. In thecontext of General Relativity, one can “fine tune” the modelof the experimental input, i.e. the Energy-momentum tensor(postulate some “new matter” source, etc.; that’s easy) or “ad-just” the T = κG theory, i.e. the theoretical framework. Now,that’s another story; twitching with the cosmological constant(e.g. [Was Einstein right all along?], p.18) is not enough!

An attempt to merge QFT and general relativity is madeby incorporating the concept of information to the very foun-dations of space-time-matter viewed as an interaction process(see also Ch.8 §3).

The principle involves in an essential way the dichotomy“observer-system observed”, which is the new key feature ofquantum physics (I guess, despite what some people say; see[Gosson], p.35).

An avenue for involving new cosmological phenomena (darkmatter etc.) opens up when taking into account entropy, sinceprobably the first thing to do is to try to interpret mass andinertia etc. as an entropic effect, while keeping in mind thatgravity has a clustering effect, so it is in fact an organizationalprinciple at the level of space ([Penrose], Ch.30; [D1]).

[email protected]

18 INTRODUCTION

3.2. String theory and entangled thumbs! The prosand cons regarding String Theory are a great divide: one thumbup and the other one down!

To start with the bad news, the Nobel laureate David Gross’exasperation regarding String Theory “We don’t know whatwe are talking about.” [ST], it is due to so many decades ofunfulfilled hope; string theory looked like a promising theory“... and promising ... and promising”, yet without any soondelivery, in terms of experimental predictions or verification.

It is well agreed: new ideas are needed [ST, D1], startingwith String Theory’s lack of explanations regarding “ ... wherespace and time come from”, while providing a sense of an aca-demic exercise since the equations “describe nothing we couldrecognize”.

The “good news” is: the idea (within string theory) counts!The Feynman philosophy pointed towards the mathematicalstructure I call Feynman Process, as a representation of a Feyn-man causal structure, and the benefits of using Riemann sur-faces, a historical load we have to reexamine (e.g. “fewer”equivalent transitions which are naturally relativistic since in-teractions are not point-wise localized etc.) are overcome bythe complexity leading to “rigidity”; on top of this they donot poses a “computer friendly interface” (In fact they do, butunder the guise of ribbon graphs etc.).

But the killer feature is that they need to float in somebackground space (cheaper by the dozen!).

What can we do to save the day? Comparing with the prin-ciples of DWT, the missing conceptual principle is the dualitybetween internal and external DOFs. With Riemann surfaces,internal DOFs come as vertex operators, and after representingthe Riemann surfaces “prop” (PROP), one gets a “clean” alge-braic structure: Vertex Operator Algebra (VOA). What lacksis a “graph differential” allowing to insert / collapse EDOFs, induality with a corresponding differential (L-infinity structure?)of the VOA.


The present stated principles cope nicely with the simpler,toy model structure of graphs. The difficulty of putting a space-time structure on them, in order to have Poincare invarianceand therefore relativity, is avoided by categorifying classicalphysics first 3. The idea is to “forget” manifolds, and have acategorical substitute for the phase space with external symme-tries (Poincare group) dualised as internal DOFs. The currentproposal is crude, yet promising (and ... promising ... etc.).

3.3. “Paradoxes” of Quantum Mechanics. The his-torical paradoxes of quantum mechanics are revisited: a modelof the measurement “paradox” is (tentatively) suggested andthe dichotomy quantum description of system versus classicaldescription of the apparatus is de-emphasized, as rather beinga modeling issue.

The information-matter duality is introduced, as an unifi-cation beyond Einstein’s energy-matter unifying principle. Inorder to do this, thermodynamics have to be accommodatedwithin QFT. The solution: interacting systems, whether com-plex networks or fundamental particles present two fundamen-tal dualities. The dichotomy between observed and observerin correspondence with the duality between internal (“hidden”DOFs) and external (“observed” EDOFs).

The black-hole laws, estimated to play an important role inthe implementation of the theory, are discussed in the perspec-tive of a derivation from the “thermodynamics of the causalstructure” with (future) “technical help” from Loop QuantumGravity.

In the context of DWT, we speculate regarding some “co-incidences”, e.g. the number of fermion generations and thedimensionality of “space” etc., to envision deeper connectionsin the context of the DWT.

3.4. What is new, really. This, of course, is “readerdependent” and it is too soon to “classify”.

20 INTRODUCTION

The author pointed out that the concept of event and there-fore of field as a function defined point-wise on a predefined“space-time”, constitute the source of the incompatibility be-tween general relativity and QFT [Ion00].

The author proposed a global categorical picture where theinteraction is the primary concept, and the “system” (whateverit may refer to: universe etc.) is viewed as an automaton, orin mathematical terms, a representation of a geometrical cate-gory capturing causality and thought of as replacing the “old”concept of space-time. The space part consists of the objects(source - target), while the morphisms represent the causality(correlations/transitions etc.), replacing the “old” time arrows(e.g. generalized TQFTs, QFT on Feynman graphs, StringTheory on Riemann surfaces etc.)

This picture is necessarily incomplete if entropy-informationis not included, as part of the quantum “interaction” observer-observed system (quantum measurement). The important du-plex interpretation interaction = communication is introduced(or emphasized).

Including the qubit as the quanta of information and treat-ing it as a virtual particle “completes” the quantum descriptionof reality (a timid completion, for the time being). The “deter-ministic evolution of the system governed by (say) Schrodingerequation should now be compatible with the collapse of thewave function due to a measurement, since information is trans-fered from the observed system (S) to the observing system (O).In this way unitarity of the evolution of S is sacrificed, and non-local features are introduced in the new theory, yet the enlargedpicture describing the evolution of both S and O is expectedto be consistent at a closer inspection (the quantum computa-tion should be considered together with the quantum programcontrolling the quantum computation).

The generic space-time foam, beyond the particular imple-mentation using (say) triangulations etc., represents also infor-mation flow, and the destruction/creation processes (operators


or other mathematical techniques used to implement them)will affect the information content from the point of view ofthe observer’s knowledge (“transfer of spacial interface”). This(pre)concept should be compatible with the quantum laws ofblack-holes etc.

In a way, the approach replaces the usual quest for a unifiedgroupG to represent the internal symmetries of fields, function-als on space-time:

Field : “Space− T ime′′ → Rep(G)

with a search for the right implementation of a “space-time”incorporating the symmetries directly into its structure usingduality.

We must warn the reader that, in a top-down design ap-proach, the specification of the theory will start from generalprinciples towards a precise conceptual interface, explained ina high-level language. The actual implementation is almosttotally ignored in the first part (User’s Manual), except whenan existing specific mathematical theory may shade more lightregarding the ideas had in mind.

The more technical description towards a possible imple-mentation of the present ideas is deferred to the second part,The Reference Manual.

Some “clues” which seem to be important, yet prone forspeculations and alternative approaches are deferred to the An-nex 3.

Overall is seems that the main point is that “information/entropy” is the missing dual aspect of the traditional approach(energy-matter etc.). In some sense a “fast implementation”approach consists in “doubling String Theory” to incorporatethe entropy/ information current (think of the Yes/No-corollaas ST’s pair of pants), allowing for duality between external(topological distinct RS) and internal DOFS (vertex operators),by implementing insertion / elimination operations (QDR withduality). We believe that ST is not “so ill-formulated” ([Woit],

22 INTRODUCTION

p.18), but rather not conceptually understood in the past. Inorder to do so, physicists and mathematicians alike should re-visit the conceptual aspects first, relaxing the old “strong cul-ture where everything you say has to be very precise and youhave to be able to rigorously prove it.” (loc. cit. p.19). And... let Computer Science join the efforts 3!

4. Acknowledgments

Bibliography. It rather reflects the author’s (more recent)reading prefferences, than providing the optimal or primarysources. In several places the exposition and new ideas (in thecurrent presentation form, but some persistently haunting theauthor since the “glass age” 4) were prompted by the reading ofa specific article/book; the citation was meant to acknowledgeit. When the author’s idea was “confirmed” by the reading,the further plead for the idea was “left as a burden” to thecorresponding author/source.

Dedications. The book is dedicated to my family. It wasmade possible by my teachers, too many to be mentioned here.

But it’s too soon to look back! The final picture is not clear,but the work to be done is; let’s get it started! (see the AnnexA).

About the author. With a Mathematics, Computer Sci-ence and (self taught) Physics background 5, his work “off shell”often contains a (superposition) mixture of “What if” besidessome “pure” ideas, not being under a “technicalities constraint”

3For computational purposes “upgrade the technology” (Riemansurfaces).

4“Voi chi entrate qui, lasciate ogni esperanza”, but beware of author’shumor!

5Not quite the list from [‘t Hooft]!

4. ACKNOWLEDGMENTS 23

(“on shell”). Ultimately we learn from mistakes too; and some-times lough at somebody else’s mistakes (and cry!) ... it’s Artsand Sciences after all!

Part 1

User’s Manual

6

An important stage in the conception of the present projectwas realizing that a further unification is needed, expressedas usual by a new missing Unifying Principle. Bellow we re-produce the description of the main project of VIReQuEST[VIRequest-UP], with minor changes, as it was conceived atthe inauguration of the Virtual Institute for Research in Quan-tum Entropy and Space-Time A. It is the source of the DWTProgram, as described in the present version, stating the “hints”and also “hopes” that such a principle is still out-there.

6Not too many formulas in this part!

CHAPTER 1

The Search for a New Unifying Principle

Major breakthroughs are based on a change in perspectivedue to unifying principles.

I will try to backup this statement and revisit some folk-lore fundamental questions and theoretical difficulties (”para-doxes”) which in the author’s opinion should be solved as aresult of a unification steaming from a new principle. At thisstage (proposal level) we are able to “list” what seem to be themajor pieces of a puzzle: a theory including the benefits of,and built with the technology of the present quantum theories(Quantum Mechanics/Quantum Field Theory, Quantum Grav-ity etc.) while releasing the conceptual “tension” of the “mea-surement paradox”. In my opinion, we do not always need ex-press contradictions between experiment and theory (nowadayswhat theory predicts, say string theory, lies “safely” outside theexperimental range). The understanding (“revelation”) maycome from a new way of looking at the same “technical tools”(e.g. Special Relativity - see §2.1, etc.).

1. Models, models, models!

Recall that we model reality and we do not know whatreality IS; many books have been written on the subject, so Iwill only mention a few relevant names: Kant, Mach etc. andrevisit a few points (questions), briefly:

1.1. What do we mean when we ask “What is time?”Implicitly we refer to a concept within a theory (framework or

27

28 1. THE SEARCH FOR A NEW UNIFYING PRINCIPLE

context) which usually belongs to a specific community or per-son’s knowledge (KDBMS :- ), linked (pointed to) via a tag likeNewton, Einstein, Heisenberg, Feynman etc.. Or, when asking“What is an Electron?”, the answer ... depends on “hiddenvariables”; “Electron” may refer to the corresponding particlein Lorentz’s theory, or the de Broglie’s wave, Dirac’s spinor etc.(bad scenario: the context changes with the opponent’s tenta-tives to avoid our “theoretical hits”). Even worse still, it can bequite misleading when “explaining” quantum mechanics, andin the same statement making use of the term “electron” to re-fer to the quantum description and then to ponder in classicalterms about it.

In this sense, there are many meanings one can call “time”(or “space” etc.) within various theories; so one has to becareful about the implied context.

1.2. “Is it a particle or a wave?” The “electron” forinstance, is very well modeled as a particle by a few theories,when it comes to a certain range of experiments, yet there is aneed for other theories modeling the “electron” as a wave be-cause of another class of experiments. Overall quantum theoryhas a unified explanation for “all” experiments (of a certainkind) and the Complementarity Principle may be thought of asa “Two classical charts atlas of Quantum Mechanics”.

1.3. Interpretations of quantum mechanics. Why dowe need to “interpret the result” of a quantum mechanics com-putation in classical terms?

Classical mechanics is contained within quantum mechanics([LL], p.12), and it is not just a “limit” (Correspondence prin-ciple). Indeed, the measurement process involves a “Q-probe”(quantum probe: microsystem, elementary particle etc.) in-teracting with the measuring apparatus (usually a macrosys-tem) and the result of the experiment itself is modeled, orat least used by the experimental physicist (or processed by

1. MODELS, MODELS, MODELS! 29

some software!), in classical terms (we acknowledge macroscop-ical events: dots on a screen, beeps in a counter, bobbles in achamber etc.). Even a Stern-Gerlach experiment (i.e. involv-ing “internal states”) involves the interaction of a Q-probe (theelectron) with a magnetic field (macrosystem!) AND a detec-tor (beeps on 2 counters, providing the input to a classicalgate/computation). So, in a way Quantum Mechanics is a phe-nomenological theory! (beyond the Kantian statement of thetype “we only model phenomena ...”).

1.4. The main “lesson”. from above is that there areimplicit channels of information which are present, yet prob-ably not correctly (or completely) modeled within the corre-sponding theory! The role of observer in classical physics isthat of “user”. A crucial objective is to have a unique descrip-tion independent of observer (covariance; classical heritage).This is no longer tenable in quantum mechanics: “results” de-pend not only on “what” we observe, but also on “how”, whichin turn depends on what do you intend to do with “the result”.Nevertheless we are still looking for a “standard” in these pro-liferation of “encoder-decoder” business.

A unifying point of view, as a “slogan”, if one has in mindthe unification alluded to above, is that “All is quantum com-puting” (see also “Feynman processor” [Milburn] etc.), i.e.any interaction, whether system-system (Einstein ≈ “I like (!)to think I don’t have to look at the Moon for it to exist”),system-observer (quantum phenomenon), observer-observer (gen-uine communication!) are of the “same kind” (Which kind?that is the question!;-)


2. Three ... revolutions: returning to principles!

Let us consider Newton’s simplifying picture of Kepler’sLaws as a start for scientific modeling of (mechanical) phenom-ena. 1

2.1. First revolution: Special Relativity - gave a new lookat the technical tools already available at that time: Minkowskispace, Lorentz contraction, conformal invariance of Maxwell’sequations etc. Yet the conceptual break-through consists in“understanding” the meaning (and building various connec-tions with other concepts - networking): the unification of spaceand time (technically already done by Lorentz and Poincare- see [V], p.25 - but ... “What is it that we are doing?”was probably the lurking question of the day). The unifica-tion was derived in an “axiomatic” manner from the principlec = constant (corresponding to a constant Lorentz metric, orrather its conformal class). A probably more important princi-ple is the equivalence between mass and energy:

Principle I : E = mc2.

A “simple equation” yet with huge implications (Nagasaki, Hi-roshima :-( ).

2.2. The second revolution: General Relativity orQuantum Mechanics? In the author’s opinion, QM is TheRevolution, changing the way physics is done (see 1.4). Gen-eral Relativity is a “jewel” amongst mathematical-physics the-ories, again starting from a principle, the equivalence betweenaccelerations, gravitational or not (or masses: inertial or grav-itational):

Principle II : mg = ma

1Or ... is it “a culminating point of the scientific revolution of theseventeenth century”? [Katz], p.425

2. THREE ... REVOLUTIONS: RETURNING TO PRINCIPLES! 31

General Relativity “upgrades” the Newtonian geometro-dynamicdescription “force of some kind=centripetal force”:

Force = Mass × Acceleration

to a pure geometric description (space and time were alreadymerged) “matter tensor ∼ geometry tensor”:

Matter Tensor = κ Einstein Tensor

which, beyond the new “technical tools” (semi-Riemannian spaces,Ricci curvature etc.) amounts to passing from a description ofdynamics as “curved motion in flat (universal) space” to “flatmotion (geodesics) in curved space(-time)” (again as a figure ofspeech). In other words, taking a phenomenon (gravitationalforce) from the LHS of Newton’s principle and incorporating itinto the RHS as Einstein’s tensor (essentially the average cur-vature; κ denotes the gravitational constant). The “trick” pro-liferated: then came Kaluza-Klein, attempting the same withelectro-magnetism. It did not work as well, since “internal de-grees of freedom” could not be well accommodated as externaldegrees of freedom (i.e. dimensions of space-time). The alter-native was to build degrees of freedom outside the “obvious”ones, leading to Gauge Theory etc. Meanwhile the technologyadvanced and string theory is capable of such feats, introducing“real” dimensions (for a grand total of 11? 21? etc.). Some ofthem, of course, need to be “hidden” from every-day “access”by compactification, declaring them small enough (just anothermodel for space-time).

In the “phenomenological camp” the opposite tendency maybe noticed (in the spirit of quantum mechanics; see 1.3): let thedegrees of freedom (and states) be “internal” (abstract) ... andVertex Operator Algebras appeared! (or maybe to tame TheMonster? :-)

So, where is the third?

2.3. Space-Time: Is “motion” possible? We do notneed Zeno’s paradox (see [Katz], p.56) to claim that motion


is not possible ([LL], p.14) 2. Of course, we have to specify inwhich theory; in quantum mechanics, of course, since otherwiseclassical mechanics deals great with motion/continuous evolu-tion/dynamics (Poisson manifolds etc.), and we’ve learned notto talk about what reality IS, but only modestly (?) about our(scientific community at large?) best model about it/Him/She(one preconceived model for each specific “glasses” we chooseto wear) .

In quantum mechanics there are “states” and “transitions”,as in a sort of a “complexified” Markov process, where, amaz-ingly, the possibility of having a result in two ways may can-cel each other’s contribution (“indecision”), rather then buildup the probability! (to model mathematically this feature, wechoose superposition + interference, implemented as a lineartheory over complex numbers).

The incompatibility between knowing the position ANDmomentum at the same time, for the same direction (Heisen-berg’s uncertainty principle), conceptually refutes classical tra-jectories altogether (but still refers to classical concepts!). 3

If we insist in adopting QM to investigate the “motion pro-cess” and still have a “classical understanding” of what theelectron “does” in a two slit experiment we have to concludethat “IT” goes through both holes simultaneously! (The “bestapproximating” classical statement for the quantum occasion).So, “Is motion possible?” Well, ...

2.4. What is an “Event”? The differences in approach(Classical Mechanics, GR, QM/QFT) start with the concept of“event” (see [Ion00]).

2Zeno’s Arrow paradox seems to urge for Lorentzian contraction atleast.

3From a constructive point of view, one can always rewrite a C++object-oriented program in a modern version of FORTRAN, or obtain QMas an emergent theory [A], or based on classical keywords [BM]etc.


For Newton the “event” is a “particle” (existence) “some-where in time”; these three concepts, existence, space and time,are “absolute”, i.e. independent of observer and of each other.

For Einstein, “existence” is still “absolute”, although the“event” occurs in space-time (still “absolute”, even after theadvent of GR). After the QM lesson, we should agree that wemodel correlations: A interacting with B produces a C (sayelectron in a magnetic field yielding a beep on the up or adown counter; and the observer?). There is a missing factorhere to be explained later on, appearing in a parallel betweenquantum and classical computation - see [VIRequest-CS]).

To implement “correlations” one needs to define the “states”and “transitions” (e.g. using categories: objects and mor-phisms). There is usually a “time-ordering” issue here: statesfirst, then transitions. This may be thought of as developing thetheory starting from the “free case” (inertial reference frames/free theory in scattering method etc.) and then adding “inter-actions” (all frames/scattering matrix etc). It is essentially theold Newton’s goal (and Descartes’ methodology) of represent-ing functions (or theories) as power series or theories as otherseries: indexed by Feynman graphs, Riemann surfaces etc., i.e.building the “big processor” out of “micro components”), andcalled perturbation theory (yet this is not the whole story - seeIntroduction §2).

2.5. Quantum Field Theory. In QFT we have a contin-uum of degrees of freedom (the values of the field) only becausewe strongly believe in a given continuous space-time. Roughly,QFT is an “upgrade” of QM as a (complexified) Markov Pro-cess, where the complete graph represented (a matrix indeed),is replaced with a class of graphs and the complex numbers ascoefficients are replaced with operators (propagator) (a figureof speech again ... and again ...)

Then came Feynman (bringing QFT to the masses, thankyou! :-) - see [Zee], p.41) and introduced what we will call The


Automaton Picture: states and transitions, whether these arepaths in space-time (external DOF) or transitions in internalspace (IDOF).

Since we ultimately look for transition amplitudes of aninteraction in the context of a framework based on the freecase (classical in essence: we know how many particles go in,and what comes out, in classical terms), sum up the amplitudesfor all possible “scenarios” (correlation function as a sum overFeynman diagrams). This is a basis in the “transition space”(space of all “paths”). In some sense, the analog of a partitionof unity is the partition function.

The “problem” is, that if we believe “motion” is possiblein a continuous space-time, then we end up with too many“paths”, divergent integrals etc. Physicists have learned quicklyhow not to step in quick send, while mathematicians had a hardtime building the bridge over the “swamp of infinities” (con-stant/variable, infrared/ultraviolet, important/negligible etc.:-)).

2.6. External/Internal Degrees of Freedom: The Au-tomaton Picture. The natural way to “solve” the problem oftoo many paths (and infinities?) (let’s just cut the Gordianknot already!) is to realize that all we need is a category of“paths” and an action allowing to build a representation ofthis “Feynman category” with suited coefficients correspond-ing to the internal degrees of freedom (phenomenology) had inmind (hopefully towards a GUT, encompassing all of them, butgravity).

By now it appears that gravity is an organizational prin-ciple within the space-time description (GR), rather than anexchange interaction. Trying to push the beautiful particle-field picture from scalar and vector fields to spin 2 tensor fields(... obstinate ?) could be the “take a bigger hammer” (stringtheory ... M-theory?) approach to “crash the nutshell”, ap-proach which looked so repellent to some (Grothendieck etc.).


It worked with Fermat’s Theorem, right? But, “What’s takingso long?” (see [Are-we-nearly-there-yet?]).

... or rather try to implement gravity as a pairing betweenthe Feynman category, which captures the causality (there isNO universal time at the micro scale, so that we have to dealwith time ordered products / OPEs etc.) and the “coefficientcategory” which captures the macro-behavior/hidden details(see §1.3) in an adjunction which trades additional externaldegrees of freedom (e.g. applying the homology differential, i.e.insertion of an edge [Ion03]) for additional internal degrees offreedom (technical details should not clutter the picture at thispoint!).

This should be done, perhaps, in conjunction with a modelfor the information flow (see §1.4), since there are several macrosys-tems involved, and an experiment, like a quantum computa-tion, involves classical read/write operations subject to classicallogic/laws ... (see [VIRequest-CS]).

No matter what the specific implementation will be (e.g.using graphs, networks, categories etc.), it will capture the ideaof automaton (states and transitions; e.g. cellular automatain [Wolfram] or [Cells are circuits, too!] etc.), implementa-tion written in one’s favorite object-and-relations oriented highlevel language.

2.7. Is there a “time”, after all? Indeed “time” is THEdelicate concept; or rather a plethora of interconnected con-cepts! We all like to ponder on the fundamental questions,especially as adolescents or young researchers trying to findnew ways ... (see “Time’s Up, Einstein”, by Josh McHugh,Wired 06/2005, p.122). It was the analysis of what time is,that led Einstein to a clear picture unifying Newton’s univer-sal space with his universal time. Even at that stage, onecould ponder on a hidden assumption Einstein implicitly made:transitivity of synchronization (“the early years”: challengingeverything). It may indeed fail in GR, if there is no local


time (non-integrable orthogonal distribution to a Killing vec-tor field etc. - see [O’Neill]), and instead of spending $200,000on a “Michelson-Morley experiment” trying to reintroduce the“ether” (! see “Catching the cosmic wind” by Marcus Chow,New Scientist, 2 April 2005, p.30 ... or rather not) one mightrather test the above mentioned possibility, which definitelyholds true at some level of accuracy (the “ergodic principle”:if it is not forbidden by the laws of physics, if will happen ...eventually): the problem is “In what conditions?” ...

So, better investing more into theoretical research (ratherthen building bigger muscles for physicists to smash every-thing), and maybe find out that the “ether” is ... the “Higgsfield” 3 ... or not!

But since we aim at a deeper model (beyond SM or ST),where “events” are “pure correlations”, such issues are sec-ondary. One lesson learned from Special Relativity is that thereis a causal cone; events can be spatial separated (no causal cor-relation possible - not talking about entanglement yet ...), or ifcausally correlated, than they must be time-separated. Yes, a“proper time”, is a different concept (“continuity of existence”),rather playing the role of a parameter (implementation depen-dent); as opposed to “laboratory time” in quantum mechanicsetc.

So, what we need is a “Causal Structure” and that is pre-cisely what a Feynman Category provides!

Feynman Category =⇒ Causal Structure.

In bombastic terms:1) Event: NO! Correlation: Yes!2) Time: NO! Causality: Yes!Of course, we do not request a total “removal” of time from

the “old” theories (we still wake up at 7, take the kids to schoolat 9 etc.); what’s been said is well said. We request its limita-tions be acknowledged.


If a causality structure is given, then to benefit from thepresent and past theories one has to deal with embedding itin a classical 4d manifold (d=11, 21?), as some “backgroundspace”; or at least, after representing it in one way or another(e.g. decorating punctures on Riemann surfaces with operators,VOAs etc.) one has to come up with an Operator ProductExpansion (OPE) as a much more complicated issue that theusual 1-parameter group of unitary transformations capturingthe dynamical evolution, or “time flow”.

What is left of the idea of 4-coordinates as a “... start-ing point of the mathematical treatment” ([V],p.24)? Firstof all, one should postpone the “mathematical treatment” un-til the “design” of the theory at conceptual level is complete(or at least satisfactory; “implementation specifications” of thephysics model), then let the (“implementation”) specialists im-plement the theory ... (another story! we would not have hadQED a few decades ago, right? It had to be done fast, no time(...) to wait for mathematicians to be pleased with a “rigorous,i.e. mathematical, implementation”).

What I am advocating for is a “device independent, portableinterface” between math and physics models (author-independentimplementations, user friendly :-)) ...

Then there are some holistic questions ... There are 3-pairsof non-commuting observables representing external degrees offreedom (q1,p1, etc.). Why 3-d? why 3 generations? etc.; thesecould be “separability questions”, telling theories apart, but wefeel there is more to it than it meets the eye ... (see Ch.9 §3).

At the pragmatic level of VOAs, the substitute for Lorentzor Poincare transformations (or conformal invariance) is hav-ing a built in Virasoro algebra, which in a sense is the simplestgraded Lie algebra ... What is its deeper message? (Sure, dif-feomorphisms of the circle: quantum phase?; a way to removethe “continuum” from the quantum complex phase? ...).


3. A “New” Principle

Returning to General Relativity, perhaps the most impor-tant consequence, beyond expansion of universe and Hubble’sconstant, is the concept of black hole. The unification of GRand quantum theory was initiated by S. Hawking as an exten-sion of GR incorporating the black hole radiation. Since then,three laws have been identified (see [Smolin], p.92).

The first law relates temperature, as a measure of energyper DOF, with acceleration as a measure of the interaction(Newton’s sense):

Unruh′s Law : Temperature /~ = Acceleration/c.

It expresses a principle, therefore the simplest (physicist fa-vorite) way is enough: linear relation. Together with Einstein’sequivalence principle, it suggests that there is an energy distri-bution for the 2-point gravitational correlation (in our quantumdiscrete picture).

The second law:

Bekstein′s Law : ~ Entropy =1

κArea/(8π),

relates Entropy, as a quantity of information needed to com-pletely specify a state (the “quantum memory size”) and Area,which in a discrete (geometric) model should be thought of asa measure of the possible In/Out interactions (“quantum chan-nel capacity”?). Beyond the “global statement”, adequate forstating an equivalence principle, there should be here a “lo-cal/discrete” Stokes Theorem at work ... (?)

It is reassuring to find that Lee Smolin mentions implicitlysuch a “would-be” principle: “one pixel corresponds to fourPlank areas” ([Smolin], p.90), although it could rather be “oneinteraction qubit corresponds to four Plank areas” ... (?)

Later on (p.102), he derives some conceptual implicationswhich are evaluated as not admissible, IF there is no theory to

3. A “NEW” PRINCIPLE 39

back them up (we have learned allot from the old story: “Eu-clid’s Parallels”, axiom or not? Let’s derive the “unbelievableconsequences” first, then decide how to build the theory).

Finally the third law relating temperature and mass, butin an opposite way as the first law, is:

Hawking′s Law : Temperature = k/Mass,

or alternatively:

Mass = kβ

(with an eye on the entropy: Boltzmann’s correspondence etc.).

It refers to the radiation capability of a black hole (‘‘densityof I/O-interactions”?), rather then its energy distribution perDOF.

The situation is reminiscent of Newton’s position when sim-plifying Kepler’s laws ... so let’s look for a “new” unifyingprinciple! (we do have a “situation” here, right?).

3.1. “Mind versus matter”. Recall that E = mc2 (Prin-ciple I), in a sense, unified energy and matter.

Quantizing energy (bound states):

Principle III : Energy = hFrequency (= ~ω)

should correspond to quantizing information in some sense (Whatabout unbounded states?).

The new Unifying Equivalence (Super)Principle will be la-beled “Mind versus Matter” to convey its scope. It states acorrespondence between energy and information, both quan-tized/discrete:

New Super-Principle IV: qubit↔ ~ (S(qubit) = ~)

aiming (so far) to unify the “observer” and “observed” of quan-tum physics, explaining the “measurement paradox”, and whynot, providing an interface between the “safe” science and theother “believe-it-or-not” areas of investigation (conscience etc.- You name it :-)).


The idea is that a transfer/fluctuation of a unit of energyshould correspond to a quantum bit of information (“Preciselyin what sense?” ... a very good question indeed!). An addi-tional DOF (E/I) (internal, i.e. type of particle, or external(!), i.e. space-time “location”) changes the partition functiondescribing the distribution of amplitudes of probabilities in away similar to a black hole “leaking” a qubit of information.The theory should naturally incorporate the black hole lawsin the context of a General Relativity relocated from its natu-ral habitat (manifolds with a metric/Lagrangian) to the realmof Feynman Processes (representations of Feynman Categories:string/M-theory rephrased and background free/mass genera-tion upgraded). ... It is still missing, but in conception! (seealso Ch.8 §3)

3.2. Are black holes prototypical? Again it is reassur-ing that the idea of the above Unifying Super-Principle, in oneform or another, is present in the remarkable book [Smolin](p.101): “There is something incomplete about a law whichasserts a balance or an exchange between two very dissimilarthings.”. Paying too much attention to its draw backs is not al-ways a good idea (p.102). Yes, if one would just claimE = mc2,would not be enough ... But again, a theory starts with an idea,a new principle (1% of work), and then one designs the theorytop-down (99% of the work - Edison).

So, Lee Smolin is talking about a balance between “atoms”and “geometry”, but only in gauge theories there is a clearcut distinction between external DOF (space-time) and internalDOF (implementing the type of particle as a representation ofa gauge group and then “marrying” them as a principle bundleetc.).

Moreover, a distinction between “atoms” and “geometry”leads back to an “absolute space-time” point of view. This isno longer true in a Feynman Theory (FPI adapted to FeynmanProcesses as representations of Feynman Categories) where an

3. A “NEW” PRINCIPLE 41

insertion of a new graph should be thought of as “adding ge-ometry” (and also as a change of scale!) but then, under thefunctorial adjunction, in this way new internal DOFs are intro-duced.

Now my “bet” regarding the two profound questions from[Smolin], p.102, is:

(A) Yes, there is an “atomic structure” of the geome-try of space-time (i.e. leads to a better model - see 1 - e.g.PROPs or LQG’s “grains of space-time”), and our UnifyingSuper-Principle generalizes in a sense the idea behind Bek-stein’s Law. Indeed, in a discrete Feynman Category model,“area” corresponds to the number of interactions (or some moresophisticated concept depending on the author-dependent im-plementation, yet “counting” is the safest way to proceed -combinatorics, what else?), which from Unruh’s law, “carry” acertain energy; roughly speaking a “space-time event” A → Bhas a double role of both interaction channel (s/t or ... neither!)and information channel.

B) Yes, the Digital World Theory incorporating the theoryof information (Shannon, quantum computing etc.) on top of aFeynman Theory, will have as natural consequences the blackhole radiation laws (... and much, much more :-) ).

How to switch from black holes, thought of as “prototypi-cal” when it comes to “global” quantum aspects, to the generalcase, say in terms of Feynman graphs? In a brainstorming way:

Horizon→ Hidden causally ∼= No interactions→

→ Non− connected Feyman Graphs ...(?).

Then: “black hole radiation (interactions) appear” → “insert-ing non-connected FG in another FG” ... !? (not implementa-tion ready :-))

It is too soon for technicalities :-)


4. Conclusions

The following table contains the main “evolutionary steps”of the key fundamental concepts:Newton Space Time Particle xor wave N.A.Einstein Space-Time Particle xor wave ObserverHeisenberg Space Time Particle/Wave & ObserverDirac Space-Time Particle/Wave & ObserverFeynman Path Integral QuantizationFolklore Representations of Feynman CategoriesDWT 1.0 Representations of Extended Causal Structures

Here “Extended Causal Structures” refers to the incorpo-ration of the concepts of entropy and information processing[VIRequest-CS] in order to unify the classical interactions“particle-particle” and “particle-observer” modeled by Quan-tum Theory with “observer-observer”, i.e. genuine communica-tions (for symmetry reasons at least, but hoping that it wouldlead to a better understanding of “reality”, for example of themeasurement paradox).

How to put together all the above “design constraints” ina coherent theory, is another story ...

Then came The Institute (idea) :-) 4. It is meant to be theWeb Process aiming to stimulate the upbringing of the DigitalWorld Theory (“The Quantum Matrix” representing The Holo-gram) by taking advantage of the Butterfly effect (in a teacupor in our local inflationary bubble of universe ...?).

4See Annex A - VIReQuEST.

CHAPTER 2

Miscellaneous: a “warm-up”

The ideas and comments included bellow as a start up, willbe revisited later in more detail; the present chapter is only awarm-up ...

1. On the measurement “paradox”

Where the deterministic description becomes probabilistic?The state of a quantum system (QS) represents our knowl-

edge (model) of the system and its evolution is modeled in adeterministic way as being governed by Schrodinger’s equation.On the other hand it should come as no surprise that our knowl-edge of the system changes “abruptly” when a measurement isperformed. What changes is our description of the system.

Ultimately we only model reality and our models are notthe reality itself (phenomena versus reality - “a la” Kant). Butthe change of the state due to interactions between subsystemsshould also be thought of as “relative measurements”.

Deterministic or not?Hidden variables which are not local can be implemented

to provide a classical interface to quantum mechanics whichgives us “peace of mind” (we’re in control - reality evolves ina predictable way). This is just a way to disguise the sameconsequences, which are probabilistic in nature.

We are used to translations from one language into another(and thankful, as it disseminates knowledge), but some lan-guages have a future ...

43

44 2. MISCELLANEOUS: A “WARM-UP”

2. Modeling a complex process

We prefer to talk about “processes” rather than of systems,to emphasize the role of the evolution/transition in the contextof the Automaton Picture (Path Model).

2.1. The duality: internal - external. A process cap-tures the duality between a system (existence) and its evolution(change).

This in turn reflects on the two inseparable components of amodern approach to modeling reality: software and hardware!One can always trade one for the other!

To “fix a scale” amounts to splitting between I/E DOFs,corresponding to a chosen “modeling resolution” etc. A changeof scale operation appears in various contexts: partition ofstates in the micro-macro description (µM -correspondence),collapsing EDOFs and averaging over IDOFs (colored subgraphs)etc.

Once a scale is fixed, one has to model the componentsof the system (e.g. attaching vertex operators, associating amacro observable on state space cells etc.) and model the in-teractions between components: the type of interaction (e.g.class of graphs for a φ3-theory) as a “perturbation” of a “freetheory” (e.g. Feynman rules once propagators are selected).

2.2. Structural modeling (SM). Modeling, as a (hu-man) process for reflecting a part of reality into an artificalconstruct, is a cyclic process when including the experimentalfeedback (a learning process in a broad sense!).

Structural modeling adds emphasis on adding structure (sayalgebraic) to the usual quantitative/numerical capabilities anyscientific model should have. The structure consists in relationsbetween objects/concepts/variables, leading naturally to PathModels (see [DelM] or [SL] etc.).

Some of its key features, although almost universal, appearin different (specific) guises in various “places”.

3. WHAT IS THE INFORMATION FLOW? 45

A path model for instance, with multiple independent ob-served values and multiple dependent observed values ([SL]p.4), is essentially an IO-computing process.

Confirmatory factor models (loc. cit.) consisting of ob-served variables that are hypothesized to measure latent vari-ables, are essentially micro-macro correspondences of the Boltz-mann type.

The observed variables may be thought of as “readable”, i.e.global variables as opposed to latent variables ([SL]), or “hid-den observables” in QM, which are “internal variables”, or “lo-cal variables” which need not be directly accessible (althoughfor testing purposes one can try to read/print them; exceptwhen quantum computing/interaction is concerned, eavesdrop-ping/observation will alter the state etc.).

2.3. Why Path Models? The use of Path Models is at-tributed to Sewell Write (1918), a biologist ([SL]). Path anal-ysis is used to test relationships.

Now QM teaches us that there is nothing else but corre-lations, so designing Path Models is rightfully termed causalmodeling. The researcher, in an early stage of the design pro-cess, specifies a model “a priori” based on theoretical considera-tions. This model specification, involving the available relevanttheory, research and information, towards the developing of atheoretical model, is a critical part of SM (loc. cit.).

3. What is the Information Flow?

The Einstein-Podolski-Rosen paradox (EPR-paradox) ex-hibits the inconsistency of QM under adjunction of classicalassumptions/ interpretations: non-locality as an “action at dis-tance” (classical view of force/field etc. producing a change ata spatially 1 separated distant point). The role of informationis not considered, since it plays no role in the classical physics:

1Declaring “spatial”, we mean “not causal”; the paradox prone condi-tions are half-set!


it’s “all” about “motion” and the number of particles (NOP)does not change!

In the teleportation setup the crucial concept is that ofinformation flow. The quantum model (within the DWT atleast) generalizes the concept of causality into the concept ofcorrelation which incorporates correlations due to informationexchange which might not be local 2. One then distinguishesa t-correlation (time correlation = classical causality) and s-correlation, referring to/modeling space-correlated subsystems,which from the observer’s external (macro) point of view aredistributed in “space”. The need for this dichotomy steemsfrom limiting the quantum computation as being sequential orparallel (in an outsider’s description).

Now, although from the observer’s point of view the NOPdoes changes (creation of particle-antiparticle pairs), there issomething that does not change: “information is conserved”,the number of qubits is conserved, and appears as a “motion”up and down relative to the external global time of (or descrip-tion made by) the observer. This is the information flow. Theinformation flow carries s and t-correlations!

The root of the incompatibility between the basic assump-tions of classical mechanics and (relativistic) quantum mechan-ics, and exposed by the EPR-paradox as a non-locality, is thenon-constancy of the NOPs relative to a classical external de-scription, usually based on a local or global “time”.

I.e., in the S−O interaction, “straightening” the s/t-correlationsof the S − output that O perceives as Input, therefore having“to decode the message” into “spatial” or “temporal” (processthe info as sequential or parallel; ultimately label consecutivelyblocks of symbols for which the order is indiscernible), the S-computation (the information flow) is not always an “s/t-grid”.And on of the basic Redemeister’s moves, straitening an S into

2Not even local to the system under study and involving the observer’sintervention.

4. THE “UNDERSTANDING” OF QM 47

an I does not work in the world of quantum measurements(Measuring the observables A, then B then A, is not equiva-lent to measuring A twice Ch. 6 §1).

The non-constancy of the NOPs, in the spirit of the Gen-eral Relativity, means that “Space-Time” “evolves” (is not just“geometry”) and since in the spirit of QM the evolution is notdeterministic (if accounting for measurements, or at least inthe spirit of DWT), then “Space-Time” cannot be “fixed” asin GR (“pure geometry”) or as in String Theory (unable tosurvive without a background space manifold). Therefore, inthe spirit of DWT at least, “Space-Time” with its IDOFs isthe “vacuum”, which is a resolution of the point/qubit. The“singularity resolution”, or call it the “experimental scale”, de-pends on the questions (input) O provides to S, and as in anycomputation, classical or quantum, requires various amounts ofmemory being used ...

4. The “understanding” of QM

There is an extensive literature on the physicists’ ways of“understanding” quantum mechanics. A first key issue is themeaning itself of the term “understanding”; it refers to the com-patibility between QM and our classical model used to perceivereality, for example assessing the position of a (macro)pointer ofa macro-apparatus etc. To properly study this relation, one hasto model and study the quantum-2-classical (Q-C) computinginterface (e.g. in our suggested ABE-model of measurement).

The three ways presented in [Stapp] (Bohm’s interpreta-tion, branching universes, Heisenberg’s model) attempt to dealwith the “quantum jump” (collapse of wave function etc.) inthe context of the “macroscopic possibilities previously gen-erated by the deterministic laws of motion.” (loc. cit., p.19).Indeed, limiting to modeling motion will not allow escaping thedeterministic realm, even that of QM at the level of Schrodinger’sequation. The (more) complete picture has to include the dy-namics of information/entropy of the S − O-system (process).


5. Is it all in the brain?

Ultimately there is nothing special about O in the above“dual system” (S-O), except (we need to emphasize) that thedescription is “order sensitive”: model(S−O) 6= model(O−S),for a given pair S and O. In the broad picture any of the Sand O could be “machine” or “human”.

When it comes to the “measurement paradox”, the O-componentof the above S → O-system (process) will achieve a conver-sion from quantum computation (communication) to classicalcomputation (communication), usually associated with an I/O-operation. It does not matter if O is a human or a macro-apparatus, “macro” meaning that it “behaves” (i.e. it is accu-rately modeled) as a classical system.

Usually O is pictured as a human staring at the pointer ofan instrument. The “conversion” Q→ C was already in place,done by the apparatus. The point is, our eyes are not “goodenough” to perceive (read) a quantum communication (com-putation), and this is why we are provided with a conscienceperforming classical computations. But for difficult decision-related tasks, we probably are endow with a sub conscienceperforming quantum computations (nice try, right? :-). Andwe do have trouble “reading the results”, i.e. what our “deepinner self” inspires us to do. The “crudest conversion” seemsto occur when we wake up; the nonsense we remember of ourdreams is reminiscent of a measurement of a superposition ofstates/thoughts (quantum computation results, perhaps).

It is conceivable, continuing the above speculation, thatwe will find the way to by-pass the classical interface whenthe merger between human and (quantum) machine will beachieved: The Human 2.0 [Human 2.0]. Then the new possi-bilities are hard to imagine ... 3

3Any ideas, Mr. Spielberg?

6. CLASSICAL OR QUANTUM MECHANICS? ... WHAT’S THE DIFFERENCE ANYWAY!49

6. Classical or quantum mechanics?... what’s the difference anyway!

The so called correspondence rule (Dirac quantization) isimpressive, but perhaps a bit misleading ...

To understand the “difference” between CM and QM onehas to overcome the difference in “language” used to encodethem, by bringing them on a common ground (see also 4.4.1).

Note that CM, say in Newton’s formulation, is an “affinetheory”. Galileo’s relativity group which allows to realize spaceas a homogeneous space, represents not only independence ofcoordinate systems, but due to translations it represents the“free theory” encoded in the second order differential equa-tion of motion. The external forces (or associated potentials)appear as perturbations, providing a framework for modelinginteractions.

Now quantum mechanics appears from the start as an ef-fective theory: states and transitions implemented as a lineartheory, i.e. homogeneous. There is a zero vector, but which isnot a state; in fact states are “rays” (lines), so that its true ge-ometry is projective and when representing observables one al-ternatively has a projective representation on the original spaceof states, and the need for a central extension to relate the tworepresentations is clear etc..

To compare CM and QM one has to translate CM say intoa homogeneous theory first. The simplest way is by addingone ficticious dimension to space and consider the embeddingof Newtonian space as R3 × 1. Then Galileo group is real-ized as a group of linear transformation on R4 (see [AI], Ch.3,p.152). Now the Newtonian space appears as a projective space(almost: without the North pole!) and that the “larger space”is a compactification of the Newtonian space! One can pro-ceed to implement the theory in operator form moving from


the classical observed values, say energy, as scalars to an eigen-value form, since the “states” (position etc.) are now “rays” inR4. Observables will be operators etc.

In this “quantum-like” formulation CM is a linear theoryand can be “better” compared with QM.

The first major and crucial difference is the lack of super-position, while the second is the absence of interference.

To admit “superposition” for CM would mean “delocaliza-tion”: a system which is a “particle” could be at two places atone time, i.e. the system would be distributed in space, yet theparts would not be independent, but s-correlated! Of course,this would be a step towards explaining the two-slit experiment,since now a particle could “split” and its two s-correlated partswould go through the two distinct openings in the wall (nots-correlated) and then “recombine” on the measurement plate...

A moment’s thought reveals that this is the right way toenhance the power of our models regarding reality. One shouldnot “encode” all our knowledge at the level of “states” and pos-tulate a classical and deterministic dynamic law, but should bemore cautious and distribute the possibilities between statesand the transitions which are supposed to model the “ways”a cause “determines” and effect (Path Models). A law of dy-namics is still required on top of this, as in the prototypicalapproach of Feynman’s regarding high energy quantum phe-nomena (FPI).

Even better, one should allow for “partial knowledge” (in-formation) about the system (mixed states) and make the tran-sition to a statistic description first, before comparing withquantum mechanics. This step (“difference”) is essentially theadjunction sets-vector spaces together with a 0, 1 to [0, 1]change of logic’s coefficients (probabilities).

To accept within CM the second crucial difference, “inter-ference” (say by complexifying everything) is much harder (al-most paradoxical). It seems to enable what is left to be able

7. TO THINK OR NOT TO THINK CLASSICALLY? 51

to explain the fringes in the above experiment, which may bethought of as the generic (prototypical) quantum phenomenon.

Then, where are Heisenberg’s uncertainty relations in thistransgression from classical to “quantum”? Would it be enoughto “add” a compactified ghost-like dimension: an S1 complexphase? (R3 × 1, i).

7. To think or not to think classically?

The classical meaning of these assumptions is revealed bylooking at their dual role in the context of interaction-communi-cation duplex interpretation.

Classical information (communication) consists of stringseither 1 (x)or 0 (Z2); same with classical mechanics (inter-action), which is based on functions (∀x∃!y), which leads todeterministic models etc.

Quantum information (COM) deals with strings of qubitsconsisting of a superposition (complex linear combination) of1 and 0 )(“just” change the “coefficients”: CZ2!). This trans-forms the classical “exclusive choice” into an “offer” of possi-bilities (of evolution/correlation between cause and effect), butalso allows a “fine tuning” mechanics to ballance positive andnegative contributions, “pros and cons”, into the model: a mul-titude of paths/ways a cause may produce an effect. Viewedin this way, it is a natural upgrade of the bold “determinis-tic way” of classical science (“we can understand everything,in principle”), to a more humble, realistic point of view: “still“dubito, ergo ...” modeling”.

A priori we don’t really know (or we are not so sure), howevent A led to B, but we have some hints ... “this way ORthat way” (NOT the “God-like confidence”: this XOR that).We wish “tertium non datur”, but ... the simple/naive pictureabout reality is history by now ...

Then, how should “We” think, if our models and appa-ratuses are quantum (deep down)? When we read the quan-tum information we project onto a measurement basis [BZ]


because ... (?) or we really don’t have to!? (n.a.) Indeed, whyshould we “reduce”/collapse/ etc. the quantum description?Well, obviously because we can’t process QI (not well anyway),since we are not equipped with quantum computing capabili-ties/algorithms/logic (yet), since ordinarily we wouldn’t benefitfrom it ... Or do we?

The fact that the system is in a state (message symbol) of1xor0 is a “property” of the system, but having the system in astate of 1or0 is not [BZ], because we don’t like them. It seemsthat “property” really means “classical property”. This reflectsthe fact that in an S → O interaction/ communication, if S ismodeled as a quantum system, yet O as a classical system, thenand S − property is not an O − property (how else?)...

Ultimately, the question is when O should “think in a quan-tum way” and when not! We should get used to Q-think sincethe quantum world (from nano-technology to astrophysics andbeyond, homeopathy, cold fusion, cryptology etc.) reveals un-tapped possibilities reminiscent of miracles (“How is that pos-sible?”, i.e. conceivable, since otherwise perfectly “real”!).

Well, a good (small) step for mankind would be to “up-grade” the curriculum ... (probably I shouldn’t go there!).Teaching 400 years old physics and 2000 years old math inschool [Smolin-3], when the knowledge accumulates exponen-tially, is leaving everybody behind!

Solutions? Plenty! For example, if you would like to learnphysics, go to ’t Hooft’s web site 4 and ... teach yourself! (seealso [‘t Hooft]).

On the other hand, why hurry so much? “Do we really haveto?” (“Langsam, aber sicher!”).

4http://www.phys.uu.nl/thooft/theorist.html How to become a goodphysicist?

8. ADD A PINCH OF ... EINSTEIN RELATIVITY! 53

8. Add a pinch of ... Einstein relativity!

Now the “genuine” relativity, Einstein relativity that is, canbe thought of as another “projectivisation effect”.

Again add a ghost-like dimension to space and describeGalilean motion in the resulting projective space: relativityseems to emerge (Minkowski space-time) ... (?).

Some say that this “concoction”, that is mixing quantummechanics and relativity in a coherent way, yields QuantumField Theory! It would be instructive to see a “preview” of this“prediction” in such a simplified way (theoretical “gedankenexperiment”) ...

Part 2

Reference Manual

CHAPTER 3

The Principles of DWT

The mathematical implementation of DWT is based on rep-resentations of Feynman Causal Structures (FCS), called Feyn-man Processes. The name should suggest that FPI approach toQFT may be thought of as a complexified and enriched versionof Markov processes. QFT, as a 2nd quantization, is in a sensea “micro theory”, when compared to Quantum Mechanics as aneffective and macro-theory. Alternatively, QM is a (0+1)-QFT[Zee], i.e. “space-time” is collapsed to a point. A FeynmanCausal Structure plays the role of a resolution of a point.

The above alluded to representation is rather a pairing,allowing to incorporate IDOFs and enabling the I/E-duality.

FCS are essentially PROPs with additional structure, gen-eralizing QFT (algebras over the Feynman PROP), CFT andVOAS (algebras over the Segal PROP) and TQFTs viewed asalgebras over cobordism categories etc.

1. Why Path Models?

The reasons steem from the main corollaries:- Leads to Automaton Picture (universal computing model

etc.)- Enables both “local” and “global” (i.e. non-local) features

of the theory- Extends QM: implemented as matrices over complex num-

bers, it is a Feynman theory in disguise (complexified Markovprocess; but not enriched!).

57

58 3. THE PRINCIPLES OF DWT

- Why the IO-computer model? because, as Feynman saidit [F1], modeling a quantum system starts by splitting the uni-verse in two parts (system and non-system) and modeling theinterface between the system and the rest of the universe (ex-ternal influences). We add to this suggestive image the thirdcomponent, the observer (“modeler”), which tries to controland make sense of the In’s and Out’s. It leads to the ABE-model of measurement.

2. The Causal Resolution

Modeling reality is management of DOFs (computationalresources).

The “perturbative approach” delineates the “free” from the“interactions”, at each stage in the resolution.

The “point” is the system overall, without any parts distin-guished, i.e. the theory at the corresponding “scale” is effective.Various “resolutions” distinguish subsystems (parts of the partsof the parts etc.) modeled as “points”, as opposed to interac-tions which are modeled as (multiple) arrows (small category,rather then graph, but to keep the terminology simple, we willstill use the term graph).

Mathematically, an example of a causal resolution of a sys-tem is a double complex Gn,m [FI1], bigraded by the numberof internal and external vertices of the source and target of theinteraction graphs. The axiomatic version includes the otherusual examples.

The (asymptotically) Free Theory amounts to the exter-nal structure (external points - we will avoid the term “legs”)while the internal “virtual” points with the internal arrows cor-respond to (model) the interaction(s).

In the Lagrangian FPI picture, splitting the LagrangianL = Lfree + Lint amounts, via Wicks Theorem to giving theclass of graphs and the propagator as the inverse to the qua-dratic part, once the representation is given via an action andthe Feynman rules.

3. THE FUNDAMENTAL PRINCIPLES OF DWT 59

The free classical theory amounts to Galileo’ relativity (sym-metries determining a homogeneous space).

The role of graphs, besides being the building blocks oftransition processes:

Γ−Γ→ Γ+

is to encode the In/Out data corresponding to preparation ap-paratus and observation (measurement) apparatus (again, afterrepresentation, i.e. “coloring” the vertices with operators etc.)

In this way “space-time” is the “vacuum” from which pairsof particle and anti-particles pop in and out. Our need to “fillthe space” with say R3 between particles should be satisfiedby retaining its symmetries (Galileo, Poincare etc.), since usu-ally Space-Time comes as a homogeneous space. Then one can“convert” the EDOF per particle (point/object etc.) into ID-OFs, towards a unification of symmetry groups (internal andexternal). At this stage it is productive to simplify the pic-ture and assume that there is a universal IDOF per point,namely a qubit C × C, with its symmetry group U(2). Think-ing in this way allows for a direct connection with the CS-interpretation. On the other hand, due to the “coincidences”of structure and relations between quaternions and the otherkey groups of transformations (SL2, SU(2), SL(1, 1), Diff(S1)etc.) there are various ways (“enough resources”) to connectwith the traditional picture (gauge groups, conformal symme-tries, Poincare/Lorentz symmetry).

3. The Fundamental Principles of DWT

The 1st Fundamental Principle of DWT was explained above§2: a System is modeled as a Feynman Process, i.e. a represen-tation of a Causal Resolution of DOFs (QDR).

As suggested by the lows of radiation of black holes Ch.1,there is a new unifying principle going beyond Einstein’s equiv-alence between energy and matter (see Ch.6 §6.2). This (prin-ciple related to the) missing “4th Law” (see [D1]) implies in

60 3. THE PRINCIPLES OF DWT

a concrete way the equivalence between space-time and mat-ter/energy, or from the alternative point of view, the equiva-lence between information and energy (matter/space-time etc.).

More precisely, the 2nd Fundamental Principle of DWTstates the duality between external DOFs, determined by theQDR and internal DOFs, given by a functorial representation(as a Feynman Process):

Feynman process ∼= duality(causal structure,matter&energy).

The familiar EDOFs allow to describe the external dynamics of(an observed) system, i.e. its motion. IDOFs allow to describethe internal dynamics, i.e. its evolution in internal state-space.

Although duality blurs the distinction between particlesand fields, i.e. particles, modeled as (labeled) vertices and inter-actions modeled as relations between vertices (Feynman rulesand all that), the external dynamics primarily concerns the dy-namics of particles/vertices while internal dynamics concernsthe dynamics of internal states and types of particles.

Now, it is natural to assume that the external dynamics iscontrolled by the Action Principle (energy/Hamiltonian depen-dent), while the internal dynamics is controlled by the EntropyPrinciple.

The 3rd FP of DWT states the balance between energy Eand quantum entropy I :

F + TI = E = mc2,

where F is the Helmholtz free energy (see partition function§4), E is the relativistic energy, T is the internal temperatureand I is the quantum information charge/entropy (see Ch.6).

Such a conservation law should be obtained from a symme-try principle via the Noether Theorem.

A quantitative development of the above ideas will be de-ferred till after reexamining the classical and quantum infor-mation flow and entropy (Ch.5 and Ch. 6).

CHAPTER 4

Causality: Information versus Time flow

The dynamics is usually associated with “time”, and mod-eled as a “time flow” (1-parameter, continuous or discrete, semi-group of transformations etc.).

The DWT realizes the importance of the concept of “infor-mation flow”. Roughly put, the causality models interactions,and while there is no “foliation” allowing to define a “globaltime, there is a current of quantum information “flowing” in aquantum process (quantum communication). The observer inthe SO-model (system - observer) implements a “global time”as a labeling of the information acquired (interaction experi-enced in a S-S-interaction). It is also a “flow”: the flow ofsymbols/states is the IO-information flow at the level of thecorresponding interface (S-O, O-S etc.).

Due to our similarity in information processing (roughlythe same hardware!), humans (etc.) agree on a “simultane-ity” facilitated by the high speed of light, when averaging overquantum fluctuations which were irrelevant for us in the past(i.e. our previous “communications”!).

A quantum fluctuation, say a creation of a particle-antiparti-cle pair, viewed as a creation of additional internal DOFs, in theDWT is considered equally a creation of “space-time” (there isnothing “in between” particles, besides the potential creationof additional I/EDOFs!).

Classical information and Shannon entropy can be inves-tigated in terms of decision trees. Keeping in mind that aFeynman process is an enriched (“multi-paths”) complexified

61

62 4. CAUSALITY: INFORMATION VERSUS TIME FLOW

(changing coefficients from real to complex numbers) Markovprocess (focus on states and transitions), enlarge the class oftrees as “models” to the class of graphs (networks, of course)and replace probabilities associated to edges (paths) with am-plitudes of probability, to enable creation and destruction ofquantum information (qubits).

Then the Shannon’s entropy which at the level of trees isa residue of a more powerful invariant (to be explained bellow)“upgrades” to von Newmann entropy which is still a residueof the corresponding quantum invariant upgrading the classicalinvariant.

Recall that allowing graphs as mode general models, intro-duces loops (feedback), which provide the quantum correctionsto the classical approximation/picture.

In the diagrams that represent quantum processes (con-tributions to the resolution of a quantum process), quantuminformation (entropy) “flows”, as intuitively “felt” for examplein [Co1].

The definitions and exemplifications implementing the above“project description” will be given as marginal comments on[BZ].

The categorical implementation based on monoidal/braidedcategories with duality is deferred to the Reference Manual.

CHAPTER 5

Modeling classical information flow

Classical information (logic) deals with elementary sym-bols consisting of either 1 XOR 0 (exclusive “or”). Same withclassical mechanics: for any x there is unique y, i.e. func-tions. This leads to deterministic models (no “creation” op-eration/coproducts) governed by the classical messages Z2N(sequences of 0 and 1).

Quantum information needs elementary symbols which arecomplex linear combinations of 1 AND 0 at the same “time”,called qubits. This allows to create dichotomies; it’s an “of-fer of possibilities”, a choice given, leading to an escape fromfate!. Quantum mechanically it allows a truly “dubito ergocogito”, since it allows for multiple contributions (ways), i.e.paths, for “one cause” to produce an effect. For an objectoriented (categorical) mind, such a correlation is described asa Hom(cause, effect) in categorical language; the physics in-terpretation is clear: the set/class of paths, as Feynman wasthinking.

What is truly new is that some alternative ways may di-minish a future possibility (destructive interference).

1. Measurement: a “paradox”, or what?

Now will this help understand the measuring paradox? Math-ematically, measuring (an R-operation - see Penrose), is a “pro-jection into a measurement basis” [BZ]. With the interaction-communication parallel in mind, to “read out” the result of the

63

64 5. MODELING CLASSICAL INFORMATION FLOW

quantum communication (the interaction/ correlation cause-effect) we need to “classically decode”! Why? may be becausewe do not want (actually “need”) to process quantum infor-mation, or we use devices incapable to do so (bulky classicalapparatuses; but if a polarized filter is followed by an otherone, and another one etc. in a sequence of measurements, thiswould not happen ... except: what’s the use for us? we werenot “eavesdropping” on what the result was, and we are reallycurious with our classical curious mind!). We and our macro-apparatuses are equipped only with classical computing algo-rithms and logic, be it IO-level only or not. When we claimthat “properties” (and we mean classical properties) do not ex-ist prior to th measurement we refer to the comparison of thesystem’s behavior with classical M-observables of our choice.

Of course there is no 0 or 1 M-values prior to measurement,yet the quantum property of the system (part of the holographicworld, yet reasonably accurately modeled as aHom(cause, effect))is there (what is “is” ... well, that’s not my story!), whether weread about it or not!

It is a “hidden property”, since it is not classically mea-sured (no classical IO) by us. The same projection, trunca-tion etc., occurs when a QS → CO interaction/communicationtakes place, involving a quantum system QS and a classicalobserver CO (our conscience or apparatus).

To model and predict the outcome (behavior) of such a“combo” one may chose the ABE-eavesdropping process, prob-ably leading to a logic which is an extension of the (irreducible)quantum and totally reducible classical logic (see Ch.7), as itwas suggested elsewhere (§3, Diagram 1).

What can we do about it? ... this is the question! With ourclassical hardware (body) we indeed are confined to classicalsoftware, for the day-time and everyday activities at least; butI believe that the integration of acquired information is doneotherwise, more efficiently ...

2. SHANNON OR VON NEWMANN? 65

2. Shannon or Von Newmann?

Now, to use Shannon information to measure quantum in-formation would be surprising, as it was not designed to do so!Yet the ideas, some implicit, true, are deep enough to be usedwithout change; all we need to do is to upgrade the implemen-tation: “change coefficients” and change the trees (forests) into... (more general) graphs! This is quite natural, since quantumphenomena can be often well modeled as “loop corrections”(i.e. don’t forget the feedback, even if it is back in time! ...that is, relative to our communication’s time!)

Then a correspondence emerges relating quantum entropyand classical entropy, represented by Shannon’s and Von New-mann’s mathematical implementations:

Q− info

reduction // C − info

vN −Entropy

? // S −Entropy.

Now in both cases entropy is an invariant of the µM -correspondence(M-parts or µ-states; e.g. Boltzmann correspondence), whichis described usually in terms of partitions, but which can bemodeled by (decision) trees or more general graphs (see also§3.3):

ρ− states

vN−Entropy

// µ− states

S−entropy

< M >ρ?// < M >µ (M − cells)

This will disclose a deeper underlying invariant, we call Infor-mation Charge, possibly related with Kolmogorov-Sinai entropyas a limiting case, for which the usual “global entropy” appearsas a residue sensitive to the external structure of the tree/graph(see §6.1):

H(Res(tree)) = |tr(QI(tree))|2, SvN (graph) = tr(QI(graph)).


The quantum information charge flows through the network asa current subject to the usual Noether formalism: it is dueto symmetries of the Lagrangian controlling the dynamics ofthe network, expressed in the language of category theory asbraided categories with duality.

To be more precise, we need to “creatively” review Shan-non’s approach to information theory, with an eye on [BZ],turned critical for liveliness, definitely not to offend in any way!

3. Shannon’s measure of information

Since a comparison “quantum - classical” is always useful,we will consider both quantum and classical descriptions attimes.

3.1. What is a (good model of a) quantum sys-tem”? Assume a quantum system of “simple type” (H) (theHamiltonian forms a complete set of observables) is modeledby the (prepared) state ψ revealing Ei as a classical measure-ment of the quantum (classical) observable H (E), energy,with a probability distribution π = pi. Then pi = |λi|2,where λi are the probability amplitudes relative to a mea-surement basis ψi (i.e. we know how to filter/refine/partitionthe system ...) ψ =

∑

i λiψi. The expectation value of E is< E >ψ=< ψ|H |ψ >.

The probability distribution can be alternatively introducednot using amplitudes as above, but also using a partition func-tion Z(β) depending on a global parameter β (the “tempera-ture”), under the assumption of a “normal distribution” law(see §4).

3.2. What is a (good model of a ) classical system?Now the Boltzmann correspondence for a classical system, inview of the linearization of classical physics (Ch.2 §6), can beinterpreted as follows. The classical system is in a superpositionψ =

∑

i niψi with bits ni ∈ BZ2 as coefficients.

3. SHANNON’S MEASURE OF INFORMATION 67

The classical urn can be used as a typical example. Theurn is the (classical) system with N parts (subsystems), thecolor is the M-observable, with m = Spec(M − observable)possible colors which determines a partition of the state spaceΩ (mN states): P = Ωi with |Ωi| = ni, where i ∈ [m] =1...m. Now shake the urn (i.e. apply an unknown evolution,decoupling the system and the observer by a transfer of entropy:decoherence).

Then “listen” to (interact with) the system: extract theballs one by one. In fact the model changes (NOP not con-stant), but if accounting for the balls extracted (message re-ceived in this S − O-communication), then NOP is conservedin the larger Maxwellian box (see Maxwell’s demon) withN−kballs on one side and k on the other of the one-way permeablescreen. To understand the information implications, we shouldthink of the balls as information carriers (symbols), and thesequence of colors as a message received by the observer. It isa classical information source, but with memory, characterizedby the probability distribution π = pii=1..m of occurrence ofthe “emitted” symbols (states).

3.3. Decision trees. A statistical approach, like the oneabove, inevitably uses averages and the limit of large numbers.We are looking to bridge the gap with “particle physics”, andtherefore think about a measurement as an exchange of a “in-fon” carrying the “information charge” of I qubits.

As a provisional stage, think of the associated partitionfunction of the Urn as a linear combination of labeled binarytrees:

Z(d) =∑

t tree

ct/|Aut(t)| t,

where d is the maximum depth (“number of questions”), forexample d = NH , where H is the Shannon entropy:

H = −∑

pi log pi.


The leaves (terminals) represent length N sequences. The num-ber of typical sequences, i.e. those realizing an actual (exper-imental) probability distribution “close” to the given one π,is:

W ≈ 2NH.

¿From the point of view of QM, a decision tree as above, withvertices corresponding to questions and descendent edges la-beled with Y/N, represents a sequence of measurements hav-ing the questions as observables with outcomes Y/N (spin-like).The Shannon entropy is sensitive to the external structure ofthe tree, not at its particular structure:

H = H(Res(t)),

where Res(t) is called the “residue” of the tree, representingthe external structure of the tree, i.e. the corolla resultingfrom collapsing all interior vertices and edges.

To relate with the Feynman formalism, the leaves (finalevents/ block messages) may be identified with paths join-ing the root (“source”) and particular outcome (“target”) (see§6.0.5).

A more general framework has to allow all trees (classical),not just D-ary trees (“Brownian motion in an D-dimensionallattice).

Shannon entropy (information) represents the minimum num-ber of questions to determine any such particular path (se-quence), i.e. the information of the longest path, in other wordsthe info contained in the “directions from root to leaf”.

Intuitively speaking, if the path is the cause and the partic-ular sequence is the effect, then there is unique cause determin-ing the effect. In the quantum world one should allow loops,and mimic the paths analysis (see what follows) with ampli-tudes; Von Newman entropy must be used (obtained) in placeof Shannon entropy.

The Shannon info density is the info per “jump” or tran-sition in such a “Brownian path”. Think of the “space” as a


The Measurement Process

Preparation Observation

EncoderAlice

C−int. ))SSSSSSSSSSSSSSSSS

Q−Interaction // // DecoderBob

C−int.uukkkkkkkkkkkkkkkkk

Eavesdropping

Eve

Experimenter

(1)

Figure 1. The ABE-model of the measure-ment process

quiver, similar to Markov chains. If there is a source withoutmemory, it is called a Bernoulli scheme.

In the quantum case, the quantum event is the message re-ceived, and depends on the type of measurement performed (i.e.what questions were asked). The answers, i.e. the particularmessage received, can be predicted with a certain probability,depending on the observer’s prior knowledge about the sys-tem. Recall that, in the Hom model, a measurement consistsof preparation, interaction and observation:

System → Observer : Hom(cause, effect).

If the “cause” itself is an interaction, it leads to the ABE-model:The “Level 1” of structure consists of the SO-interaction/

communication, as above. The “Level 2” represents a classical“eavesdropping” on the quantum process from Level 1.


We will revisit Shannon’s theory of classical informationand try to refine it in terms of information flow in graphs.

3.4. Shannon’s entropy: a framework. A simple frame-work to model an interaction/communication (system/informationsource) is a µM -correspondence (e.g. Boltzmann, etc.). It con-sists of a state space Ω (states/symbols), together with a par-tition P = Ωii∈I , such that the indices are in a (1:1?) corre-spondence with the values of an M-observable, say the energyE:

IΩ

E

##FFFFF

FFFF

P // Range.

There are various levels of mathematical sophistication imple-menting the idea: foliations of the state space (e.g. Hamiltonianactions etc.), vector bundles, groupoids and moduli spaces etc.We will focus on the underlying idea and work on the above toymodel: the relation (quotient space) determined by a (measur-able) function defined on a finite set (measure space relative tothe counting measure):

Ωpr→ “Spec(E)′′.

The statistical approach modeling “our independence” regard-ing the system (we do not control it!), starts from assuming aprobability distribution π = pii∈I representing our best esti-mate regarding the measurement outcomes (symbols received).

The goal is to measure the additional information needed tocharacterize the events, i.e. to specify the “address” of elementsof Ω (memory needed to store the address).

Shannon was looking for an invariant depending on theprobability distribution π:

H(p1, ..., pn), n = |I |,which in turn may be assumed to be correlated to the partitionP (e.g. normal distributions etc.).


3.4.1. It’s all about ... colorful decisions! Our goal is tointroduce a finer invariant on decision trees which will produceShannon’s entropy under the equivalence relation determinedby the residue, i.e. the external structure of the colored tree:

H(Res(tree)) = Q(tree).

A colored tree is labeled by subsets in the state space, andthe edges cary probabilities determined by the values on theirvertices:

e = Ωi → Ωf , p(e) = |Ωf |/|Ωi|, S(e) = − ln p(e).

The invariant Q is finer since here the “Feynman rule” involvesa trivial propagator: ω = 1 (see §6.0.5).

Extend the probability function defined initially on edges,functorially, on paths:

p(v1 → v2 → ...→ vk) = Πp(vi → vi+1).

Similarly, S has an additive extension,The residue of a colored path in a colored tree is

Res(γ : Ωi → Ωf) = (Ωi,Ωf),

i.e. the colored external structure of the colored tree. Thenp(γ) = p(Res(γ)).

The alternative notation:

∂t = Res(t)

will be used, as a hint to the role of boundary of the externalstructure, when the graph is interpreted as a cobordism 1.

At this point we would like to coin a fancy formula, triviallysatisfied here, as a beacon for later developments of a quantumversion of the theory. If r is the root of a tree labeled with

1This role will be exploited in connection with holographic theoriesCh.10 and black hole entropy Ch.9 §4.


the total state space and t a terminal leaf labeled with the cor-responding “favorable events”, then the associated probabilityis:

p(t) =

∫

γ∈Hom(r,t)

e−S(γ) Dγ.

3.4.2. Shannon’s axioms for classical entropy. Classical en-tropy H(p1, ..., pn), defined as a function of the probability dis-tribution, is assumed to satisfy the following requirements [Sh](see also [BZ], p.3):

1) It is continuous in pi;2) (Normalization) If π is the equiprobable distribution, i.e.

pi = 1/n, then H is an increasing function of n;3) If the partition of the state space is refined at the ith

outcome, i.e. Ωi = A ∪ B is a disjoint union, with associatedprobabilities a+ b = pi, then

H(p1, ..., a, b, ..., pn) = H(p1, ..., pn) + piH(a/pn, b/pn).(2)

Shannon shows [Sh] that only

H(p1, ..., pn) = −∑

pi ln pi

satisfies the above axioms.At this point we elaborate a different approach based on

the idea of “scaling of DOFs”, or the “quantum dot resolu-tion” (QDR), i.e. relating the operation of collapsing/insertionof subgraphs (DOFs) with the information loss/gain. Mathe-matically speaking this bring up front the coalgebra structurepresent in QFT renormalization, as introduced by Kreimer anddeveloped together with Connes [Kreimer, CK1, CK2]. Therelation with the resolution point of view comes from [Ion03,Ion04-1], which ties Kreimer’s coalgebra structure with Kont-sevich graph homology [K92] (see also [FI1]).


3.4.3. An invariant of trees. In the above example, withn = 2 and i = 1 for simplicity, consider the correspondingcolored trees t (left) and t′ (right):

Ω

~~

AAA

AAAA

Ω

AAA

AAAA

A

~~

Ω1 Ω2 Ω1

~~

AAA

AAAA

Ω2

A B

and let γ be the subtree of t′ with root Ω1. Then collapsing γyields t. This is expressed by saying that

γ → t′ → t, t = t′/γ

is an extension of graphs and t is a quotient of t′.Then the above relation can be “restated” (anticipating a

bit) as the Euler-Poincare mapping property:

H(Res(t′)) = H(Res(t)) +H(Res(γ)), (3)

where now H should be replaced with a relative entropy:

H(p1...pn|p),∑

pi = p

such that, if p = 1 then H(p1...pn|p) = H(p1...pn) (see 6).Let’s define a function of colored trees Q(t) and require that

it satisfies two axioms:1) It is an Euler-Poincare mapping, i.e. given an extention

as above:

Q(t′) = Q(γ) +Q(t)

2) Satisfies the normalization condition:

Q(corrola) = p · (−∑

pi ln pi).

Our colored trees are assumed to satisfy a conservation con-dition. The set labeling a parent is the disjoint union of sets


on its children (descendents), i.e. no “entropy production” atthis stage.

As a consequence of our simple “Feynman rule”, the “prob-ability flow” is conserved, i.e. at each branching, the sum ofprobabilities of the output edges equals the probability of theinput edge.

An Euler-Poincare map as above is an abstract version ofa “dimension function”; other examples: Euler characteristicfrom geometry etc. It is perhaps the “thermodynamic weight”foreseen by Birkoff and von Newmann (see §3).

Then the divergence of the information flow yields Shan-non’s relative entropyH via a divergence theorem (Stokes/dualityagain 2).

Hologram Theorem 3.1. 3

H(∂t) = Q(t)

The above properties/axioms come from the correspondencebetween the probability distribution and partition function §4,under the FPI interpretation §6.0.5.

In order to clarify our point of view to the reader (and toourselves), we will take a detour to statistics and thermodynam-ics, in order to link them to the information theory approachto entropy.

4. Probabilities and Partition Functions

A common framework is provided by the µM -correspondence,i.e. the multiplicity of micro states (partition) correspondingto a particular observation ([Entropy], p.255), classical or not.One may bear in mind the urn example (see bellow), for theclassical case, or loc. cit. ([Entropy], p.255).

2H = dQ - see the path algebra derivation interpretation of the entropy§7.2.2

3See Ch.10

4. PROBABILITIES AND PARTITION FUNCTIONS 75

Imagine a system modeled as consisting ofN non-interactingsubsystems (“particles”), exhibiting n possible levels of energy(the M-observable), for a total of nN possible M-states (macrostates). In the CS dual language, the M-states of the particlesare “symbols” distinguished by the observer at this “resolu-tion”).

4.1. Micro and macro states. The following combina-torial discussion has a dual role: studying one system composedof N non-interacting subsystems with state space Ω, which wewill refer to as the “space-like” point of view (prefered in thissection), or studying N copies of one system, i.e. an ensemble,referred as the “time-like” point of view (see 5.3.1).

Let Ω denote the possible micro states (µ-states) of one par-ticle, usually associated both with external and internal DOFs.

The M-observable E : Ω → R determines the possible inter-nal micro states E = Eii∈I of one particle. The M-observableE foliates the µ-state space of a particle, determining a par-tition P = Ωii∈I = Ω/E. We will call Ω/E the internalmicro-states (indexed by I) or, in view of our usually “big ap-paratuses”, the macro-state space of the particle (M -states).

Remark 4.1. Indeed, we tend to “project” our apparatuses’soutcomes (our “artificial eyes and ears” etc.) as internal DOFs(e.g. spin etc.), while the “remaining relations”, the so called“motion”, be attributed to “space-time” as modeling the exter-nal DOFs.

The members of the partition denote the corresponding in-ternal macro-states (M -states), and the state space is an “ex-tension” (bundle/fibration etc.) of internal macro-states byexternal micro states (addresses: partly known and partly un-known in a measurement):

Ωi → ΩE→ E ∼= I.

We will say that the particle is of type (Ω, E) (or just E).


Now energy is a special M -observable: it “discriminates”states according to their probability of occurrence, leading tothe so called Boltzmann’s distributions in correspondence topartition functions (see 4).

Definition 4.1. An M-observable is a Hamiltonian for aµM -correspondence iff the associated partition is compatiblewith the equiprobable distribution partition.

In other words, the observable factors through the probabil-ity density. Then one can reduce the analysis to the analysis ofthe moduli space Ω/E together with the associated probabilitydensity π.

A system of N-particles has as state space Hom([N ],Ω, theµ-state space of the system.

States fall into “categories”, represented by the type of thepartition under the M -observation E:

Hom([N ],R)

Hom([N ],Ω)

E∗

66mmmmmmmmmmmm∃!

col// Hom([N ], I).

E∗

OO

N/col// N − partitions.

A µ-state µ is mapped to col(µ) representing the outcome ofthe E measurement of the n-th particle (“color”). An M -stateis the associated partition of N :

M = N/col(µ) = mii∈I ,say (m1, ..., mn) in the example from [Entropy], §12.4, p.255,where I = [n]. Of course, the “completeness condition” holds∑

mi = N and:

|Hom([N ],Ω/E)| = nN , |Ω/E| = n.

The number of µ-states µ of an M -state M is:

WN (M) = |(N/col col)−1(M)|.


In the above example (I = [n]):

WN (m) = N !/m!, m! = m1!...mn!.

We will say that E foliates the state space into cells (ad-dresses) ΩN/E, which correspond to macroscopical states (M-states). To the M-observable E we may associate the decisiontree (corolla) (Ω, (Ω1, ...,Ωn)), where the leaves correspond tothe possible measurements E = Eii∈I.

The expectation value of E is:

< E >=∑

niEi =

∫

[N ]

M∗(µ).

The large numbers assumption relates the combinatorial de-scription with the “limiting description” in terms of probabili-ties distributions (see 5.3.1):

M − states : π = limN→∞

([N ]/M∗(µ))/N.

relating the combinatorial probabilities ni/N , a projective spacepoint associated to the combinatorial M-state (see 4.4.1), andthe probability distribution π = (p1, ..., pn), as a continuouslimit M-state, which is also a projective space point, due to thenormalization

∑

pi = 1.

4.2. The Law of Large Numbers. The correspondencebetween the combinatorial description of the micro-macro states(Boltzmann) correspondence and the stochastic description interms of a probability distribution is essentially the content ofthe Law of Large Numbers.

The probability of a particle to be in the i-th cell, i.e. itsµ-state to belong to Ωi is pci = mi/N .

Then the (combinatorial) probability distribution associ-ated to the M -state M of an N -ensemble of particles of (Ω, E)is πN = (pc1, ..., p

cn) (I = [n] for readability).

Now if considering systems with more and more particles ofthe given type in a sequence of macro states M(N) such that the


associated probabilities (projective points) converge (assumingthe large numbers probabilities exist):

π = limNπN ,

then, in the above example, with the Stirling approximationused:

limNWN (NπN)1N = eH(limpiN ),

where H(π) denotes the Shannon entropy.So, with e = limn(1 + 1/n)n in mind

limNW (N )1N = eH lim

N

will be interpreted in terms of the classical information chargefunction on trees §6.1.

4.3. Shannon entropy and typical configurations. Nowassume that the macroscopical knowledge about this “modulispace” Ω/E is given via the associated probability distributionπ = pi, where pi = mi/N , defining the so called typical cate-gory (“generic category”).

The probability of such a category is:

p(m) = WN (m)/nN .

Then Shannon entropy is the large numbers limit (via Stirling’sformula):

limN lnWN (m)/N = −∑

i

pi ln pi.

It represents the amount of information per particle necessaryto identify the “address” (category) of a state of the system.

4.4. Partition function. The “continuous limit” (prob-ability distribution) may be encoded in the partition functionof the system.

Let < E >=∑

piEi be the expectation value of E, thoughtof as a constraint.


Define the partition function associated to E ([F1]):

Z(β) =∑

e−βEi .

Then there is β such that

pi = e−βEi/Z(β).

They maximize Shannon’s entropy H(π).Indeed a Lagrange multipliers method applied to the opti-

mization problem with two constraints:

max H(π) = −∑

pi ln pi,∑

piEi = E,∑

pi = 1,

yields:∂piH = βpi + γ => pi = e−γ+βE ,

and therefore defining:

Z =∑

e−βEi => γ = lnZ, pi = e−βEi/Z,

with β determined by the energy constraint.Then

H = βE + γ, γ = lnZ,

or alternatively, after defining the Helmholtz free energy by Z =e−βF , obtaining the Gibbs entropy ([Entropy], p.256; H(π) =S(E)):

H = −βF + βE.

The equivalent relation TH + F = E will lead to one of theFundamental Principles of DWT (Ch.3).

Now, why “free energy” and “reduced energy”? because weare in a projective space situation relating a “free theory” witha reduced theory, as we will explain next.

4.4.1. Probabilities and real projective space. Probability dis-tributions are points of a projective space due to the normal-ization (constraint) condition

∑

pi = 1. At this point a com-parison with quantum mechanics is instructive; remember theMarkov-2-Feynman “upgrade”: complexify and enrich the the-ory, no “free-2-reduced theory reduction”, since both are theo-ries defined in projective space!


CM/Statsitics QM

States π = (p1, ..., pn), ψ = (c1, ..., cn),

π =∑

piei ψ =∑

ciψi

Normalization ||π||1 = 1 ||ψ||2 = 1

The Bohr correspondence rule:

pi = |ci|2

is the link between the classical (“real” as in R) and quantumpictures (“complex”, although in fact real!).

In QM the state is a vector ψ which can be represented asa linear combination of pure states (measurement basis) withcoefficients amplitudes of probability, i.e. according to Bohr’sstatistical interpretation, complex numbers whose modulus arethe probabilities of measuring the corresponding outcome (rel-ative to an M-observable). The quantum states are in fact raysin the projective space of the corresponding Hilbert space, rep-resented (ambiguously modulo a phase) by a vector of L2-norm1: ||ψ||2 = 1.

So the probability distribution π = (p1, ..., pn) is the clas-sical analog of a state, except it is normalized relative to theL1-norm:

||π||1 =∑

|pi| = 1.

If one considers arbitrary basis elements of the correspondingray in Rn+, then its norm is the partition function:

Z = (Z1, ...,Zn),Z = ||Z||1, pi = Zi/Z;


of course, still under the assumption that probabilities (or Zis)are determined by the energy levels (see 4.1):

Zi = e−βEi .

In some sense (leading to the 3rd FP-DWT Ch.3), the internaldynamics, i.e. the evolution of the state π, is governed by theMaximum Entropy Principle (max H(π)) 4.

The free energy is the energy counterpart in the “free the-ory” in the Rn+ space. (?)

Remark 4.2. In view of the above discussion §4.4.1, thedifference between classical (statistics) and quantum mechanicsis that the former is an L1-norm real projective space theory(“analysis-like”), while the latter is an L2-complex projectivespace theory (“geometry”). But the “hart” of QM/QFT is theannihilation-creation not only of particles and anti-particles,but also of “possibilities”/histories (superposition with inter-ference).

4.5. Taking absolute values and limits: old habits!Why is statistics a (internal) dynamics in R+? As if a Z2-graded version (Y/N with probabilities, i.e. “double the bit”...) might be the missing link (intermediary stage) between theclassical picture (logic) and quantum picture (logic) ... Is thisrelated to considering a projective representations (via centralextensions) and linking the “projective space” picture with thefree theory?

Also Shannon entropy is a classical limit of the combina-torial description (counting) of configurations (permutations/symmetries, Young tableaux, braiding, spin and statistics etc.).Stirling’s approximation links the two:

lnn! ≈ n lnn− n + 1,

providing an asymptotic conceptual correspondence, as in quan-tization prescriptions (e.g. Dirac quantization etc.). What are

4What about the “external dynamics”? (FPI & localization ...)


the “true concepts”, after the “old habit” of taking limits isabandoned?

5. The thermodynamics legacy: a hidden message?

To better understand the role of entropy in physics, besidesconnecting with information theory, it is mandatory to reviewwhat thermodynamics has to say about it.

The following “recall” has a pronounced subjective flavor,in view of the Quantum Dot Resolution philosophy of DWT.

An informal thermodynamics primer may be found at [ThD](who has time for books anymore ...). “If we restrict ourselvesto crystals ...” (p.2), lattices or our discrete QDR, the mainkeywords suggest a certain hierarchy:

Entropy

ttiiiiiiiiiiiiiiiii

**TTTTTTTTTTTTTTTT

Internal (Reduced)

**UUUUUUUUUUUUUUUUExternal (Free)

ttjjjjjjjjjjjjjjj

Energy & Enthalpy

This cannot be a coincidence; it must be the duality at work(internal / external) together with the dual description projec-tive / affine space.

5.1. Internal Energy U . We will rephrase [ThD] p.2.5.1.1. What internal energy is not. Neglecting external en-

ergy (e.g. gravitational potential) due to interactions, i.e. dueto the “visible structure” at a given “resolution”, and discard-ing internal energy which “never changes”, i.e. it is constantduring our modeling process (no evolution, e.g. “mass”?), whatis left are “vibrations ...”, i.e. the dynamics of some internalDOFs.

Again thinking of “energy” as an M-observable “foliating”the µ-state space into equally probable events, we understand

5. THE THERMODYNAMICS LEGACY: A HIDDEN MESSAGE? 83

the temperature T as referring to the external contribution toDOFs, i.e. the number of vertices of the graphs (via Boltz-mann’s constant), while the “total internal energy” being dueto the pairing with the “colors/labels” associated with vertices,i.e. providing the types of subsystems via their symmetries: in-ternal DOFs:

U = 1/2 T ×EDOF × IDOF.

A thermodynamic macro-state is usually given by “the numberof atoms N , the pressure p and the temperature T” (loc. cit.p.3).

We will speculate on “pressure” p later on (5.2.1). Now,what is the dynamics?

5.1.2. Is it heat “virtual” work? A change may be due toheat transfer Q or “mechanical” workW , or changing the num-ber of subsystems N ! Assume for the moment N is constant,to focus on heat transfer. The First Law of Thermodynamicsstates:

dU = dQ− dW.

As a working hypothesis one may try to “explain” heat transfervia “work”, more precisely as an interaction process:

Subsystem (a)dQ //

dU

''OOOOOOOOOOOSubsystem (b)

−dW

dU77ooooooooooo

i.e. “reduce” the heat transfer to a “virtual work”:1) Convert internal energy dU into work “at (1)”;2) Transfer work −dW “from (a) to (b)” via interaction;3) Convert work −dW into internal energy “at (b)”.

5.1.3. Vacuum causal bubbles. A possible quantitative im-plementation is suggested next. Consider the two subsystemsS1 and S2 “point-like”, i.e. no structure is externally visibleat this stage, so the considered energy is internal. Insert asubgraph γ1 for S1, i.e. “resolve the singularity” at (a) as a


“virtual” generation of structure (vacuum causal bubble). Instep 2 above, redistribute the structure, by changing the clus-tering of DOFs, say into γ2 containing the point representingS2, so that the remaining (only) point will represent S1.

Then collapse γ1 to obtain the “same” configuration, at thelevel of number of points at least.

What is the actual “math code” for the above, is yet to beseen, but the simultaneous presence of collapsing and insertinggraphs with an identity result at the level of external structureis reminiscent of a chain homotopy:

ds+ sd = id,

where d is a graph homology differential (collapsing an edge)while s corresponds to insertion of an edge ...

5.2. Enthalpy and free energy. We will try to addresssome possible lines of thought, keeping in mind the quantumjumps picture Ch.8 §1.2 and the radiation laws of black holeswhich tend to “redefine” the surface area and volume (§4).

Remark 5.1. If graphs model Space-Time, what is the “di-mension” of Space-Time? The full structure is a 3-category(think 3-simplices) with a Poincare duality with “reduces” thepicture to graphs plus order information and duality. At thegeometry level, Hodge duality is involved (see also Ch.9 §3.

5.2.1. Pressure and volume. The pressure and volume asmacroscopical classical observables must be reconsidered in thecontext of a discrete causal structure. The classical mechanicalpicture is clear and quantization will carry it to the quantumrealm, but still in the framework of a continuum space-time;what happens if a granular space-time (QDR) is used instead?

“Any mechanical work must change the volume”, since “some-thing” must move! Is it due to a change in position or mo-mentum, since in the discrete picture v is not dx/dt (i.e. use


Poincare-Cartan form rather then a Lagrangian). Either:

work

''OOOOOOOOOOOO// ∆ position

∆ momentum

dynamics

66mmmmmmmmmmmmm

or work as a change corresponding to some action, is momen-tum times the position change.

If pressure p, as a measure of interaction per unit of surface,is constant:

dW = pdV, p = constant.

5.2.2. Enthalpy and free energy. The state function mea-suring the energy needed to “form a substance” is called en-thalpy [ThD]:

H = U + pV.

It measures the “heat of formation”, whether it is a chemicalreaction or a particle interaction.

2H2

AAA

AAAA

A e+

???

????

?

• // 2H2O • γ //

O2

>>e−

??

To find study equilibrium states, a minimum principle will in-volve both the enthalpy and entropy, since the change in en-tropy occurs parallel to the change in enthalpy.

Remark 5.2. This “balance” between energy and entropy,competition for energy between internal energy and external in-formation needs. It will later lead to an new equivalence prin-ciple between information and energy §6.2, allowing to accountfor a conversion between internal and external DOFs.


To study the equilibrium states of purely mechanical sys-tems, i.e. composed of non-interacting subsystems, the appro-priate state function to minimize is enthalpy.

To study the equilibrium of thermodynamic systems, i.e.many interacting subsystems, one should minimize the free en-thalpy, also called Gibbs energy:

G = H − TS (G = U + pV − TS).

Remark 5.3. The last formula should be compared with theaction term representing the change in the momentum potentialin the Hamilton-Jacobi formulation [Gosson], p.12:

Φ(r, t) = Φ(r′, t′) +

∫

γpdq −Hdt.

It suggests that the above thermodynamical potential is an in-ternal version of the external momentum potential, in this al-ternative (field-like) formulation of Hamiltonian dynamics.

The pV term (better d(pV ) = pdV + dpV ) is related to theexternal description of the system (geometry and dynamics ofEDOFs), while TS term (better TdS) accounts for the changesin the internal description.

Remark 5.4. The similarity with pdq −Hdt reflects upon“time” as a global and macroscopical description (symbols =labels) of “internal changes”, which are not “motion” (externalchanges).

The entropy S of reversible thermodynamic processes is re-lated to heat transfer Q and temperature T :

dS = dQ/T.

Then Gibbs energy is essentially:

G = U +W −Q,

i.e. an account for various energy contributions: internal andexternal (In/Out).


For constant volume and variable pressure, we learn thatthe appropriate state function to minimize is the free energy,called also Helmholtz energy:

F = U − TS, (F = G− pV ).

5.2.3. They’re just state functions after all. These, U,H,G, F ,are all state functions, also called thermodynamic potentials andonly one is needed for a complete description from the pointof view of thermodynamics. This implies that the probabilitydistribution, viewed as a complete state of the system, is deter-mined by the corresponding foliation by any of these “energyfunctions” (see §4.1).

Remark 5.5. When “relaxing” the constraint p ∼ dq/dt weprobably should add an additional state function to account forthe duality between internal and external DOFs. A temptingway to do so is to pair them as a complex number, probablyrelated to the “upgrade” from probabilities (identified by an en-ergy M-observable) to amplitudes of probabilities (identified bytwo state functions).

5.2.4. The energy conservation: a homotopy? Since pV term(or pdV, dp V etc.) represents work, the Gibbs energy leads toa balance equation:

E −W = U − TS.

Now the macroscopic equilibrium point of view, i.e. “ther-mostatics”, needs only one state function for a complete de-scription. In other words, a complete family of independent M-observables consists of only one state function, i.e. a function ofthe type “energy” which separates the fibers of the microstatespace relative to probability distributions (classical stochasticstates).

The duality between internal and external DOFs will berepresented by a balance between energy and entropy. Thiswill involve two “a priori” independent state functions, U re-sponsible for controlling IDOFs and TS controlling EDOFs;


the work term W = pV should be thought of as an externalsource, while Q = TS represents an internal source:

E − U = d(W −Q) = pdV − TdS.

5.2.5. To minimize or to maximize, this is the question. Tofind the equilibrium M-states, i.e. to solve the “ThermostaticsProblem” (thermo - since one state function suffices, and stat-ics because, well, no dynamics is involved), minimize the freeenergy F = U − TS or maximize entropy S ...

To calculate the canonical probability distribution π (M-state: the stochastic state or classical information source) cor-responding to a state function E (some energy M-observable),one can use either (1) the combinatorial picture counting themicro-states per M-state 4.1, or (2) give the partition functionwhich will determine the M-state via the Maximum EntropyPrinciple (see §4). The large numbers assumption states thatthe second is the continuum limit of the first (see 4.1).

Although any state function describes the equilibrium of athermodynamic system (the Thermostatics Problem), for crys-tals, the most convenient is the free enthalpy [ThD], p.6.

Clearly there are too many “If ... Then ...”, i.e. a case-by-case analysis when dealing with complex systems, which is... expected in a way from such “expert systems” :-) Still thereshould be a unifying point of view (framework/tool kit etc.)with a more “friendly user interface” ...

For example, if energy is constant (constraint) maximizeentropy (partition function correspondence). If energy varies“slowly” it means we are studying the “dynamics” (or kine-matic?) of equilibrium states. Outside of the equilibrium is“hic sunt tiggers” land ... (see [Entropy]). Taking into ac-count I/EDOFs in duality brings some hope though ...

5.3. Entropy - statistical considerations. Broadly speak-ing the attribute (prefix) “Macro” entails forgetting labels andaveraging quantities (“statistics”) while “micro” refers to indi-vidual quantities, carefully labeled (“addresses” etc.)


5.3.1. Ensembles. Given a (type) of system, an ensemble isa collection of copies of the system. If the system is an urn,say with N balls (internal structure - subsystems with EDOfsignored but with IDOFs modeling colors), then a collection ofν such urns or rather a repeated investigation of one such sys-tem, idealized, is an ensemble. Even a dice, with no explicitinternal structure (N = 1 subsystems, yet with a non-trivialinternal space with a measurement basis with 6 elements), canbe “cloned” in a gedanken experiment, or rather through sev-eral times and studied under the scientific assumption that thebehavior is autonomous (astrology has a different point of view,of course).

The alternative “copies” or “sequences of “time chains”ω1, ω2... (Markov chains etc.) invites the computer science in-terpretation of states as symbols and ν-ensembles as messagesof length ν. This duality may be sketched as a diagram:

Combinatorics of One System Source of Information

ν − Ensemble (Message of length N)

T ime chains .

Then the statistic entropy in the combinatorial picture corre-sponds to the (classical/quantum) channel’s capacity.

There are two possible large numbers limits. One is the emcontinuum structure limit limN→∞, with the purpose of “sim-plifying” the combinatorial “chaos” a human mind can’t handle(avoids?), starting with Newton’s endeavors in mechanics andbrought to an end by the quantum theory. It is useful (manda-tory?) when attempting to model “space” (EDOFs etc.) bycontinuum structures (manifolds etc.).

The 2nd limit is the large numbers limit limν→∞ which isagain useful in approximating the original discrete structureof time, consisting of communications exchanging quanta ofinformation, in order to “simplify” the description towards a


continuum model for “time” (rescaling a long ν sequence to[0, 1] is a sort of a “zoom out” to smooth out irregularities andthe fine grain structure).

The relations between the various “boundary theories”, in-cluding the “classical corner” limN limν , will not be discussedhere.

¿From the three types of ensembles, micro canonical (iso-lated system: fixed energy and number of particles), canonicalensemble (in contact with a heat bath at fixed temperaturewith a fixed number of particles) and grand canonical ensemble(allowing for exchange of heat and number of particles), onlythe last one is general enough to be considered in a commonframework for energy and information.

As a technical point, considering ν “copies” of a system withstate space Ω, or rather states ωi ∈ Ω, i ∈ [ν] again correspondsto Hom([ν],Ω), which may be interpreted as a “trajectory”in state space. If the system has N subsystems (“space-likedimension”) then the “space-time trajectory” is an element ofHom([ν] × [N ] → Ω). If the time-order is irrelevant, then asequence of states such that νω systems of the ensemble are inthe state ω is a (homological) chain in C0(Ω) = ∑ω∈Ω νωwith finite support of “volume” ν.

5.3.2. Statistical entropy. The statistical definition of en-tropy is via Boltzmann’s equation

S = klnW

where W represents the number of µ-states per fixed M -state.It can be derived either using the continuum limit, as aboveN → ∞ or the large numbers limit as in [Stat1]. The entropyof an ensemble

∑

ω νω is

Ων = ν!/∏

νi!, ln Ων ≈∑

−ν∑

pi lnpi = νH(π),

where pi = νi/ν, and the Shannon’s entropy (per symbol) isobtained via Stirling’s approximation.


Conceptually we are in a F = ma situation, where S be-longs to the idealized framework (information source) while Wis a precise theoretical concept (counting states) correspondingto a particular / actual type of system.

Notably S itself is not a state function, but TS is, termwhich enters the energy balance.

5.4. What do you say, Mr. Feynman? ¿From Helmholtz

free energy F we may calculate everything else ([F1], [Stat1]4.3), via the relation with the partition function:

Z = e−F/(kBT ).

5.4.1. Energy: dark or white? The corresponding energybalance E = F +TS reflects the duality between the “externaldynamics”, associated with info/order etc., which we will call“white energy” in contrast with the terms affecting the “inter-nal dynamics” of “not visible” (internal/averaged etc.) DOFswe speculate may provide the “correction terms” in a quantumversion of General Relativity, as an alternative to the “missingmatter” from the energy-momentum side.

5.4.2. The partition function - revisited. So what is the par-tition function just a “normalization constant” or a function?Neither; in the context of QFT it is a generating function, buthere we will turn off interaction for now:

Z(E ; β) =∑

j∈Ω

e−βEj .

It depends on the chosen state function, say energy E, andtemperature kBT = 1/β (patience please ...).

As explained above 4.4.1, the probability equipartition as-sumption gives the correspondence between energy and Boltz-mann’s distribution via constraint entropy maximization:

(Eii∈I, Z)max H(π)

∑

piEi=<E>∑

pi=1

// (π = pii∈I , β)


The converse is direct; given π and β, Ei = −1/β lnpi.Therefore, if viewing the energy as fibrating the state space

Ω (the “extension/bundle” picture), then the partition functionis a way to redistribute the “energy gaps”:

ΩE→ Spec(E)

e−β(·)

→ R,

where Z = (Z1, ..., Zm) (say |I | = m) is interpreted as a coor-dinate function on the M -state space Ω/E ∼= Spec(E), and thenormalization constant is ||Z||1 (see 4.4.1). So temperature, orpreferably β, is essentially an infinitesimal generator ...

Now the correspondence between the projective space de-scription in terms of probabilities π and the “free theory” interms of E is:

(Z : Spec(E) → R) 7→ (π, β).

Then the relevant state function is not entropy, but rather TSin some convenient coordinates:

Q(Z) = Q(Spec(E), β) = H(π)/β.

Alternative coordinates Xi = − lnpi = β(Ei − F ) or Yi =Xi/β = Ei−F (departure from the free energy) should also beconsidered:

Q(Y, β) =∑

i

Yie−βYi .

If Boltzmann’s probability distribution is obtained maximizingentropy as a function defined on the projective space of theM -state space of the system or ensemble (type of info source):

H : P 1Spec(E)R → R

under constant energy as a constraint, then in general, whatshould be maximized in the free energy picture? Free energy isthe only qualified candidate!


5.4.3. What is energy after all? Clearly the probability de-scription lives in the projective space; but what about en-ergy? It is always positive, and failures of this requirementare dreaded and quickly fixed (on shell at least).

But being positive is a symptom of projectivity (the quan-tity is “infinitesimal” already, can’t take a ln any more).

So, we should reexamine relativity in an attempt to traceback this feature ...

5.4.4. The other observables: piece of cake. The other ther-modynamic observables can be derived from the partition func-tion [Stat1]:

< E >= − 1

Z

∂Z

β, cV =

∂ < E >

T, etc.

Furthermore [F1], p.7:

H = −∂F∂T

, etc.

or [Entropy] p.256:

H(π) = lnZ(β) + βE, dH = βdE.

We should remark that entropy has contributions from inter-actions Sf (formation entropy) as well as from configurationsSc = H (entropy of mixing) §5, yet here (interactions off) onlyHc is “visible”. Formation entropy corresponds to “coloredconfigurations” (internal DOFs) while entropy of mixing corre-sponds to labeled configurations (“position-momentum” etc.)

The relation with Poincare-Cartan form is intriguing:

PCF ∼ pV − TH ∼ −Sf − Sc?

towards a similarity (speculation: nothing new, right?) be-tween the role of Planck’s constant (energy quanta) and Boltz-mann’s constant (information quanta!?).


6. Feynman path integrals and entropy: graphsinvariants!

Although we are in the realm of “tertium non datur” (prob-abilities don’t destructively interact, trees etc.), the sum overpossibilities idea, which is the mile-stone of Feynman’s ap-proach naturally applies to ... decision trees.

One may think of a of a sequence of answers to a sequenceof questions represented by a decision tree as outcomes of mea-surements, or computations, in an algorithm: going from “Start”to “Stop” via various branches. For example, if the root repre-sents the question (measurement of) A, then depending on thesubsequent questions, B independent on the A result, or B orC depending on the A-outcome, we will have different labeledgraphs (or just trees):

A >

>>A

>>>

A(BB) : B B A(BC) : B C

The probability distribution - partition function correspondencemay be extended to such graphs, yielding none other then theFeynman Path Integral!

6.0.5. FPI and probabilities. If a probability distribution π(stochasticM -state), which may be viewed as a colored corolla,corresponds to the partition function describing the statisticsof the system §4, then a colored decision graph corresponds tothe amplitude via Feynman rule under the FPI formalism:

6. FEYNMAN PATH INTEGRALS AND ENTROPY: GRAPHS INVARIANTS!95

BoltzmannCorrespondence

ProbabilityDistribution

PartitionFunction

π Z

FPI

Feynman diagram Feynman rule Green function

Γ W (Γ).

In a sense Feynman Path Integral formalism generalizesBoltzmann’s correspondence between probability distributionsand partition functions.

The colored corolla is the transition function of the (mini-mal) Start-Stop Automaton (STA):

•

~~

AAA

AAAA

A

• ... •(4)

If the Input is the probability p then the Output is one of theprobabilities p · pi.

Recall that under the Boltzmann correspondence pi = Zi/Zwhere the partition function Z = (Z1, ..., Zn), Zi = e−βEi cor-responds to an M -state (E , β) (or rather M -observable E).


Proposition 6.1. The expectation value of the energy de-termines the entropic state function:

H/β =< E > .

Proof.

< E >=∑

i

piEi =∑

pi(− lnpi)/β = H(π)/β.

Theorem 6.1. The Boltzmann correspondence extends fromcorollas to Feynman Path Integral (sum) on graphs, relating theentropy H and expectation value of energy < E >:

HT =< E > .

Indeed, the expectation value of the “algorithm” (the decisiongraph), is:

< M >=∑

γ:r→i

p(γ)Mi =∑

p(γ) lne−S(γ) = −∑

p(γ) lnp(γ)/S?

Mi is the “info contribution” (action) flowing through the pathγ. For an elementary edge (channel) of probability p, the flowis:

W (• p→ •) = − ln p/β, p = e−βW ,

where β corresponds to the information capacity of the chanel(propagator in the QFT framework).

If sources are present, then a balance equation holds be-tween the Input Probability pIn (information) and Output Prob-ability (information):

pOut = pIn +W (• p→ •),in complete analogy with the “momentum flow” determined bythe generating function (action) (see [Gosson]):

Q(r′, t′) = Q(r, t) +W (r, t; r′, t′).

So the role of β, the inverse of the thermodynamic tempera-ture, “localizes” in the general framework (QFT and quantum


information) to be related with the channel information capac-ity.

The difference between the classical and quantum theoryat this point is at a flip of a switch away: turn i on (andloops/feedback; more in Ch.6).

But before that, we will elaborate on the content of thetheorem on the tree from the example of Section 3.4.3.

6.1. Entropy and Information Charge/Potential. Thecorrespondence between entropy and Feynman Path Integral isa natural generalization of the Boltzmann correspondence be-tween a probability distribution and the associated partitionfunction, as explained above 6.0.5. But the role of “temper-ature” was understated, so we will review the “enhanced pic-ture”.

The probability distribution is a state belonging to the pro-jective space. In order to compare it with the “energy pic-ture” (partition function) one needs an additional parameter:


β, playing the role of the inverse of temperature in thermody-namic.

Projective Space

BoltzmannCorrespondence

// Path Space Formalism

(β, π) Z(β) : Spec(E) → R

β, 1

pi

1

Ei

pi Zi = eβEi

Decission graphs

Res

OO

QDR

FPI

OO

(β,H(π))H(E)·T (E)=<E>

// < E >=∑

iEie−βEi .

A normalization of the above correspondence means assuming0 free energy F = 0, i.e. normalizing the partition functionZ = ||Z||1 = 1 (one can always take advantage of the gaugetransformations Ei → Ei + C in the RHS). Then pi = Zi andEi = −(lnpi)/β yields the inverse of the correspondence asso-ciating to a partition function, under fixed energy expectationvalue, the probabilities maximizing Shannon entropy,

π · E =< E >= H(π) · T, max limπH(π),

∑

i

pi = 1,∑

i

piEi =< E > .


6.1.1. The “right invariant” goes by the right rule! We claimthat H(π) is not the right invariant for the combinatorial pic-ture, but HT , the state function from thermodynamics, is.But this will change the rule labeling decision trees, to a gen-uine “relative” version: do not associate probabilities to edges,rather associate number of states νi to vertices! For example,if ν = |Ω| is the number (measure) of the µ-state space andνi = |Ωi| is the number of micro-states corresponding to themacro-state Ωi, then LHS is the probability edge of the corollarepresenting the µ−M -correspondence, with the probabilitiespi = ν/νi labeling (coloring) the edges, while the RHS is theconfiguration edge (address branching instruction) of the samecorolla, where the number of configurations ν(i) (or some otherEP-map: number of qubits, quantum dimension of a modularcategory / representation of a quantum group etc.) labels (isapplied to) the vertices:

• pi→ •, ν• → νi• , pi = νi · ν−1.

The “other data” (vertex probabilities or edge probabilities)can be readily computed in either formalism.

The probability formalism is a relative one (moduli space/ projective space), therefore not appropriate to a “Noetherinformation current” picture, as the particle formalism (“par-ticle=state”; forget IDOFs for now).

To relate TH and H , we will stare a few moments to theEquation 3, in the special case of a previous example 3.4.3, inorder to relate them:

TH(

ν

~~

AAA

AAAA

A

ν1 ν2

) = ν H(

1ν1/ν

ν2/ν

???

????

• •)

= −ν1 ln ν1 − ν2 ln ν2 + ν ln ν.

since conservation of number of states yields ν = ν1+ν2. So THis additive (EP-map or a “functor” in the categorical picture:


action or energy) due to Stokes Theorem in disguise:

TH(ν(ν1, ν2)) = H(ν1, ν2) −H(ν).

In other words:

H =∂(TH)

ν,

which, as we shall see, makes me wonder whether ν plays therole of β!

Now the ν’s are the appropriate candidates for an Euler-Poincare mapping due to the conservation of number of states(and information current etc).

Indeed, equation 3 now reads:

TH(Γ) = TH(Γ/γ) + TH(γ).

But how to derive in an axiomatic manner H(ν) = −ν ln ν? Infact this is the “wrong formula”! With Stirling’s approximationin mind, the illumination comes: the right (hidden) quantity isν lnν − ν, since due to the conservation equation:

−∑

i

(νi ln νi − νi) + (ν ln ν − ν) = −∑

i

νi ln νi − ν lnν !

Definition 6.1. The information potential of a labeled ver-tex is:

Q(W ) =

∫ W

0lnxdx = [x lnx− x]W0 = W lnW −W.

The Stirling approximation was “stretched” a bit, to avoidthe unpleasant constant term of 1 (although we abhor limits,we used it; hmm ...?).

Now extend Q additively (on the “tensor algebra”):

Q(ν1, ..., νn) =∑

Q(νi).

¿From the definition it follows that:

Q(ν) =

∫ n

1

∫ ν(i)

1

ln(x)dxdi


(just one more formula to think about). Then TH is the EP-map sensitive to the residue of the tree (graph):

TH = dQ : TH(ν → ν′) = Q(ν)−Q(ν′).

Speculating on the relation with temperature and heat transferis tempting, but it will be postponed (not too much!); (n)or theinterpretation of Q as a generating function of a flow ...

6.1.2. Invariants: discrete versus continuum. So, not theentropy H is the key, but rather the “informational energy”TH with its defining “potential” Q(ν), which measures theinformation charge stored by “adding the ν DOFs”; since also

dQ

dν= lnν

TH may be thought of as a measure of the information flow(“gradient” / differential, what’s the difference when there is ametric :-)) and the well known thermodynamic relation betweenentropy and heat is obtained:

H =dQ

T.

The continuous invariant Q, corresponding to the large num-bers limit, is just the infinitesimal measure of the internal sym-metry of the Quantum Dot measured by the discrete invariant:

C(ν) =∑

ln νi! = ln |Aut(P)|,where an internal symmetry of the partition of Ω is a permu-tation of the elements of Ω preserving the partition.

They are related via Stirling’s approximation:

C(ν) ≈ Q(ν) (

N∑

0

≈∫ N

0). (5)

6.2. Concluding speculations. This is not entirely sur-prising (but pleasant indeed!), since the Feynman path Integralformalism was originally invented as a way to tame the combi-natorics of the perturbation approach in QFT while the work


horse of high energy physics, gauge groups and symmetry, withits representation theory, invokes the Young tableaux machin-ery for grinding permutations etc.

So what are temperature and entropy? Inseparable dual(micro-macro) aspects of information theory regarding a sys-tem.

The relation with the levels of energy:

Ei = −T lnpi

suggests to think of Si = − ln pi as a information capacity(flux). Indeed, since pi = νi − ν:

Si = S(ν → νi) = ln ν − ln νi, Ei = TSi

and therefore:

Q(ν) =

∫ ν

0

lnxdx =

∫ ν

0

S(• → 1)d•,

which is yet another relation open for speculations ... Think of

I(ν) = − lnν = S(ν → 1)

as the information charge corresponding to the information po-tential Q. Then

I = βE (6)

is the missing Unifying Principle, relating energy and informa-tion (Ch. 3).

But then, since temperature determines the energy per DOF:

< E >= kBT,

the relation with entropy suggests a “quantization condition”must be present; is it

1 qubit = β < E >?

Is it related to a PCF / Maslov index formalism? (see 3.3).There are many other tempting questions regarding this

“puzzle”; we will record them for later use ... Is the QDR finite?

7. RELATIVE ENTROPY AND INFORMATION CHANNELS 103

Is kB related to a quanta of information? Is β a “couplingconstant”? etc.

7. Relative entropy and information channels

Conditional probabilities are related to relative entropy andsequences of measurements. Classical logic is clearly inadequateto handle the corresponding tree representations of measure-ments (as decision trees), since it is sensitive to the externalstructure only.

Consider for example two binary-outcome measurements AandB, with associated probabilities p(a0), p(a1) and p(b0), p(b1).Then “refining” the outcomes of A with the measurement of B,or B first then A yields two distinct decision trees, denoted tAB(shown bellow) and tBA:

A

zzzzz

zzz

!!DDDD

DDDD

B

xxxx

xxxx

x

B

##GGGG

GGGGG

a0b0 a0b1 a1b0 a1b1.

The measurements are independent if the joint probability isthe product of probabilities; then the conditional probability isdefined by:

p(ai ∧ bj) = p(bj|ai)p(ai).This relation implicitly refers to a path A→ ai → (aibj) of thetAB decision tree.

Now tAB 6= tBA and a finer invariant will distinguish be-tween the two. An example to be considered is the measure-ment of spin/polarization §6.

The “probability” is the classical information flow in de-cision trees (without spatial correlation: S-correlation), whilethe amplitudes of probability represent the quantum informa-tion flow in graphs, where S-correlation is possible in causal


structure, and responsible for phenomena like entanglementand teleportation etc.

Back to decision trees, their external structure is the clas-sical shadow of the quantum non-commutative reality:

Res(tAB ) = Res(tBA).

The Shannon entropy H and the information potential Q areclassical, being unable to distinguish the internal structure. Aquantization is necessary, beyond the extension of H (or TH)from trees to graphs with loops colored by complex amplitudes.The general picture is a quantum computing with objects onvertices and unitary operators on edges, as a categorification ofthe classical picture where one counts DOFs on vertices ν = |Ω|and applies a boolean calculus to the corresponding probabili-ties (roughly speaking).

As a guiding idea, the quantum entropy is essentially VonNewmann’s entropy applied to quantum computing, which inturn is an upgrade of quantum mechanics, with its Hilbertspaces and unitary operators replaced by (braided) monoidalcategories with duality (e.g. type II-factors [S]; [Ion01]; [Co2]etc.), leading to representations of PROPs (QFT, CFT), rep-resentations of cobordism categories (TQFTs, HQFTs) and fi-nally to representations of causal structures (Feynman process).

7.1. Heisenberg commutation relations. We believethat, within the above conceptual framework, the informationflow counterpart of Heisenberg Canonical Commutation Rela-tions is a “normalization version” of:

tBA − tBA 6= 0.

Now quantum measurements can be more intuitively modeled asrepresentations of trees, including the “mother of all quantumexperiments”, the double slit experiment, which should allowthe derivation of the Heisenberg CCR.


7.2. What is relative entropy. We will proceed to an-alyzing equation (5) from [BZ], p.4 (n = 2 will suffice):

H(p(a0), p(a1b0), p(a1b1)) = H(p(a0), p(a1))+p(a1)H(p(b0|a1), p(b1|a1)).

Conservation of states implies:

p(a1) = p(a1b0) + p(a1b1),

while independence of A and B (“commutation assumption”),implies:

p(a1bi) = p(a1)p(bi|a1).

Now this defines p(ai|bj) in the case of the joint probabilitydistributions. But what is p(ai|bj), really?

7.2.1. Transition cobordisms. A measurement of A followedby a measurement of B is a graph cobordism between the out-comes of A and the outcomes of B, called the In/Out boundarypoints. In fact A plays the role of a source or preparation stage,while B is the actual measurement, or target of the measure-ment process (interaction or communication).

In the above joint case scenario, the graph is a bipartitegraph with inputs ai and outputs bj. The edges aibj are labeledby the conditional probabilities tij = p(bj|ai), representing theentries of the transition matrix T (B|A), of this Markov process.

We will denote with Hom(ai, bj) the set of paths (or thevector space with the corresponding basis). It corresponds to acomponent of the graph, called a communication channel. Thenumber of channels is the degree of A: dim(A).

A micro-macro correspondence associated to an observableA with probability distribution π is the Markov process 1 → A,with transition probabilities

T (1, ai) = p(ai) = p(ai|1).

The associated colored graph is the corolla 4.

Definition 7.1. An IO-process is the colored graph cobor-dism < B|T |A > represented by the following composition:

1 → AT→ B → 1.


Anticipating complexification,

B∗ ∼= (B → 1), ∼= (1 → B).

Of course T (bj, 1) = 1 for all j.7.2.2. Entropy as a derivation. Note that the functionH(p) =

−p lnp is a derivation:

H(pq) = −pq ln(pq) = H(p)q + pH(q).

This property allows for the following constructive definition ofthe entropy.

Definition 7.2. The entropy of the Markov process T =T (B|A) is the matrix H with the (i, j) entry the entropy of the(ai, bj) communication channel:

H : T → R, Hij =∑

γ∈Hom(ai,bj)

H(γ),

where H(γ) is the value of the extension of H(p→) = −p lnp as

a derivation on the path algebra of T (quiver).The total entropy of the Markov process is

H(T ) = ||H||,the norm of the matrix of channel entropies.

The entropy of an elementary measurement A (macro-stateetc.) is

H(A) = H(1 → A) = −∑

i

pi lnpi = H(1 → A→ 1) = H(A∗A).

The entropy is determined by the information potential:

H = Id+ I, H(p→) = p+ I(p), I(p) = −

∫ p

0ln zdz.

As a corollary, written in terms of Q, the entropy of a measure-ment (macro-state) A is:

Q(∑

i

pi) −∑

i

Q(pi) = 1 −∑

(−pi lnpi + pi) = H(A).


7.2.3. Entropy as a functor. Recall that matrices may beviewed as representations of quivers, as done above in the caseof entropy. It will inherit the main property of matrices: His a functor on the category of Markov processes, viewed ascomposable graph cobordism:

H(T1 T2) = H(T1) H(T2).

7.2.4. Relative entropy.

Definition 7.3. The relative entropy of the Markov processT : A→ B with input data A, is the average entropy:

H(B|A) = ||H(T )|A > || =∑

i,j

p(ai)Hij.

Then we have the following property (compare [BZ], p.4).

Proposition 7.1.

H(< B|T |A >) = H(A) +H(B|A)

= H(1 → A) +H(A→ B) +H(B → 1).

For example, with n = 2 and factoring

Γ = A (Id⊗B), Γ = (p1, p2) (1, (q1, q2)),

the above property reproduces the equation (4) from [BZ]:

H(Γ) = H(A) + p1H(1) + p2H(B) = H(A) +H(Id⊗ B|A).

7.2.5. Entropy in the Energy Picture. The correspondencebetween probabilities and energy levels via partition functionallows in particular to go back and forth from an interpretationin terms of information theory (“group level”?) to an interpre-tation in terms of energy (infinitesimal level).

Changing variables z = e−βE yields:

Q(pi→) =

∫ pi

0= −β2

∫ ∞

Ei

Ee−βE = −βP (Ei),

revealing a connection with an underlying spectral measure.


7.3. Overview: Z2/R/C/H-physics. The formalism in-troduced within the framework of Markov processes, is viewedas a real Feynman path integral approach (probability the-ory/Markov chains - “R-FPI). The relations with ClassicalMechanics in its particle-wave dual formulations (Lagrangianor Hamilton-Jacobi) and with Quantum Mechanics/QFT as acomplexified version of Markov chains, are open for specula-tions. Classical mechanics focuses on external DOFs via PCFpdx−Hdt, while statistical mechanics and thermodynamics fo-cus on internal (classical) DOFs via −Edt ∼ βE, allowing fora channel capacity versus local time analogy.

The entropyH is an infinitesimal character on the edge col-ored path algebra (Probability Picture) while TH is a (charac-ter?) on the vertex colored path algebra. The correspondencebetween the discrete (combinatorial or micro-state space) andcontinuum (partition function or Energy Picture) frameworksamounts to the approximation

∑ ∼∫

under the correspon-dence νi = Zi.

Proceeding from classical deterministic mechanics, with itsTrue/False- deterministic approach, via statistical mechanicswith a sort of “fuzzy logic” towards quantum mechanics, ismarked by a change of coefficients; from Z2 to probabilitiesR and to amplitude of probabilities C. There are repeatedattempts to further proceed and use quaternions H to incorpo-rate the 3D-space, so elusive to a rational explanation for “why3D?” (see also Ch.9 §3.1). A “good sign” is that a qubit is notjust a complexification of a bit, but rather its double ... is itH = C ⊕ C the right number system? It is non-commutative,but manageable, with a lot of potential regarding the symme-tries due to “coincidences” reminiscent of the fact that “vectorphysics” is just the algebra of imaginary quaternions.

One of the distinguishing features of classical physics andinformation theory is the presence of a global time (“clock”)corresponding to the existence of a unique factorization of tran-sition processes allowing to separate “parallel” and “sequential”


(at least locally). In the quantum world such a factorizationmay be “artificial”/non-canonical, or at least correspond to ad-ditional data, as the need and role for Operator Product Ex-pansions advocates: in general there is no global or even localtime in an arbitrary causal process. Or may be a local “timecut” exists locally at a vertex: In=Past and Out=Future ...Anyways, what “flows” is information (Information Picture) orenergy (the Energy Picture).

So, there are two sides of the coin, the Information Pictureand the Energy Picture, related via information capacity / time:

< H >= β < E > .

Ultimately, the mere mechanical motion we call “time” is nomatch for the “time we experience”. But then “time”, as in-formation/causality etc, flows again ... “E pur si muove!

CHAPTER 6

Modeling quantum information flow

Chapter 5 dealt with the classical theory, although the newideas are not dependent on the classical context. The differencebetween the classical and quantum theory at this point is a flipof a switch away: turn i on with a Wick rotation (imaginarytemperature!?) and ... allow for feedback (loops).

1. Constructivism: the critique of the critique

As a preliminary “program” in developing the quantumanalogs, we will comment on the critique from [BZ] of Shan-non’s entropy formalism.

The Shannon postulates should be understood as “hiding”the Path Model approach, and therefore they are not inappli-cable to QM, but need “upgrading”.

As any “inspired generalizations”, it should be conceptu-ally simple: sum over histories remains, taking care of the “vis-ible structure” (graph-cobordism), while the trace (“internalresidue”) should take care of the internal counterpart of thepossible hidden histories.

In general the B outcomes bj may depend on the A out-comes, so the joint experiment is not the general case; it alreadyassumes a sort of independence of B and A.

In the simples possible case, a quantum jump, measuringposition or momentum should be modeled as txp 6= tpx, leadingto Heisenberg’s CCR. Similarly for the double slit experiment,with its alternative: left or right hole?

111

112 6. MODELING QUANTUM INFORMATION FLOW

With quantum measurements as operator versions of tran-sitions A→ B (7.2.1), and “uncertainty” (or information gain)related to von Newmann entropy S, the Heisenberg CCR shouldfollow as

S(t[A,B]) = ~.

The interpretation of QM will look natural in terms of decisiontrees, extending the lapidary justification: “It just works thisway!”.

Adding interactions requires including graphs with loops(feedback). So the upgraded QM-Model is a representation ofgraphs; from Heisenberg’s matrices to Feynman diagrams (fromQM to QFT).

Res(tA) < − > Spec(A).

But, still ... “QM, where is thy heart?”; probably the superpo-sition with destruction of information (not just products butalso coproducts too).

And ... “QM, where is thy soul?”; probably the inevitablenon-commutativity of information gain, as a consequence of“conservation”: if you acquire (“take”) information, you drain(change/perturb) the system (see the ABE-model of measure-ment Figure 1 §3).

Therefore Information gain does depend on the order ofacquisition! Also, in the quantum realm, beware of inequalities,since they hide brutal truncations; look for the missing terms(“neutrinos”).

The dual pictures (Information / Energy) shed differentkeywords on the same IO-process 1 → A → B → 1: measure-ment or interaction.

Source→filter

encoder

transitioncommunication→

screendecoder → Target.

Then, some of the so called “paradoxes” regarding our knowl-edge of th system’s properties (“real” or not “real” etc.), for ex-ample comparing different successions of measurementsABA 6=

2. QUANTUM ENTROPY 113

AA, should be analyzed in view of the relative quantum entropyand information flow:

Teleportation Diagram

• •

• •

OO

•

OO

•

??~~~~~~~ •

__@@@@@@@

??~~~~~~~ •

OO

•

OO

•

OO

It is no longer surprising that a B intervening measurementmay drastically change the “function” of the gate, leading from“idle check” AA to teleportation! (see e.g. [Co3] etc.). Thefundamental concepts involved are: S and T -correlation (par-allel and sequential computations), which cannot be reduced toa local space-time description while denying “spooky actions”...

The two examples of [BZ] should be reanalyzed in this light(§3).

2. Quantum entropy

The ideas introduced in Chapter 5 will be implemented inthe quantum context as a change of coefficients from real tocomplex. The Bohr interpretation is the projection from oneformalism to the other: p = |ψ|.

2.1. Quantum transitions. Now we will turn internalDOFs on.

Outcomes ai of an M-observable will become eigenvalues ofthe associated operator with the corresponding measurementbasis. We will overload the notation, without distinguishingbetween ai the eigenvalue and |ai > the eigenvector.

114 6. MODELING QUANTUM INFORMATION FLOW

The quantum analog of a transition A → B is a repre-sentation of a graph-cobordism, called a VO-graph, where thevertices are labeled by states and edges by operators:

aiUij→ bj, p = | < ai|Uij|bj > .

The definition hints towards vertex operator algebras, as al-gebras over a PROP (functor from the PROP to a modularcategory or so).

2.2. Mixed and pure states. The relation between A =(1 → A) and mixed states ρ =

∑

i piPψ will be investigatedlater on.

2.3. Von Newmann entropy. The analog of the classi-cal entropy H , extending Shannon’s entropy to the categoryof graph cobordisms, should be an extension of Von Newmannentropy:

S(ρ) = tr(ρ lnρ),

leading to FPI:

S(1 → AU→ B → 1) =

∑

γ=∏

e

S(γ)?

2.4. Non-commutativity of measurements. In the clas-sical case (see 7) H(tAB) = H(tBA), where tAB = tA ⊗ tBetc., under the assumption that A is independent of B, i.e.p(bj|ai) = p(bj|1), which is equivalent to the condition that thejoint probability is independent of order:

p(1 → ai → bj) = p(aibj) = p(ai)p(bj) = p(bjai) = p(1 → bj → ai).

Now in general S(tAB) 6= S(tBA).As an exemplification, consider the polarization gedanken

measurement experiment from [BZ], p.7.First, one should not consider just A, a measurement of the

polarization in the direction of the z-axis, followed by B themeasurement of the polarization in the direction of the x-axis,

4. QUANTUM INFORMATION 115

but rather the 2-D vector space V =< sin θMx + cos θMz >and a transition from V into itself:

VU→ V.

As a guide in comparing with the classical information, inthe quantum world, the analog of the correspondence betweenprobability (p = | < bj|U |ai > |) and energy (p = e−iE/Z), isthe correspondence between states ψ (amplitudes) and phase.

3. Two good examples

Instead of the whole theory, “we” will analyze the “twogood examples” 1 from loc. cit..

3.1. Example 1. (left to the reader).

3.2. Example 2. (left to the reader).

4. Quantum information

The “total” info content of a quantum system is ... the sys-tem itself! What we need, I think, is a measure of this quantuminteracting/communication, i.e. a measure of the quantum in-formation sent (quantum information capacity etc.).

1Was it Gelfand? ...

CHAPTER 7

Classical and Quantum logic

The above mathematical representation of information flowhas some interesting connections with the quantum logic intro-duced by Birkhoff and von Newmann quite a while ago [BN].

1. Propositional calculus

Since then it became quite clear (loc. cit. §4 p.825) that“The central idea is that physical quantities are related, ...” andit seems strange it took so long until Feynman interpretationgave the “categorical flavor” to quantum physics, although theappropriate language (category theory) was in place at the time(of Birkhoff and von Newmann). Also Markov’s work ... (etc.).

The mark of classical mathematics was still too strong aninfluence to “break away” with tradition; even Birkhoff and vonNewmann focused on the correspondence between “observation-spaces” and “subsets of phase-space”, i.e. at the level of objects(loc. cit. §6, p.826). Not the objects alone, but together withthe relations should be set into a (functorial”) correspondence,i.e. data and programs or on the other side states and correla-tions.

The (A) “subsets in phase space” should be thought of asprograms and data, i.e. quantum computation (I/O and pro-cessing) or quantum interaction at the level of mathematicalmodels, while (B) “experimental propositions” would ratherrefer to measurements consisting of preparation (I: “source” of

117

118 7. CLASSICAL AND QUANTUM LOGIC

input states/particles/qubits etc.) and observation (O: “tar-get”/recording apparatus etc.), with a quantum process/ inter-action between the two. A “physical quantity” is this triple;e.g. “spin” needs source and axis and Stern-Gerlach magnetetc.

2. The lattice: creation and annihilation!

The propositions of quantum mechanics were related withthe structure of a lattice. It is tempting to speculate (at thisearly stage) that the idea is a “premonition” (“hides”?) of astronger claim, that the language/logic reflects the representa-tion of a causal structure as explained before, i.e. of a Feynmanprocess. In other words, “Does QM represent the (quantum)communication process?”, i.e. leading to representations of (de-cision) trees, PoSets (lattices), graphs etc.?

It is striking that the trademark of QFT, i.e. creation andannihilation operators, formally correspond to the structure oflattice (creation and annihilation of DOFs, what else is there?).

It is also indicative of an important connection (math pathmodel - information interpretation/formal language/automata)that “classical” distributivity fails in quantum logic (when mea-surements corresponding to M -observables) do not commute,but a substitute exists: the modular identity (p.832). A PathModel interpretation will be considered elsewhere (§5.1).

The classical distributive property “... is the characteristicproperty of set-combination.” (field of sets is a Boolean al-gebra) (p.831). But “sets” are not realistic models; reality isholographic (!) and we can’t really tell if “something” (object,event? etc.) is in isolation (“part of a set”) ...

Somehow “set theory” and “holographic world” are deeplyincompatible. Categories with objects and relations (we don’tneed the philosophy of set theory to build it) is a much moreflexible and high level language; therefore better suited to de-scribe our models.

4. RELATION TO PROJECTIVE GEOMETRY 119

3. The modular identity and ... entropy!

Returning to the modular identity, it is “a consequenceof the assumption of that there exists a numerical dimension-function d(a)”:

D1 : Ifa > b => d(a) > d(b),

D2 : d(a) + d(b) = d(a ∩ b) + d(a∪ b).The “hint” is clear: “D1-D2 partially describe the formal prop-erties of probability” and the modular identity “is closely re-lated to the existence of “a priori thermo-dynamic weight ofstates.” (p. 832-833).

Birkhoff and von Nwmann were thinking (probably), aboutentropy!

Indeed, a “dimension-function” is a special case of an Euler-Poincare map on extensions, similar to the above finer invariant(the “charge” of the quantum information flow/current). Itsresidue is the (classical/quantum) entropy

H(lattice/tree/forest/graph)

(colored with probabilities or amplitudes), depending only onthe external structure (terminals and roots).

4. Relation to projective geometry

Now any lattice of finite dimensions are products of projec-tive geometries and a finite Boolean algebra. This looks likea nice “marriage” between classical and quantum mechanics,providing the necessary framework to explain the measurement“paradox”.

The transition from a “quantum” to a “classical” descrip-tion would correspond to a “truncation” (projection) droppingthe “projective part”. It is at least a mathematical candidateto implement the “quantum-2-classical interface”.


The correspondence between projective geometries and skew-fields suggests a connection with the moduli space of non-commutative differential forms (left action of F on T •(F ) etc.).The relation with the “homogenization” process of classical me-chanics (from “affine”/free geometry to “linear/“reduced” ge-ometry) will be investigated elsewhere. The role of the Rie-mann sphere, viewed as a bifield will be explained elsewhere.For now, let’s note that the qubit has a certain redundancy,and its projective space is the bifield (Riemann sphere), as thebasic I/O quantum memory unit.

Complementarity, implemented in loc. cit. p.835 as aninvolution, will be interpreted as an antipode (of a sphere orHopf algebra).

5. Conclusions

How many ”logics” are there? We learn from [BN] (p.836) that “... one can construct many different models fora propositional calculus in quantum mechanics, which cannotbe differentiated by know criteria.” at that time, at least. Ifthese “quantum logics” are derived from Feynman processes,i.e. representations of Feynman causal structures, then one maytry to classify them using homological algebra (see [Ion04-1]etc.).

In any case one should not move to the continuous-dimensionalcase and stay within the graded-objects realm (keywords: skewfields, projective geometry and Grassmanians ... differentialforms and non-commutative differential forms etc.).

While statements of quantum mechanics correspond to aprojective geometry (see Ch.2 §6), classical dynamics descrip-tions constitute a Boolean algebra. A “classical explanation”of quantum mechanics “facts” should belong to an extension ofone logic by the other.

Now the classical (commutative) case decomposes into 1-dimensional independent constituents (characters), while quan-tum mechanics has a “greater logical coherence [BN], p.836,

5. CONCLUSIONS 121

leading to the impossibility of measuring different quantities in-dependently. As explained elsewhere, the µM -correspondenceis now implemented by decision graphs, extending the usualfiner joint partitions due to independent (commuting) M-observables.The coherence of quantum mechanics, mathematically imple-mented by Heisenberg CCRs, appears at the level of physicalinterpretation as “uncertainty” or at the level of computer sci-ence level as different entropy of the information flow graphstAB and tBA (see §2.4).

This interpretation allows to address the two questions in[BN], §18, p.837.

5.1. What is the experimental meaning? In a sensethe distributive property of classical lattices corresponds toprobabilities (real/ classical information/ entropy: R), whilethe modular identity of quantum lattices corresponds to ampli-tudes of probability (complex/quantum information/entropy).

5.1.1. Creation or annihilation? As a corollary quantuminformation (qubits) can be branched and merged (quantumgates for graphs, partition of unity for trees etc.), which mayappear to an observer as “created and destroyed” (meet andjoin lattice operations), while classical information can only bedestroyed (entropy increased - Second Law of Thermodynam-ics).

The fundamental role of creation and annihilation operators(bialgebra structure) becomes more and more clear ...

5.1.2. 2nd Law finally tamed! Also the “limits” of the 2ndLaw provide relief from a “thermodynamical death”. to thecontrary, the Quantum 2nd Law of Quantum Information al-lows for diversity and complexity to emerge from “chaos”. Thenthe “arrow of time and entropy” becomes a subjective matter ofthe more complex observer “looking down” to the less complexsystem ...

5.2. What about “tertium”: “datur”, or not? Ata more technical level, the most objectionable classical logic


assumption is [BN], p.837:

a′ ∪ b = 1 =⇒ a ⊂ b,

where a′ denotes the complement of a. The dual relation is:

a ∩ b′ = 0 (False) =⇒ a ⊂ b,

enabling the proof by contradiction (“tertium non datur”!).With hind site we view this relation as hinting to a split de-

composition (direct sum). In contrast, in the quantum world,the ‘factorization” has an “overlap”, being analog to the at-las of trivialization of a complex line bundle ([CK2]). In thealgebraic-geometric language the “points” of the algebra of ob-servables/states is not a split extension, but it consists of twosubalgebras. Correspondingly, the quantization always involvesa central extension before the “deformation step” (global de-formation extending the infinitesimal deformation).

So, “tertium datur” after all! It’s “black or white or ...gray” (Strictly speaking, “reality” is complex: God is playing... qubits!).

CHAPTER 8

Quantum dots and bits

Now what to do with a resolution of degrees of freedom, asa new comer in the plethora of mathematical-physics models?We have to tie anchor it in both classical and quantum physics(see §3), but first, it needs a name; how about: “the QuantumDot Resolution” (QDR)?

The term quantum dot originates from nanoscale physics[QD], referring to tiny spatial regions of size 100-200 nm (loc.cit. p.12) big enough to be considered “macroscopic objects”(“artificial atoms”), yet small enough to exhibit quantum ef-fects (only understood and described using quantum mechan-ics).

1. Quantum dots: theory and practice

The quantum dots may be thought of as corresponding toexternal DOFs “colored” by qubits in a representation of athe causal structure (“space-time”) given by a resolution of a“point”. A quantitative correspondence between the experi-mental quantum dot and the theoretical one would be worthpursuing.

1.1. What are position and momentum? The usualconjugate (dual) concepts of position and momentum exhibita conceptual incompatibility even at a classical level: the ex-istence of a position denies in a sense the possibility of mo-tion. The mathematics (derivative as a limit) poses of course

123

124 8. QUANTUM DOTS AND BITS

no problem, yet it clearly is misleading in the light of the quan-tum phenomena.

So back to an old question: “How is motion possible?” orrather “How to model motion, really?”.

The analysis of information gain during a classical mea-surement of position and momentum as a joint measurement,reveals (or claims) that the a choice of the order of measure-ment, i.e. position first x(t) and momentum next p(t+ ∆t) orthe other way around, should be irrelevant.

But the instantaneous momentum (theoretical concept, af-ter all) can be approximated (what else is out there?) as a pairof position measurements, x(t) and x(t+ ∆t) (see §3.2.4). It isa classical space-time correlation.

To approach a Path Model description (since there is no ab-solute events nor space-time [Ion00] etc., except correlations),we propose a matrix model based on the concept of quantumjump.

1.2. Quantum jumps. It should be thought of a “quan-tum space-time correlation”, in analogy with the above descrip-tion. Mathematically it can be modeled as follows, making useof an ambient manifold for representation purposes (the intrin-sic description will entail shortly).

For a fixed map r : Γ0 → M in C(M) representing an em-bedding of a causal event into the ambient manifold M , sayRiemannian (classical state-space), define the “conjugate vari-able” p : Γ1 → TM as the matrix of tangent vectors at theith vertex in the direction of the (unique) geodesics to the jth

vertex:

pij = Ti(ri → rj), “rj − r′′i ,

if there is a link between i and j.If Γ is an edge, then it should be thought of as a “quan-

tum jump”, so we move away from the classical “local pic-ture” (“manifolds”) towards a “quantum picture” (Path Model)

1. QUANTUM DOTS: THEORY AND PRACTICE 125

based on transition “functions” (quivers, Markov process, FPIetc.).

Then the analog of momentum as a “conjugate variable”(dual?) should be “delocalize”, but when the points are “close”we should recover the (or a) tangent vector. In this picture the“dimension of the tangent space” varies with k(r) the connec-tivity degree of the quantum dot, but we hope that the “effec-tive dimension” (in some sense) will be at most the dimensionof M ([FI2]; see Ch.10).

Indeed, Γ allows to implement, in a sense, a stronger du-ality between the classical Hamiltonian/Lagrangian “conjugatevariables” (later we will analyze the role of the Lagrangian),due to the correlation between Γ0 and Γ1, i.e. between “freetheory” and “interactions” (Lfree and Lint).

For simplicity we will use the notationX = (r, p) : Γ → TMwhenever convenient. Note that r and p are “coupled”, but inthe general case, for example when considering the Poincare-Cartan form on the tangent bundle extending the Lagrangianform (see [AI], p.283), they are not.

Now the (generalized) Feynman rule (the “amplitude”) maybe defined:

< FΓ(ω), X >=∏

ωri(∏

pij),

where ωri =< ω, r >i= ωi(ri).The

∏

j pij is (like?) a k-volume form at the correspondingquantum dot.

The big product (over i) is a form on the configurationspace of Γ with coefficients in (M, η).

Now integrate over the big configuration space to get the(generalized) Feynman integral

KΓ(ω) =

∫

C(M )< FΓ(ω), X > DX

as a pairing between “big forms” (“amplitudes”) and “big chains”(“paths”) with kernel ω (“multi-propagator”) for a given graph(intrinsic “interaction pattern”; Wilson chamber observable!).


The relation with renormalization as a change of variablescorresponding to averaging over unimportant variables ([QD],p.3, “... elimination of uninteresting degrees of freedom”, i.e.collapsing subgraphs (n.a.) etc.) and involving a transfer ofstructure process is considered in [FI2].

The information content/entropy of the underlying graphis related to the total quantum information gain of the pro-cess (quantum communication), having a classical componentdetermined by Shannon entropy and a quantum component de-termined by Von Newmann entropy (see Ch. 6).

1.3. Position and momentum incompatibility. Re-turning to the roots, Heisenberg uncertainty relation applied toposition and momentum expresses (also) the conceptual incon-sistency between “being in one place” and “moving” (see also“How is motion possible?”). In the transition picture of quan-tum jumps the concept of “motion” (change/transition etc.)is central, yet incorporating the concept of “position” (state)(again the duality object / morphism etc.):

xp→ y.

But “existence” actually refers to knowledge, so “position change”will be interpreted as an “information flow” from x to y. At thequantum level, it is modeled by a transition amplitude (com-plex) from x to y, which reflects into a classical probability oftransition from x to y. The probability density of the position(or wave function) changes, let us say in a classical (limit) wayfrom certain at x ((1, x) → (0, y) configuration) to certain at y((0, x) → (1, y)).

An intuitive picture is provided by the analogy with anelectric circuit, where the edges of the Feynman graph (vari-ous types of particles) are the elements of the quantum circuit,the quantum information (unitary operators) flows through theelements and the classical observables are associated with theIn/Out measurements. This is just pictorially explaining the

2. REVERSIBILITY: INFORMATION FLOW AND TIME REVERSAL127

quantum interaction/computing duality: a quantum system isa quantum computer (not too programmable, though) etc.

The main addition to the old Copenhagen interpretation ofthe wave function as the amplitude of probability, is its role ofinformation flow. In the context of a discrete model (DOFs)as representing space-time itself (“all there is”) this leads toquantum information charge and current etc. The mathemat-ics is similar to the Hamilton-Jacoby theory describing Hamil-tonian mechanics as a dynamics of the momentum amplitudep = ∇rΦ [Gosson], explaining the natural classical interpreta-tion of Quantum Mechanics in the Bohmian formulation.

The interpretation of the “anomalous” term in the radialpart of the Schrodinger’s equation will be related with the (clas-sical) Shannon entropy.

Reinterpreting position and momentum is just the first stepin a massive change: “categorify classical mechanics” §3, inorder to facilitate the transition to quantum physics (Feynmanprocesses) in its relation with statistical mechanics (Markovchains and all that), as a guide to the representation theory ofFeynman processes.

2. Reversibility: information flow and time reversal

Andreev Billiards [QD], p.131, constitute an interestingchallenge to explain in terms of information flow backwardsrelative to the external global time. An electron propagatingin a metal trying to enter a superconducting mirror, is forced toretrace its path, as if time is reversed! It is called the Andreevreflection.

The first lesson we learn is that one has to model particle-antiparticle pairs for completeness, even if a “local” experiment,not of EPR-type involving entanglement, concerns just the par-ticle ([QD], p.134).

2.1. Category theory and information flow. Monoidalcategories with duality are the perfect framework for modeling


information flow ([Ion01]; [Co2]). Hilbert spaces and unitaryoperators represent just the global/bulky approach, where thereis just one (simple) object, the Hilbert space and duality, as apairing between H and its dual H∗, is provided by the innerproduct. Unfortunately they “hide” the underlying algebraicstructures (simplex/quivers and their representation) allowingfor a natural implementation of the automaton picture: Markovchains and Feynman processes.

In particular, the annihilation operation implemented byduality in a category with duality can be interpreted as a “timemirror”; it “bounces” the information flow back in time (Ch.6), the global, observer’s time that is. Is there a connection withAndreev’s billiards ([QD], p.131)? It would be surprising notto! The author suspects that the quantum dot resolution, oncethe theory matures, will provide a nice mathematical frame-work to model such “strange quantum phenomena” (Dirac’ssea of particles and the “vacuum” etc.).

3. Categorifying classical mechanics

To facilitate and improve the “quantization process” a cat-egorical description of classical mechanics would be useful as ahalf-step across the corresponding “conceptual gap”.

3.1. Phase spaces: classical and quantum. A startingpoint is to model the “phase space” as a category, merging“existence” and “motion”, i.e. position and momentum, into a

“transition” xp→ x′. Classically there is a configuration first

M and the classical phase space of “potential motions” is Ω =T ∗M .

This is a local /set theoretical point of view (set first thenfunctions). The categorical point of view merges the two in-separable aspects of change, which classically were abstractedinto the clear-cut dichotomy position-momentum (absolute ex-istence and instantaneous change!?).

3. CATEGORIFYING CLASSICAL MECHANICS 129

The quantum phase-space is thought of as being modeledas QΩ = Hom(Category,Manifold), 1 in view of the classical-quantum connection. The “category of models” C consists ofvarious types of causal structures. For example: configurationsof graphs in a given space-time, or if disregarding interactions,just the space of configurations of N -particles Hom([N ],M).

An element of the quantum state space is a representationof an edge, signifying a quantum transition in external “space-

time”, i.e. γ(e) = x1p→ x′, with γ ∈ Hom(e,M) (forget the

internal DOFs for now). This is a “bit” of a quantum trajec-tory (external q-bit!?), to have an analog term for the classicaltrajectory.

3.2. Classical limits. The correspondence between thequantum bits of trajectory (quantum jumps γ as elements (func-tors) of the quantum spate-space QΩ) and classical bits of tra-jectory (points of the state space Ω) is plainly QC : QΩ → Ω:

Hom(C,M) ∋ (xp→ x′) 7→ (

x+ x′

2, p) ∈ T ∗M.

The relation with the usual quantization and classical limit,at this crude level of description, is still worth investigating;maybe this classical correspondence is related with the “ ...embarrassing “middle rules” in the “Feynman type” approxi-mations.” [Gosson], p.148.

3.2.1. ... and generating functions. What is the relationbetween propagators and generating functions W (x, p; x′, p′)?(see for instance [Gosson], “Generating functions” p.133). Thereshould be a quantum propagator, i.e. the classical categoricalanalog of the generating function, under the correspondence:

QW (xp→ x′) = W (x, p; x′, p′), p = p′,

provided the initial momentum equals the final momentum (e.g.[Gosson]).

1To avoid loosing the big picture, a “high-level notation” is used here.


In general, the following rule seems “natural”:

W (x, p; x′, p′) = W (x(p+p′)/2→ x′).

3.2.2. ... and actions. Although there is a fine distinc-tion between generating functions and actions (see [Gosson],p.141), at this stage we may think of the generating functionW as an action:

W (x, p; x′, p′) =

∫

γ

pdx−Hdt,

where PCF = pdx−Hdt is the Poincare-Cartan form.In the free particle case, assuming M = Rn (affine space):

W = m/2(x− x′)2/(t− t′) = m(x′ − x

t′ − t)2 × (t′ − t).

It is of the form Kinetic Energy × time.In a relativistic picture, a quantum jump should have the

form qj = (x, t)(p,E)→ (x′, t′). Its (infinitesimal) “relativistic

work” is the Poincare-Cartan form:

W (qj) =< (p, E), (∆r,∆t)>∼ pdx−Edt = PCF.

Therefore the generating action is the action of the relativisticquantum trajectory:

W (γ) =

∫

γp∆x− E∆t,

where γ represents a sequence of quantum jumps (transitions)and the discrete PCF is:

Categorical Poincare Cartan Form (qj) = p∆x−E∆t.

Is then the action positive only

W (xp→ x′) = pdx−Edt ≥ 0?

Allowing particles and anti particles (particles traveling backin time) seems to complicate the picture ...

3. CATEGORIFYING CLASSICAL MECHANICS 131

3.2.3. ... and Heisenberg relations. Can Heisenberg uncer-tainty relations in the categorical picture be obtained from aquantization of the action?

A quantization of the action could be the requirement that“A propagator propagates a multiple of ~” (external momen-tum; no boson/fermion issue here):

W (xp→ x′) ≥ ~. (7)

Such a requirement could be interpreted as a quantization ofspace-time dynamics (see §3.3).

3.2.4. What about measurements? Returning to measure-ments, what does it mean to “observe the position or the mo-mentum” in the categorical picture? Observables should ratherbe functors than functions! To understand the conundrum, acommutative diagram might help:

QΩQC−limit //

Q−(X,P )

Ω

(X,P )−measurement

PC1

Project/Average//R

2.

Either an averaging process of the probability distribution (ifQX orQP are valued in a probabilistic projective space) shouldcompensate taking the classical limit.

At this point a simpler picture emerges in the “extreme”cases, in order to brainstorm a possible approach. Measuringprecisely the position of a quantum jump (∆x = 0), could be

interpreted as a functorial operation onp→ x

p′→, towards a∆p = |p′ − p| 6= 0, while measuring the momentum accurately,

i.e. ∆p = 0, suggests an operation on xpx′, and therefore ∆x =

|x− x′|. Now a lower bound on the quantum propagator couldlead to ∆x∆p ≥ ~ ...


3.3. Exercises. So Phase Space was transformed from theclassical T ∗M into a Feynman category C (causal structure),but the main task is still ahead (“geometric quantization”).

What is the analog of a Lagrangian manifold (see [Gosson],§4.6)?

Define a Poincare-Cartan Functor on the Phase Category(causal process) as an analog of the classical PCF form. Isit related to the TH invariant (§6.1)? Define the phase of aLagrangian subcategory as a pairing associated to the PCF. Isit related to Q?

Is Q, the information potential/charge (Definition 6.1), agenerating function for the information flow?

Q(A) = |Aut(A)|, S(A→ 1) = lnQ(A), TH(γ) =

∫

γS(γ(t))dt?

Is there a theory of the Maslov index leading to a quantizationof information flow? See [Gosson], p.169 and compare theabove formulas with:

m(γ) =

∫

γ

d(ln |W |), m(γ ⋆ γ ′) = m(γ) +m(γ ′).

The classical minimum capacity/action principle [Gosson], p.173,on energy shells < E >= constant establishes a relation withthe Maslov index:

PCF (γ) = iπm(γ),

coming from a quantization of the action on loops (see Equation7):

PCF (loop) = ~.

If a quanta of information exists (the qubit, right?) represent-ing an elementary bit of space-time (causal structure), then, ina would be discrete model of quantum gravity (see Ch. 9), isit related with the cosmological constant?

CHAPTER 9

Quantum gravity and information

Gravity is special. This is agreed, but from usually differentpoints of view. It is a phenomenon which resisted the manyattempts of unification with the quantum description of the“other forces”...

But, is it a “force”? 1

I never liked having to solve a problem by “this or thatmethod”; it limits you options of success. Continuing the tra-dition of field-particle duality, modeling forces as an exchangeof a boson between two fermions, is such an imposed methodto “solve the problem of Quantum Gravity.

But General Relativity showed that gravity is more then aforce; it cannot be separated from the fundamental conceptsof space-time-matter-energy. Then why should gravity be a“force”, like the others? If so, what is “space-time” then andwhere does the mass (energy) come from? There is nothing leftto put in their place, except any physicist’ prefered vacuum,e.g. string theory’s background space.

No, do not waste your last interaction in this way; then theMind-Matter Interface (“Final Frontier”) may be left unsolved!

The opinion that gravity is an “organizational principle”already emerged and gains credentials ([Penrose], etc.).

Since matter (IDOFs: mass or energy) determines space-time (EDOFs) producing the “illusion” of gravity (GR), we will

1Or is it I.T.? :-)

133

134 9. QUANTUM GRAVITY AND INFORMATION

try to “produce” this illusion as an entropic (informational) ef-fect, making use of the tradeoff between energy and information(I/E DOFs), as embodied in the main DWT principles (Ch. 3).

1. Artificial Intelligent Geometry

Artificial intelligence requires adaptability, the signature oflife rather then dead matter. So why not “Geometry” (E/IDOFs and their dynamics), if it aims to model Mind (Informa-tion) beyond the traditional Matter (Energy)?

The first step is to have “adaptable models”/variable ge-ometry.

1.1. Monte-Carlo simulations. As an example of a largeclasses of models using discrete space-time within the Automa-ton Picture (Path Model) we mention the Ising model [DJ],Ch.5. The lattice indexes the subsystems (N -particles), thespin is the simplest case of internal DOF (qubit) and the Hamil-tonian/Energy is the M-observable. The lattice is fixed ini-tially, but in other models (QG coupled with matter), “adap-tive methods” are used which update the matter system byupdating the triangulation (loc.cit. p.254).

Without going into details, we will just point out that theclustering technique used in the Metropolis updating concep-tually is a zoom-in/zoom-out of DOFs, suited for an imple-mentation as a QDR, with its insertion/elimination operations(subgraph = cluster, flipping the spin on a cluster amounts toflip the single qubit address in the quotient labeled graph etc.).This project will be called The Hierarchic Ising Model.

Updating geometry by changing the triangulation is notconvenient to be done using the “classical” (topological) Pach-ner moves (p.255), but inserting edges in the framework ofgraph homology (homological moves).

It was proposed by the author long-ago that the “correct”approach (i.e. efficient) is to abandon manifolds [Ion00] and to

2. COMMENTS ON LOOP QUANTUM GRAVITY 135

work with their discrete incarnations, triangulations and simpli-cial sets, prone to an abstract (functorial) reformulation. Theimmediate benefit of being readily “understood” by computersin view of computer simulations is obvious in the Monte-Carlosimulation studies.

General relativity banished the concept of fixed geometry,correlating the metric and the distribution of matter. Yet an-other step should be undertaken removing the fixed topology,and even the “continuous topology” altogether. We of course,advocate the Resolution Picture (QDR).

In this discrete framework GR should be implemented via aHilbert-Einstein metric for a link-distance (and triangle-distance?)but with the inclusion of an entropy term TH , together with acoupling between the matter fields and the information current(Project Quantum Gravitational Entropy). A convenient labo-ratory would be the Hierarchic Ising Model with its “detailedbalance” related to entropy/information flow (edge insertions= entropy production etc.). Updating geometry (EDOFs) andmatter fields (IDOFs) should be a dual process.

2. Comments on Loop Quantum Gravity

For an introduction to LQG see [Rovelli-1] (also [Rovelli-2]),or [Smolin-2] for a brief and leisurely overview.

In our opinion LQG is a useful theoretical lab for discretemathematical models of space-time. Unfortunately, it mani-festly disregards the other interactions, which is against thespirit of GR, since “matter creates space-time”, with its “il-lusion”: gravity. Claiming that the spin networks resembleFeynman diagrams, but have noting to do with fundamentalinteractions is a way-out of considering internal DOFs, besidesthe basic spins (q-bits). Of course, once the basic language isdeveloped, a higher language interface could be built on top of it... But why wait, and not acknowledge that vertices of graphs


(spin networks) are “real matter DOFs”, as in the QDR ap-proach? It is questionable if there is a need for “potential space-points”, i.e. “vacancies” as space-time vacuum states (lowestenergy levels) to be filled or occupied with matter (higher lev-els). But this looks as a “refinement” of the QDR approachwhen considering in a more precise manner the internal DOFsetc.

One of its virtue, that it derives the fact that space-timeis discrete, is just a confirmation that one should change theconceptual basis of the theory and model causal structures asdiscrete structures; infinite, but (a resolution) of “finite type”,to rephrase what Greeks were saying all-along (sort of Aristo-tle’s “potential infinity”). For this, a categorification is in order3, abandoning for good the “manifold approach to space-time”.

The missing ingredients are: the lack of scaling capabilities,no flexible internal DOFs and therefore not a framework for theduality between external and internal DOFs.

The existence of a fixed “grain of space” is reminiscent of aΛ truncation of ultra-violet infinities in renormalization theory;or of a fixed “numerical method” ... On principial (philosoph-ical) grounds, it is not advisable, even if it is “derived” fromthe initial continuum formalism. That space-time is a two stepconstruction, spin networks then foams representing the evo-lution of a spin network, is not general enough to cope withthe general case of causal structures, where information flows,but there is no local “space × time” product structure; unlessone views the foam as a causal structure and the In/Out spin-networks as models of source and target of the interaction, inthe spirit of DWT (of course :-).

It should not be hard to transfer the technical tools intothe richer QDR framework (AI-geometry and E/I-DOF dualitywith PCS-duplex interpretation), with the main goal of deriv-ing the black hole radiation laws (“Project LQG++”) as anoverall consistency check (i.e. subsuming QM, GR and ther-modynamics).

3. IS “REALITY” 1-D? 137

3. Is “reality” 1-D?

It is hard to explain why there are 3 generations of fun-damental particle, or why is space 3D (11, 21 etc.) etc. Butthis last question would be settled if the “space” (external localDOFs) would be (modeled) 1D, while “time” would be modeledas a causal structure (subsystems in interaction); you would notask “Why 1D?”, would you?

3.1. Are quaternions the new coefficient-material?This last “move” away from classical, consisting of treating R3

(or SU(2) / spin) as a local internal symmetry rather thena global “illusion”, is consistent with the categorification ofphysics and the trade between I/E DOFs. “Just” representthe causal structure attaching to vertices qubits modeled asquaternions (tensor algebra; related to twistor program?). Thelast, or rather latest, change of coefficients from C to H §7.3should do the trick!

Then, how to relate such a structure with “normal physics”?Again “changing coefficients ”; if what we are talking about be-fore is “QFT as derived functors a Feynman process on a causalstructure over the quaternions” [Ion04-1], then take “cohomol-ogy with coefficients” in a genuine space-time manifold as yourfavorite background (Calabi-Yau? 2 [FI2]); if the causal struc-ture is the punctured Riemann surfaces (Siegel) PROP, then... string theory should emerge. In other words, most of thetechnicalities are there; all we need is to take a step back (pullback) to reach for the underlying ideas forced upon us by good-practice, then leap forward, beyond our preconceived ideas ...

In order to explore the quaternionic picture, understandingthe deeper meaning of a Wick rotation is important; are we ina semi-Riemannian setup R3,1 or the more algebraic frame-work provided by quaternions (skew-field, Clifford algebras-quantization etc.)?

26 dim + 2 the string = 2× 4 ... C(G; T∗H)


If the conformal theory is rigid, how rigid is “quaternionicanalysis” (SO(3, 1) ∼= SU(2)⊕ SU(2) and Virasoro algebra)?

¿From the Feynman dimensional analysis point of view, isThe Theory (say QG) renormalizable with a 1D-space-time?

With external and internal DOFs on an equal footing, dowe still have a “no go theorem” against unifying Poincare groupwith a GUT gauge group?

Will the “Super DWT” explain the three generations ofparticles?

3.2. All good things come as triples; why? Indeed,besides Space-Time being 3D, there are 3 generations (families)of particles, 3 quarks, 3 flavors, 3 colors etc. (not to speak ofother “trinities” ...).

The author’s believe (or optimistic attitude) is that theseare coincidences, until one creative mind manages to link themwithin a totally surprising way. Then the expectations are toderive the facts from even a smaller number of fundamentalassumptions, according to the axiomatic ideal, from Greeksto Hilbert and beyond. On the other hand knowledge comesin data-bases, and is better accommodated in a hierarchic-relational structure. We should not be afraid of loops or “cir-cles” (unless really vicious); we should perfect our ways to ex-ploit them (better KDBMSs).

For example, in the Standard Model there are still a lotof particles (that’s OK: DB-types), but also a lot of processesused to derive their properties: Lagrangian terms (fields andinteractions), renormalization, mixture angles etc. (OK again:it’s a procedural DBMS).

Now the “family problem” is related with the “neutrinoproblem” (see [SciAm/SE]), solved by a model (oscillationof neutrino types) which cannot be accommodated within theStandard Model. And the key issue is mass again, as differentlevels of excitations of the same type of particles, leading to 3generations.

3. IS “REALITY” 1-D? 139

So, what is mass? See [SciAm/SE], p.33. Is is explainedvia a Lagrangian mechanism introducing a new special field,the Higgs field. The interaction of particles with the Higgsfield accounts for the mass of the really fundamental particles,beyond the mass of protons and neutrons due to internal kineticenergy of quarks.

One of the “special”, unique feature of the Higgs field is thatits lowest energy level is nonzero: “the universe is permeated bya nonzero Higgs field [SciAm/SE], p.36. It does not take muchto start speculating whether these Higgs Lagrangian terms inone theory (SM) play the role of the “grain of space-time” inanother (LQG), and might be assimilated with vacancies inthe a version of QDR of DWT where “empty EDOFs” (poten-tial) should be considered (vertices labeled by “Higgs” internalDOFs) with which the “normal folks” (electrons etc.) could“switch places” (vertices acting as vacancies). It is a way to in-troduce a “background space”, yet of a different kind comparedwith the usual vacuum state implemented in the context of ona Space-Time manifold (or sigma model etc.). Recall that thedistinction between free theory and interaction might not be soclearcut in certain Hamiltonians (Lagrangians), but once made,corresponds in QFT for example (FPI), to the “choice” of theclass of graphs and propagators via Wick Theorem, i.e. howto pair EDOfs and IDOFs ... In the QDR of DWT, due to theduality between E and I DOFs, this clearcut distinction doesno longer apply; what are the benefits? It is too soon to tell,but in principle the theory should be much more flexible to ac-commodate new procedures to derive previously “unexplainedphenomena”.

3.3. Matter or antimatter? Speaking of which there isone more unexplained “mystery”, biggest of them all ([SciAm/SE],p 31.5): “Why is the universe made of matter rather then an-timatter?” (not that it matters :-), but it’s intriguing ...).


An easy way out is to invoke an anthropocentric argument.To simplify the “picture”, let’s assume we are dealing with justone tye of particle/antiparticle.

If anti particles are particles traveling back in time (Feyn-man), and since each particle with its antiparticle is just aFeynman knot (loop) in the universe, all there is around us isa big link (entangled knots; can be quite a big mess). Whatwe experience locally is a braid (tangle etc.), which can becompleted, in our imagination at least, to just one link. Thenthe question “Why is the braid (“the mess”) “concentrated” inour local universe?” can be dealt with invoking a “stratifiedcomplexity principle” (can’t define that; it’s just a “picture”,remember?) to conclude that humans (the biggest “mess” ever:-)) must belong to that part of the universe ...

Technically inclined people would like to invoke charge-parity violation (loc. cit.). The present state of the art cannotentirely explain the amount of matter around us. Then may beentropy can help: due to CPT invariance, CP-violation can betraded for T violation, except physics (classical and even quan-tum), is traditionally T-invariant (only highly complex theoret-ical mechanisms can account for a local time-reversal violations... ?). But what about “entropy” (information flow etc.; if weintroduce a twist - indeed!- in the categorical framework: leftduality not the same as right duality); will it help? Probablyyes, in conjunction with gravity, as an organizational principleperhaps ... So, after all, particles alike ... cluster alike; the more“time” passes (or rather the degree of complexity increases), thelarger the clusters of matter particles and the larger the clus-ters of antimatter particles. Then, do we really need a “high”CP-violation? Is at the present time the clustering at the levelof cluster of galaxies? Etc., i.e. it seems there is plenty ofroom (space) to turn our theories one way or another ... but itshould be a simple and beautiful turn! (Occam, Einstein, etc.“are watching”)

4. THE LAWS OF BLACK HOLE RADIATION - REVISITED 141

4. The Laws of Black Hole Radiation - revisited

With the “ideological progress” in the present version inmind, we will review the implications at the level of the program[Ion00], since the birth of the present project [VIRequest-UP].

4.1. Entropy “resides on the surface”. Recall that theentropy is a derivation Ch.5 §7.2.2 and it is sensitive to theexternal structure of a graph (causal structure, thought of asone possible space-time).

This is consistent with the fact that “Black hole entropyresides on the surface; therefore, maybe the degrees of free-dom reside on the surface; ...” [G] p.53. Indeed the “externalinteractions” are the “legs” of a graph inserted (or collapsedsubgraph). As mentioned elsewhere, there is no “fixed” struc-ture, since it depends on the current scale. The conversionbetween internal and external DOFs implemented via insertionand elimination of EDOFs (and correspondingly averaging ofIDOFs) should be investigated in connection with the conceptof black hole horizon and its radiation laws.

4.2. QFT and black hole thermodynamics. It wassuggested earlier that black holes (BH) are “prototypical” whenit comes to the understanding of the quantum bits and dotsof reality. Recall that (current theories) characterize BH bytheir mass, angular momentum and charge and “the laws ofblack hole mechanics literally are the ordinary laws of thermo-dynamics applied to a system containing a black hole.” [W],p.9.

In a sense a BH is a “unit of spacetime”, so it may be mod-eled as ONE EDOF: the DOF resolution (QDR) “ends” there;a BH is a “simple subsystem” or rather a BH is a “collapsedsubgraph/system” (indeed collapsed!): γ ⊂ Γ → Γ/γ, with the


point the subgraph γ collapses thought of a the BH 3. The col-lapsed DOFs to a sink (all interactions are incoming) plays therole of a BH.

Now Hawking radiation (creation of a pair particle and anti-particle) is naturally modeled as an edge insertion (graph ho-mology differential [K92, FI1]), splitting the one EDOF (theBH) into ( a superposition of) two DOFs and a correlation. Toenable a “local time” (“since creation”), probably it is betterto replace the edge with a “coproduct-like” V (3 EDOFs etc.).

In any case, the “loss of quantum coherence” is a naturalfeature in the perspective of [LB]. Although the BH IDOFmight have been a pure state before the interaction (insertion),relative to “Eve’s time” (outside time description of the tran-sitions etc.), the split of BH into BH’ and Pair yields still apure state of the whole system BH’ and Pair, although eachsubsystem, BH or the Pair, are represented by a mixture.

In [LB], one of the main points is that, in general, twopure states mix after interaction; although the system is still ina pure state, the two subsystems are each a mixture. Indeed,the interaction of the two subsystems produces an “exchange ofinformation”. Traces of this information exchange are left afterthe interaction, as “memories” of the interaction experienced.As advocated throughout the book, the distinction between“system” S and “observer” O is relative; it is merely an “ori-entation issue” S → O, yet the nature of S or O is not an issue(“alive or dead” just changes the information processing ca-pability - in the extended sense: matter/energy/information).In a “system-to-system” interaction S → S (classical setup;although it should apply to “Alice → Bob” communicationequally), there is an “exchange of wave functions” if you please... The exchange is permanent in the description of the ob-server (Eve), who’s responsible of the distinction between the

3The continuum with the “empty space” for a nice and easy mathe-matical treatment seems like a waste of ... modeling ingredients!


two systems in the first place (“subjective”? yes; there is noother way! (see Ch.10 §3).

The subjectiveness of descriptions is related to the hierar-chy of the model and to the fact that there is no “universaltime”; the need for a resolution approach appears more clearly,including a certain “stratification of space”. “Spacetime” isno longer a bundle (or product Space × Time), since correla-tions are not globally sequential XOR parallel; at a certain levelof description (degree within the resolution) processes may beparallel or sequential, and also the subprocesses within a pro-cess, yet in the combined model the subprocesses might not bedirectly comparable with the outside processes to be classifiedas s/t-correlated at the level of the outside description 4

4.3. What is missing? A more successful incorporationof GR’s ideas will be possible after a more technical develop-ment of the Categorification of Mechanics Ch. 8 §3. Indeed,“Since the time of Descartes, we’ve found it very powerful tolabel points by their coordinates, ...” [G] p.54; it is time tomove on ...

4.4. Unruh’s Law. It states that an accelerating observer“feels himself immersed in a thermal bath of particles” at tem-perature T = a/2π [W], p.115. It appears paradoxical since anaccelerating observer asserts “particles are present” when aninertial observer would assert that “in reality” space is devoidof particles.

This BH Law is particularly intriguing in the context ofDWT. Acceleration or “force” is due to interactions, which are(modeled as) relations, so the higher the interaction force thelarger the “number” of elementary interactions (“quantized pic-ture”), and correspondingly the larger the number of vertices:“particles”!

4Elaborated in terms of DOFs extensions.


In GR acceleration is equivalent to gravity, which “is” space-time, i.e. the causal structure. So in the spirit of Einstein’sequivalence principle a = g (ma = mg), “gravity” correspondsto the density of DOFs (say number of vertices, with a certainweight associated to each type of vertex); so, roughly speaking,the greater the acceleration the more particles are observed(modeled) ... (what about uniform motion”? - see Ch. 10 §4).

The interacting DOFs form only the “topological picture”;“add” a Lagrangian to get geometry/dynamics. Then the link-ing distance will have a counterpart depending on the momen-tum flow etc. A possibility to implement this is, in the graphs(cobordisms) model for QDR (Gn,m see [FI1]), to use the in-ternal degree n (resolution “depth”) to control the spacetimedynamics (acceleration), while the external degreeM to controlthe “actual” number of particles ...

BH are essential in this early implementation stages, sincethey are in a sense extreme cases; a BH may be thought of asan “information sink”, or a causally terminal object which is re-sponsible for a total “conversion” of dynamics (micro/EDOFs)into thermodynamics (macro/IDOFs).

4.5. The first law of BH mechanics. The first laws,also known as the “area theorem” [W] p.138, is also strikingfrom the perspective of the DWT entropy H : a derivation sat-isfying the Hologram Theorem 3.1. It relates the area of thehorizon of a BH with the other “state functions”: mass, angu-lar momentum and charge (loc. cit. p.142):

κ∆A = 8π(∆M − Ω∆J).

The LHS is a variation of the Noether charge associated withKilling field integrated over a bifurcation surface σ:

∆

∫

σ

Q =κ

2π∆S,


with S = 14A. The analogy with the entropy H representing

the information potential:

I = −dHwith I as a Noether (information) current is striking. Especiallysince the role of the bifurcation surface, similar to the role ofthe focal point and surface in Bousso’s result (Ch. 10 §2), isthat of the elementary corolla used in the analysis of the roleof entropy (Ch. 5, §6.1):

I(p) =

∫ p

0

ln zdz ↔∫

σ

Q.

The (conceptual) mathematics is the same (symmetries, La-grangian, Noether etc.).

In fact the “difference” between the discrete picture of acorolla and focal surfaces in semi-Riemannian spaces (or BH asa terminal object/sink or semi-Riemannian elaborate construct- see the definition of a BH [W], p.134)) is rather “technolog-ical” then conceptual. Faraday introduced imaginary lines offorce through Newton’s space, and Feynman generalized them;the present author suggested to start from the Feynman pathsand derive spacetime as a classical limit ([Ion00, Ion03]). Theresolution approach comes as a more technical suggestion re-garding how to do this. It seems that the classical (manifold)spacetime is “imaginary” (too much “modeling material used”),and Faraday-Feynman lines of force/paths are the “real thing”!(Fine tuning: less analysis more algebra; ask Dennis Sullivan!5

And no R; C or better H! O?).

4.6. The Hwking effect. “A black hole will radiate ex-actly like a blackbody at temperature a/2π.” [W], p.151. Afew additional comments will mark the winding road to QDG(see 3).

5Talk at Henry Poincare Institute - 2002.


This law is usually interpreted as “surface gravity, a is notmerely a mathematical analog of temperature, it literally isthe physical temperature ogf a black hole.” (loc. cit. p.151).With hindsite, we recognize that these “physical identifica-tions” ([W], p.163) are “just” conjugate variables of the un-derlying theory: temperature and acceleration (force) T = a∗,entropy 6 and “area” (metric) S = A∗, mass and energy etc.(More in 3).

A “coincidence” fueling another speculation is that the “Hodgeduality” between entropy and area is 2S = A/(4π) where theRHS is the normalized area in 3D-units (areas of the 3D-unitsphere), while the LHS is the double of the normalized entropy;is it because of an underlying double cover? (SU(2) → SO(3))7 ...

A “difficulty” connected with this law is the so called “lossof quantum coherence”. As it was mentioned elsewhere, whenproperly taking into account the “whole system”, BH and cre-ated pair, it is a natural consequence of the “mixture” betweenits components, BH and pair (see [LB]): each of the two sub-systems are in mixed states, but the system is still in a purestate. That we have to consider all the subparts entering theprocess is natural; in connection with the entropy analysis iscalled the “generalized 2nd law” ([W], p.163). In fact thereis no “generalized 2nd law”, just a correct application of the2nd law to the IO-process (and forgetting about Eve ...). The“generalized entropy” ([W], p.166) is just the entropy of the

6Previously denoted as H .7Gauss curvature and angle?


whole process.

•

BH

γ

•e−

~~~~

~~~

e+

@@@

@@@@

• •

Regarding the direct calculation of the “entropy from first prin-ciples” ([W], p.164), it depends on the mathematical model; itwas acknowledged in [W], p.175, that physically “all the de-grees of freedom of a black hole were concentrated in a Planklength “skin” around the horizon.”, yet in a not suitable waymathematically (i.e. the semi-Riemannian / differential geome-try implementation of GR): “However, these ideas run counterto the notion in classical general relativity of the black holehorizon as being a globally defined, mathematical surface, pos-sesing no local, physical significance, and thus providing a verypoor candidate for where the em true DOFs of a black holeshould lie” (n.a. emphasis). I am glad of this confirmation;as I was saying elsewhere 8, a semi-Riemanian BH should bethought of, and it is modeled and generalized in DWT, as aterminal object (one EDOF etc.)

Now DWT is not the “toy theory” advocated in [W], p.185as a “theoretical lab” for investigating these issues, (althoughit looks like one :-). It will turn out that “Further investigationof these issues ...” ([W], p.175) leads to a insight into thequantum nature of gravity (QDG - Annex 3) and the “old semi-R” calculation seems by now ... irrelevant 9.

8It’s too much work to keep track of links!? :-)9Clearly the math model of GR blocks the understanding of the

BH physics; just apply Occam’s razor and abandon GR in continuousspacetime.


Regarding the “enormous entropy” ([W], p.164) of a BHwhen fundamental constants are restored:

S =kc3

4G~A, A = 16πM2

it seems that this is consistent with the energy-mass analogsituation, together with a huge number IDOFs (yet only oneEDOF). As we will see later (Annex 3), the energy-mass rel-ativistic relation hints to a similar interpretation for the wellknown relation between free energy and entropy:

Z = e−βE , S = β(E − F ).

In other words, E = S/β + F is a (approximative, i.e. non-relativistic) decomposition of total energy into external/freeenergy and internal/informational (potential) energy, similarto the classical decomposition of energy into kinetic energyand potential energy. When external and internal dynamicsis modeled, a grain of space-time-matter is the dynamical qubitT ∗qubit (more later on).

The appearance of negative heating capacity ([W], p.165)is natural in the contex of the full, complexified, theory whereenergy E is the modulus of quantum information (Annex 3):

I = F + iM, E = |I|, MBH ∼ 1/T = kβ.

We should not exclude (such) “radical proposals” ([W],p.183), which introduce “new physics” by keeping the exist-ing physics, yet introducing another mathematics altogether(basically representations of PROPs), not just before reachingPlank scale: it should model both micro and macro cosmos etc.

The evaporation of a classical BH ([W], p.178) should bethought in DWT of as a convertion between I and E DOFs.A BH, as one EDOF (not one IDOF! see [W], p.183), with acertain mass corresponding to lots of IDOFs, is rather a “reser-voir” but with a finite number, yet quite large, of IDOFs. Thecomplete evaporation (“empty the corresponding memory”) isjust an insertion of a subgraph (etc.).

CHAPTER 10

Conclusions and further developments

I will be brief here, since I would rather like to know thereader’s comments ([email protected] or Info@VIRequest).

The DWT theory is based on the interaction-communicationduplex interpretation of quantum phenomena, aiming to openthe way towards a unified description of phenomena involving“Mind” and “Matter-Energy”, going beyond Einstein’s unifica-tion embodied in E = mc2.

At the conceptual level, the source is a new Unifying “Super-Principle”, only sketched at this stage.

At a more technical level, the unification requires the du-ality between internal and external degrees of freedom. Itcomes on top of a new understanding of what “Space-Time”is: “forget the space(-time), all you need is a resolution”. TheQuantum Dot Resolution is a resolution of DOFs, representingthe causal structure generalizing the “old” approach based on“Space-Time” (manifolds or not).

The focus is now on the Information Flow, generalizing theusual dynamics approach via time-flow in a state space. Thecrucial concept is now the Entropy (“information charge/current”etc.), again a derivation, which is supposed to relate the dynam-ics and thermodynamics of internal and external DOFs.

Although the DWT is envisioned as being designed top-down, from principles to implementation, at both levels it drawson the currently emerging ideas and techniques of the variousexisting theories (ST, LQG etc.), viewed from the ComputerScience and Homological Algebra summits. As always, the

149

150 10. CONCLUSIONS AND FURTHER DEVELOPMENTS

“Truth” is somewhere ... “above the middle”; one theory maybenefit from the conquests of another (e.g. String Theory andLoop Quantum Gravity [SciAm/SE]), yet a “clean” new startseems to be beneficial (DWT, of course :-). Instead of look-ing for a “unique/final theory”, we rather advocate a “flexibleframework” capable of adaptation to new scientific discoveries,since information implies change, not only as a basis of quan-tum mechanics 1.

1. Dimensions: 1,2,3 ... ∞!

There are various other indications supporting the “reso-lution with duality approach. The Holographic theories estab-lishing the “same physics” at different space-time dimensions([G], p.53), could be “just” Stokes Theorem when embeddingthe resolution in a specific space-time background. The “res-olution approach” resides conceptually at the level of the co-homology of Feynman graphs [Ion04-1], while embedding thegraphs in a manifold resides at the level of cohomology with“coefficients” in a given manifold [FI2]. String Theory is such a“cochain”. An elaboration of the homological and homotopicalphysics (adapting Stasheff’s terminology) starting with the Co-homology Theory of Feynman Diagrams (Feynman Categories;see also Hopf diagrams etc.) may lead to a proper classificationof (k,d)-QFTs.

There are various indications to support the statement thatthe number of external (space-time) dimensions is ... irrelevant(“So even the number of dimensions seems not to be somethingthat you can count on” [G], p.53). In more technical terms,a certain Formality Theorem may be at work when classifyingQFTs by taking the “cohomology with coefficients” in a givenbackground space-time (Mirror symmetry realm?). But the“principal dimensions” (3+1) seam to play a distinctive role,

1Information implies “change” and different choice (non-commutativity): AB 6= BA - see Ch. 6§1

2. THE HOLOGRAPHIC UNIVERSE 151

possibly as a “shadow” of spin, the symmetries of the funda-mental information unit: the qubit (d ≥ 3 and t ≥ 1).

Regarding “time” or rather the more fundamental infor-mation flow, the PCS-interpretation of quantum phenomena(interaction/communication) suggests the existence of a certainorganizational principle at work (gravity related?) responsibleof the increased “stratified complexity” we see around us. Al-though an anthropic point of view, some “anti-entropy law”could be keeping the natural balance, giving a local meaningto the concept of (arrow of) time as the information gradientin our local bubble of universe 2.

2. The Holographic Universe

Other indications supporting and confirming our ideas (the“collective mind” at work) may be found in [B], p.74.

Indeed “information is just as crucial an ingredient.”. Theidea that world is “made” of more then matter and energy isobvious if we think of “hardware” which is “nothing” without“software” (antic even: body and soul etc.). As we learn fromloc. cit., “Indeed, a current trend, initiated by John A. Wheelerof Princeton University, is to regard the physical world as madeof information, with energy and matter as incidentals.” 3.

May be not incidental (the sward is double edged - dualityetc.), but we definitely should focus on information, to com-pensate the “lost years”.

As suggested earlier, black holes, although emerging fromGR in singular conditions, seam to be prototypical regardingwhat matter and energy “really is” 4.

2Helmholtz/Boltzmann/Eddington?3Could not find more on John Wheeler’s initiative; my ... thanks for

a link, please!4Loc. cit. p.80/3: “Entropy of the hole - a deeply mysterious concept

- equals the radiation’s entropy, which is quite mundane”


The information bounds (holographic bound etc.) naturallyappear in connection with surface terms, expressed as the Holo-graphic Theorem 3.1 intends to hint (Ch. 5), as a divergence(entropy is sensitive to the boundary structure only). Simi-larly, the 1999 R. Bousso’s result (loc. cit. p. 81), is stronglyreminiscent of the behavior of entropy, or rather informationflux on the simplest case of a corolla (decision node). The “bigstep” will consist in abandoning the continuum 5. The develop-ment of the theory (not from scratch: the ideas are portable) inthe discrete realm should be much easier and computationallyprofitable.

Ultimately, the “game of life” consists in managing the de-grees of freedom! (see Fundamental Principles of DWT Ch. 3);and it’s not clear if the resolution (QDR) is infinite or ends upin “degree X” (i.e. is there a “syzygy theorem” here?) 6.

3. Time flow is Subjective

Time does not fly [D2] nor flows (in general there is no“time” as we used to think of).

Information does flow and ... we knew this all the time(!) when we said “Time passes by”. I picture us (or quan-tum systems) “exposed” to an information flow (interactions)and depending on our capability of processing (interacting) the(whole) Input (the “present reality”) may pass through us orbesides us (opportunities missed, choices we chose not to makeetc., right?). Does this requires a “brain”? yes, of some sort(see Ch.2 §5). But the differences are quantitative rather thenqualitative (if we chose to make a modeling effort we didn’thave to in the past). It is perhaps time “to move” conscienceaway from the center of the universe ...

5Loc. cit. p.81/2-3:“Fields, ..., vary continuously ...”.6“What are the ultimate degrees of freedom? ... level X.” loc. it.

p.76/3

3. TIME FLOW IS SUBJECTIVE 153

What about the distinction between “subjective” (what we7 think/model etc.) and “objective” (outside our inside :-)?The ABE-model of measurement (and modeling too) can bespecialized to B = E and understood in terms of [LB], leadingto a blurring of this distinction. In a S1 → S2 interactionS2 is subjected to the interaction of/with S1; as a consequencetheir states change, with respect to Eve’s description. What isspecial is being Bob and Eve at the same time (... what time?8).

Nevertheless Past, Present and Future are not illusions [D2],p.83! Illya Prigogine “have contended that the subtle physicsof irreversible processes make the flow of time 9 an objectiveaspect of the world” (loc. cit. p.85). The remarkable book[Prigogine] perhaps came too soon/ahead of its time (QFT,categorical descriptions, quantum information flow etc.).

When Alice is disappointed (“She hopes for a white Christ-mas” – see [D2], p.84/2), The Past is “ ... and Bob had noidea!”. The Present is the event: the communication of Al-ice’s (partial) state to Bob. And (in) The Future “Bob knowsto take Alice to the North Pole (Vail?) for a Merry (white)Christmas”.

The message is that the system is “Alice and Bob” and... I’m Eve10 :-), therefore an external and global time alwaysexists!

Interactions or communications: what’s the difference any-ways? Subjective or objective, are relative (in fact dual) ... AsFeynman was saying “Physics starts with dividing the universeinto two parts” [F1]; that’s a “relative” thing to do (subjec-tive), right? yet unavoidable!

7Me, you etc. or correlated “we”.8Interface between conscience and sub conscience, processes and sub-

processes etc. - quite confusing!9I.e. information n.a.10The storyteller - modeler - experimenter or collective mind


4. Cosmological constant: nowadays Cinderella

The role of the cosmological constant (CC) as the “Einsteinapproved” 11 change of GR is overloaded ([KT], p.67).

The apparent innocuous change consists in moving CC fromone hand to the other: take the CC from where Einstein put it(LHS of his equation) and interprete it as a matter term; is itrelated to the Higgs field (nonzero ground state) or is it darkmatter?

There seems that presently a crisis in theoretical physicsis happening 12, despite a certain reassuring attitude that “ev-erything is under control” (“cover-up”?). Old and venerabletheories run amok for whatever corrections might be handy tosave the experimental and computational evidence that it’s notOK, and the present theories (no names) are far from beingclose to the Theory of Everything etc.

Whatever the RHS CC stands for, it corresponds to a “biz-zare new form of energy ... ” (loc. cit. p.69/1), which in DWTcould mean entropy/information etc. 13. The huge values ob-tained for the CC in some quantum theories taking into accountthe nonzero energy of empty space (Higgs field?) is conceivablyconsistent with a “missing quantization” of energy/informationand spacetime (“grains of space” etc.). Also the impossibilityof redefine the zero energy point (now “energy matters”!), isreminiscent of kinetic energy as part of the mass (“small”) andrest mass (“huge”) relationship.

We view DWT as a potential avenue to relate all thesepieces of the puzzle into a nice framework capable of explainboth the micro-cosmos (QFT - few DOFs) and the macro-cosmos (cosmology - many DOFs).

11Although renegated later!12“Today ... physicists ... explore every avenue possible ...” [KT],

p.73/3 etc.13The possibility of realizing mass as an entropic effect was mentioned

earlier.

5. WHAT NEXT? 155

GR took shape after a decade-long struggle following hispivotal observation that gravity and accelerated motion areequivalent (loc. cit. p.68). Or rather “intuition” that sucha new fundamental unifying principle will yield a rich “crop”:GR. A collective effort developing the DWT will need a fewyears only!

5. What next?

The other Einstein’s “three prejudices” (!?) regarding themodel (of spacetime/universe etc.) (a) finite, (b) static andsatisfying Mach’s principles, most notably that matter shoulddefine space (not “just” determine the metric of space) seamto be legitimate in the perspective of DWT, with a few amend-ments (e.g. finite type etc.).

But perhaps one of the “mile stones” of physics: uniformmotion, has to go! In some sense it disappeared when intro-ducing covariance, so that any coordinate system is “equally”admissible. Yet our need to have a “free theory” (even asymp-totically) and correspondingly inertial frames, uniform motionetc. is still there.

But a lump of matter traveling through space is contrary toMach’s principle ([KT], p.68/2). In DWT’s DOF picture uni-form motion is not “natural”; the resolution of a DOF/systeminto more DOFs/subsystems does not involve an ambient spaceetc. Perhaps the “future” is for theories like QCD, which im-plement confinement as the rule with asymptotic freedom, as aclassical limit, the exception.

DWT approach (“the physics of DOFs and interactions”)is consistent with such a view: the modeled universe is con-fined to relative properties between its subparts. Motion of asubpart cannot “continue indefinitely” (makes no sense), sinceinteractions are present (within a connected component; other-wise “one universe” knows nothing about “another one” etc.).In a way we return at the Aristotle’s point of view, but againat another level: essentially there are only harmonic oscillators


and “everything is cyclic”; yet processes occur cyclicly within acertain hierarchy (interacting qubits / “stratified complexity”).

If QCD is the “future”, what about Supersymmetry”? Atthe level of graphs as models, at least, Poincare duality allows toexchange vertices and edges, i.e. “particles and fields”. Thereshould be no new particles, but rather a unified framework fortwo dual points of view. 14.

6. Epilog

We should also mention Lee Smolin as the advocate of theimportance of information exchange among physical processes:“such a final theory must be concerned not with fields, not evenwith spacetime, but rather with information exchange amongphysical processes” (loc. cit. p.81/3).

We would like to emphasize that “physical” should include“alive”, and there is no clearcut distinction when comes to arti-ficial life (or intelligence; Turing test etc.): If it processes infor-mation, then it is (more or less) alive! 15. Indeed, processing(exchanging) matter and energy with the environment must be“upgraded”, in the perspective of the New Unifying Principleof Information/Matter/Energy, to include processing informa-tion; in what extent, that’s “details” ... the principle countsat this time. How much configurational information and howmuch “inert” information is there in a “grain of organized mat-ter” is a quantitative issue 16, but qualitatively reminiscent ofrest and dynamical mass in the context of relativity (E = mc2,m = m0γ etc.). Can be use the information “locked” in mat-ter beyond “merely” banding spacetime (worm wholes etc.; notthat we can already do this :-)? 17

14“Tabula rasa” is a clean start :-)15God’s children are playing ... “God”!16“A silicon microchip ... smaller than the chip’s thermodynamical

entropy”, loc. cit. p.76/2-3.17To suggest such a possibility the subtitle THE (Quantum) MATRIX

was coined ...

6. EPILOG 157

So (“redirecting” loc. cit. p.81/3), perhaps “the vision ofinformation as the stuff the world is made of will have founda worthy embodiment” in the DWT as a “would be theory”,embryonic in [Ion00], announced in 2005 [VIRequest-UP]and presently a Grant Proposal/Invitation (n.a.) 18.

Regarding “our position in the universe”, the causal struc-ture replacing the traditional “Space-Time” (QDR and its rep-resentations etc.) is our computational efficient frameworkto describe a quantum system. Yet the “whole story” is the“observed-observer” interaction-communication process (S−S,S −O or O −O), including the measurement process modeledas an S → O (or better ABE-eavesdropping) process. Withthis in mind and to end on an optimistic tone: “The Universeis teaching Us and the Miracle is: we do learn well!”.

18Please visit the VIReQuEST! - Annex A.

Part 3

A Research Diary

CHAPTER 11

Why is mass energy: E = mc2?

The relation between mass and entropy was conjectured inthe context of the interpretation of gravity as an organizationalprinciple, more precisely as an entropic effect.

We speculate on the mathematical similarity between theblack hole laws and thermodynamic laws in the context of theduality between internal and external DOFs asserted by theFundamental Principle of DWT (QDR).

Entropy enters the picture as a derivation residing on thesurface as pointed out by the Hologram Theorem 3.1. The suit-able framework is that of a Lagrangian with symmetries gen-erating a Noether current via Noether theorem (see Ch.5):

H = −dI,

withH the information potential (and associated charges: qubitsetc.).

The comparison with black hole state functions was startedin Ch. 9 §4.5. Continuing the comparison and taking as refer-ence for the BH laws [W], we claim that the flux of the infor-mation current :

∆H = Q(∂t) = −I(∂t), I(p) = −∫ p

0ln zdz.

corresponds to (with the notations from [W], p.146) the in-tegral of the Noether charge Q corresponding to the Noethercurrent j = dQ associated with the local symmetries of the

161

162 11. WHY IS MASS ENERGY: E = mc2?

vector field/flow N (loc. cit. p.146):∫

σ∆Q

which in turn yields the entropy (a is the acceleration denotedin [W] with κ and H is the entropy denoted in [W] with S):

a∆H =

∫

σ∆Q.

We interprete this as:

Theorem 0.1. The flux of the entropic current of a ter-minal object (e.g. BH or EDOF) is given by the macro stateparameters:

T∆H = ∆M − Ω∆J

where M is the mass, Ω is the surface angular momentum andJ is the volume angular momentum.

The “theorem” relates the “incoming external dynamics”(dynamics of EDOFs) with the “internal dynamics” (thermo-dynamics of IDOFs).

The analogy between entropy H and area (BH laws) isclearly justified by the duality (I/E DOFs) since both are “sur-face terms”. Although the area theorem in differential ge-ometry and 2nd of thermodynamics seem to belong to differ-ent realms (holding true in practice only approximately loc.cit. p.148), the latter is also a mathematical result at thelevel of Shannon entropy on corollas for example (focal semi-Riemannian surfaces ∼= corollas plus “waste empty space”). Atthis point we would like to “normalize entropy” and think of acorolla as a wheel:

Big H bar : H = H/2π.

The “time asymmetry” is just the orientation of the corolla(direction of the information flow): no input and n-outputs(rather a “white hole”).

11. WHY IS MASS ENERGY: E = mc2? 163

Again we will stress that a BH is a prototypical terminal ob-ject (DOF) disguised in semi-Riemannian outfit. The generalcase is an IO-communication/interaction; the cut and paste op-erations can be used to relate the general case to the “terminalcases” (duality, trace etc.).

The 1st Thermodynamics Law (1TL) is:

2πT∆H = ∆E + P∆V,

referring to the IDOFs (thermodynamics).The 1st Black Hole Law (1BH) is:

a∆H = ∆M − Ω∆J,

referring to EDOFs (dynamics).These are two coupled equations relating the dynamics and

thermodynamics; the (future) unified theory is called hyper-dynamics (see later on). There are additional correspondencesbetween temperature and acceleration (“force” or gradient of anexternal potential), namely Unruh’s effect, and between energyand mass, namely Einstein’s equivalence principle E = mc2.

Now the Unruh’s Law (effect):

2πT = a,

may be interpreted as: Temperature is the “hidden” acceler-ation 1 It relates the two frameworks. “... for a given ther-modynamic system, one typically obtains both a “physics pro-cess” and “equilibrium state” ...” [W], p.148, we interpreteas disclosing the “conversion” (duality) between internal andexternal dynamics of DOFs (dynamics and thermodynamics).

Now in order to “explain” E = Mc2, in fact that energyIS mass E = M in geometric units c = 1, h = 1, we need toestablish a connection between the two terms P∆V and −Ω∆J(loc. cit. “The term “−Ω∆J” is closely analogous to the “workterm” P∆V ...” p.148).

1I.e. T = |∇V| in the context of the duality E/I DOF.


If we eliminate for the moment 2πTH = a∆H to get:

∆E + P∆V = ∆M − Ω∆J

and use E = M we obtain is a necessary condition:

P∆V + Ω∆J = 0.

In order to justify such a relation, we hint towards generalframework: deformation theory.

Definition 0.1. The angular momentum of a DOF is

ω = ω1 + ω2.

2

Then we should have 3:

ω ∧ ω = 2ω1 ∧ ω2,

and ω a solution of the Maurer-Cartan equation 4:

dω + 1/2[ω, ω] = 0, (Dω = 0).

By differentiation one obtains (boundary has no boundary):

d(ω ∧ ω) = 0,

which translates into:

0 = d(ω1 ∧ ω2) = dω1 ∧ ω2 + ω1 ∧ dω2...PdV + ΩdJ.

This suggests a conformal setup: complex structure, CR-equations,T ∗M / spin structure, ω a curvature satisfying Bianchi identity:Dω=0 etc. 5).

2We think: ω1 = J, ω2 = Ω3Research is ... wishful thinking!4The Master Equation is always the key: it provides the duplex in-

terpretation: deformation theory (perturbative approach/ resolutions etc.)and geometry (holonomy, curvature etc.).

5Probably corresponding to the two levels of structure related via Wickrotation; see further on!

11. WHY IS MASS ENERGY: E = mc2? 165

Returning to the “two levels” (internal and external), a gen-

eral vertex interaction Inf→ Out (In/Out: external; f : inter-

nal), is a role separation “a la Newton” (F = ma or the 2-dimversion of Einstein T = κG) into a geometric framework (ourexternal dynamics level of EDOFs) and constituent framework(our internal dynamics - thermodynamics - of IDOFs).

At this point it would be better to use the graph invariant(and state function) TH rather then H itself, with a tacit as-sumption T∆H = ∆(TH), in order to “allow” temperature tovary:

d(TH) = dE + PdV, d(aH) = dM − ΩdJ.

Dually we should have:

d(aH) = dM − ΩdJ.

That we are in Maxwell’s position 6 (or rather Yang-Mills) isprobably too much to expect:

dω = 0, d∗ω = 0,

although the duality should be the main ingredient; a Hodgestructure (relating H and H now independent complex vari-ables ... twistor program?) would be nice to have. Indeed wedo have information/interaction sources (and sinks) ...

On the other hand the separation between external and in-ternal should represent a choice/model: In/Out. So the masterequation should rather be thought of in the context of non-abelian cohomology:

Dω = 0 ∂+ω = ∂−ω,

with the two 1st laws rewritten as:

d+ω = δν, d−ω = βδ(aν),

where ν = TH should remind us about the number of configu-rations picture (combinatorics) 7.

6... at Ostrogradsky’s examination: not enough time to finish ...7For the purpose of further staring at these pieces ...


The dimension these forms leave in should be 4 since allgood things (and exotic) happen in dimension 4 8 Again inthe spirit of duality (Maxwell/Yang-Mills/Hodge etc.) a Hy-persymmetry principle between space and em time should beexpected, besides the Supersymmetry between particles andfields (vertices and edges). Indeed in causal structures s andt-correlations are totally a matter of the way the process is de-scribed/represented (information may flows in all “directions”).In more technical terms, Wick rotations should be generalizedto spacial directions also, so that Space and Time 9 are finallyunified, appearing on an equal footing.

So not only the number of dimensions we like to use isirrelevant (beyond the use of the qubit as the building block),but also what is time-like or space-like is relative 10.

1. Information flow revisited

The classical discussion of Shannon’s entropy (Ch. 5) andinterpreted as a flow of information in the context of Noether’stheorem (information current and charge) are a toy model forthe general case: pi = |zi| (Bohr’s rule) (or when a unitaryIO-process is decomposed according to a spectral partition ofunity, analog of

∑

i pi = 1 etc.).The toy model of the probability flow is the colored corolla

(probability distribution π):∑

i pi = 1, H = −∆I , I(pi) =−

∫ p0 ln zdz. The information vector field w = ln z seems to

afford the role of “time” (information “Killing” vector field?)and I the role of information potential. Then the total surfaceterm H =

∫

∂t I =∫

t div(I) plays the role of information flux.Indeed, the BH entropy resides on the BH’s surface (BH −entropy = 1

4Area). The information charge is the probability

8Or 1 if working with quaternions.9Once a coordinate system is chosen to describe the information flow

is made.10The more symmetries the better!

2. 4D HYPERDYNAMICS 167

itself and∑

i pi = p is just the conservation of informationequation (ν =

∑

νi is conservation of possible states under thepartitioning of the micro state space) 11.

2. 4D Hyperdynamics

The internal/external canonical variables are P,Ω (1/2-forms) and their canonically conjugate variables (V, J); we willseek guidance from classical Hamiltonian/Lagrangian formal-ism by interpreting the work term W = PdV as pdq (momen-tum p as the canonical conjugate to position q).

Recall that in 4D p = (p, E) and q = (q, t) are conjugateand the PCF is pdq−Edt = pdq. So let’s introduce ω = (P,Ω)and ω∗ = (V, J). Also define ρ = (aH, TH) (or ρ = aH + iTH?) the information 4-current. Then aH is the “spacial” infocurrent and TH the “temporal info current” / info charge. Wehave the “PCF” ωdω = PdV − ΩdJ (recall PdV + ΩdJ = 0...).

Then the two 1st laws combine as:

ρ = dE + dM + ωdω.

Here E is not the relativistic energy, i.e. does not include mass.So E and M are on one hand the two relativistic sides of to-tal energy, while on the “DWT-side” they reflect external andinternal properties.

A complex formalism (or symplectic/Kahler etc.) shouldbind them as E = p + iM with iM Wick rotated time-liketerm, so that the relativistic equation would hold (c = 1):

|E|2 = E∗E = p2 +M2.

Then

E +M ≈ p2/(2M) +M ≈√

M2 + p2 = |E|,11Rephrase for the amplitude flow and related to Schrodinger’s

equation in polar coordinates: Bohmian mechanics interpretation andHamilton-Jacobi formalism.


to get a complex equation unifying the two 1st laws:

ρ = dE + 1/2d(ω2).

Alternatively: dF = PdV − TdH since E − W = U − TSetc. and its conjugate d∗F = ∗d ∗ F = aH + ΩdJ so thatMaxwell/YM-equations would hold:

dF = dE, d∗F = dM.

In this picture, temperature is conjugate to acceleration (“equal”as in Unruh’s Law), and mass is conjugate to energy (“equal”as in Einstein’s Law).

The conjugation must be due to the duality between in-ternal and external DOFs, which in turn requires the dualitybetween space and time (“explaining” the String Theory’s s-tduality; more on the “Virasoro picture” later on ...) which isachieved via the generalized Wick rotation (again info flow doesnot know about a “parallel-sequential projection plane”).

The mass is thus an entropy related property (informationmomentum / info-gradient, i.e. “time-like” ?) and inertia isdue to an (imaginary) “internal rotation” (quantum phase?).

A complex version of the above two equations would sim-plify the theory:

DF = E, D = d+ id∗.

Indeed the complex or symplectic formalism is required bythe dynamical qubit T ∗H (or T ∗(C ⊕ C not to include thequaternions at this point); then its symmetries Aut(qubit) =SU2 ⊕SU2 are the conformal transformations (Diff(S1) / Vi-rasoro algebra).

Now the question is: “What is the corresponding 12 (Nobelprice) Lagrangian (Hamiltonian)?”.

12See Borcherds (QFT).

4. QG AS A DEFORMATION THEORY 169

3. Space-Time duality and Hypersymmetry

That info flows in “all directions”, spacial or temporal (i.e.quantum computations may be parallel or sequential) suggestsa new gauge symmetry, a sort of generalized s/t-duality. Itallows for a better analysis of the Bohmian mechanics whichseparates Schrodinger equation into two equations, when rep-resenting the wave function solution (i.e. the amplitude of prob-ability) in “polar coordinates”. The probability p = |ψ| yields

I(|ψ|) =∫ |0 ψ| lnrdr the “real/spacial amplitude” of the in-

formation flow and the phase lnψ − lnp“imaginary/temporal”component. The complex info potential I(ψ) =

∫

lnψ shouldbe related with von Newmann entropy, playing the role of agenerating function of the info momentum flow.

The proper language for a rigorous implementation is thatof categories with duality (left 6= right probably allowing forCP-violations, yet preserving CPT).

4. QG as a Deformation Theory

The semi-classical Einstein equation:

Gab = 8π < Tab >

is an “2-dimensional” analog of Newton’s equation maa = Fa,and is only half of the “full picture”. As Einstein himself saidit, the LHS is beautiful (physics as geometry - Greek ideal), butthe RHS is not (so). Indeed, comparing with an IO-process, aNewton type of law F = ma is a label for a framework leadingto a computational machinery (e.g. RHS ma within differen-tial equations or G within semi-Riemannian geometry), capa-ble of processing an “input”, the LHS, e.g. a given “force” orenergy-momentum etc. (which in turn can be obtained from aLagrangian formulation etc.).

A “complete theory” should be a “cycle” where O = I∗

(maybe thought of as Newton’s 1st law: action=-reaction) label


the “cut cycle”:

Newton’s theory including gravity (F = kmM/r2), as the onlyfundamental force, is indeed a complete theory: the LHS forgravity as a “Coulombian force” is beautiful too!

A complete theory including Einstein’s GR equation shouldbe of the form:

tr(I) = 0 or D(ω) = 0,

expressing a conservation of information (2nd generalized law?)as a result of hypersymmetry with its EL-equations (rathersome analog DS-equation).

The analog of “average curvature is due to matter” (G = T )may be thought of as tr(R) = T ; now what is the law yieldingthe energy-momentum tensor T ? (We will see that the externaldynamics must be merged with internal dynamics, later on):

Tr(ω ∧ ω) = dω ... > Tr(TdT ∗) = 0.

Let’s keep in mind that von Newman entropy is S = tr(U lnU),and exp tr ln = det, so the null average entropy equation is:

< S >= 0 ↔ det(J(O)) = 113.

For example, QFTs (statistical mechanics etc.) are theoriessteaming from such generating equations: partition functionsas vacuum-to-vacuum process (amplitude):

< ∅ >= tr(I)

As TQFTs etc. teach us, it is better to have the theory for allHom(I, O) then to sum up everything just “to reduce categorytheory (transitions between many objects, including symme-tries) to ring theory (endomorphism ring of one huge object,and its symmetries)” (see Mitchell? etc.). That the “partitionfunction” is not convergent (in the L1-norm - see §4), should

13Relations with Jacobian conjecture, formal diffeomorphisms, “time”inversion etc.

5. STRING THEORY AS A DEFORMATION THEORY 171

not be a problem; the partition/generating function should bea graded object of finite type (see [K97, CF, Ion03]).

GR is in some sense a “deformation theory” in disguise:

G = T ... > ωdω = j,

with a curvature complex form ω tieing together internal andexternal DOFs in duality (although the RHS equation is rathera conservation equation ...).

If allowing a background metric (Minkowski; for a fleetingmoment only), then we are looking at an equation providinga perturbation of the background metric etc. In a discreteframework the “background space” role is played by “vacan-cies” perhaps ... (see LQG setup / foams).

To include QFT, which is a deformation theory (Dyson-Schwinger equation, as an Euler-Lagrange quantum dynamicalequation, is Maurer-Cartan equation in a suitable sense 14, inthe DWT QDR framework the analog of Einstein equation is:

D(ω) = 0 ↔ dω +1

2[ω, ω] = 0 G = T.

ωdω = j div(T ) = 0.

The “free theory + interactions” approach leading to the“perturbative” interpretation is implemented via Wick theo-rem as “graphs” (QDR) and Feynman rules (representation:propagators, coupling constants etc.).

5. String Theory as a Deformation Theory

To interpret ST as a deformation theory, we comment on[H].

The creation and annihilation operators A satisfy Maurer-Cartan equation (p.136):

HA+ A2 = 0

14Idea “confirmed” by experts.


where H denotes the Hamiltonian (conformal invariance: H2 =0 p.139). It represents the equation of motion (see also [Marcus],p.64). The string field A is the perturbation of a particularstring solution Φ0 (analog of the “vector potential), yieldinganother string solution (analog of the connection):

Φ = Φ0 +A,

satisfying the Master Equation Φ2 = 0 provided that the Hamil-tonian is an inner derivation:

H = adΦ0 , HA = Φ0 ∗ A+ A ∗ Φ0.

Here ∗ denotes a convolution (e.g. a convolution algebraHom(C,A)of “Feynman type” - see [Ion03]).

It is remarkable that “half” of the Hamiltonian (left/right“factorization”) provides such a solution as Φ0 = HLI , where Iis the convolution algebra identity, as in the context of torsionalgebras [?].

This last condition (H cohomological trivial) “can be viewedas the condition for the solution Φ0 to correspond to a set ofbackground fields on spacetime” loc. cit. p.138. Since H corre-sponds to the energy-matter current (with energy-momentumtensor its flux) and Φ0 the “background spacetime”, then thecondition

H = adΦ0

may be thought of an algebraic analog of Newton’s LHS forgravity (constitutive equation): “String theory is a unified the-ory of gravity ...” loc. cit. p.138.

It also plays the role of a (beautiful) LHS of Einstein’s equa-tion. Then the algebraic analog of Einstein’s equation T ∼ Gis a MC-equation:

HA = −A2.

In contrast, this is a “complete theory”.A formulation, based on the same formalism (deformation

theory) as string theory, but without a background metric noran ambient manifold topology (see loc. cit. p.141) is the QDR

5. STRING THEORY AS A DEFORMATION THEORY 173

approach of DWT. The representation of a QDR with “coef-ficients in a certain manifold” will play the role of a concretemodel (e.g. generators and relations versus group representa-tions), while the “cohomology class of the theory” will dependonly on a certain class of the manifold. What is essential, isthe “star algebra” (convolution algebra of Feynman type).

At this point we should mention that the symmetric for-mulation of deformation theory studies the perturbation ∂ =D1 −D2 (here Φ−Φ0) as the “failure” to have a “double com-plex”:

D2i = 0 ⇔ ∂2 = [D1, D2] etc.

In conclusion, the Quantum Digital Gravity 12 initiative recordedbellow should probably be formulated as a deformation theory.

CHAPTER 12

Quantum Digital Gravity

QDG consists in a few new ideas integrated in the commonknowledge, starting from the motto: “a grain of space-time-matter is the dynamical qubit”. Also, Special Relativity is a“projective theory”, reminiscent of QM, so the possible inter-pretation I = Eeiθ is kept in mind:

“QFT := SR+QM ′′.

As a first objective we would like to merge the two 1st laws(§4) which we will rewrite as (entropy is now denoted by S):

dE = TdS − PdV, dM = adS + ΩdJ.

Some additional “coincidences” will play a crucial role. ToEinstein’s Principles (equivalence of energy and matter, andequivalence of gravitational mass and inertial mass:

E2 − p2 = M2, M2 = M2g , (8)

we add and interprete Hawking’s Law and Area Theorem:

T 2 − a2 = 0, S2 −A2 = 0. (9)

We interprete them as referring to the two levels of structure,external and internal, and as in Maxwell’s case, since we thinkof gravity as an entropic effect, we postulate two charges: in-formational charge (pure inertial?) Ma and gravitational Mg

(“due to entropy”; electric charges: yes!), Now, the inertial(dynamical) charge/mass equals the gravitational charge

M2 = M2g

175

176 12. QUANTUM DIGITAL GRAVITY

since there are no free pure inertial charges (no monopoles!information comes in qubits; if there is a “Yes” there is a “No”too. This is required by the zeros in the RHSs above.

1. Energy-momentum tensors: external and internal

Now we assemble an energy-momentum tensor (EP-tensor)with a presence at the two levels (I/E) (back trace to the action- eventually).

Definition 1.1. The “classical” external energy-momentumtensor is:

TE = (t, q|q, V ), (10)

and the internal energy-momentum tensor is:

TI = (S, A|A, J). (11)

where t, q, V and S, A, J are external and respectively internalvariables of a typical object Q of a Feynman category C (to beexplained later).

The conjugate variables are E, p, P and respectively T, a,Ω,of the space of symmetries of the object: Aut(Q).

The corresponding conjugate EP-tensor is:

T ∗E = (E, p|p, P ), T ∗

I = (T, a|a,Ω). (12)

We will try to “produce” gravity as defined by the gravita-tional momentum a and correlation tensor Ω (to be related witha “metric”). Since we model “spacetime” by graphs, we do notjust have the EM-tensor of a point or body (including pressureetc.), but we also have edges representing interactions (“pres-sure”; “stress”?), and a corresponding EM-tensor controllingthe internal dynamics (temperature etc.).

Now the First Law of Hyper-Dynamics, expressing the con-servation of energy-matter-information is:

Tr(T ∗dT ) = dω, dω = dF + dM. (13)

1. ENERGY-MOMENTUM TENSORS: EXTERNAL AND INTERNAL177

Note that “gravity” will not be obtained as in Kaluza-Klein(ST etc.) using additional external dimensions (more roomat the classical level) but it is implemented by “moving it”to a completely another level (“internal”), exploiting the I/Eduality on top of s/t-duality. To implement gravity still as anexchange interaction, use the (postulated) I/E duality (hyper-symmetry) to allow the exchange of bosons not only between“external particles” (and internal DOFs), but also inter-types(to be clarified later on):

int : •I

!!BBB

BBBB

B

time/phase?// • a

ext : e−γ // //

I

==||||||||e− p

Note that the pair of 3D-momenta (p, a), representing the cur-rents of matter and information, are reminiscent of Maxwell’sEM fields: E,B. Their unification as the curvature obeyingMaxwell’s equations dF = 0 (Bianchi identity) and d∗F = jwill be attempted later on.

We think of T = (TE, TI) as a “bundle morphism” over theclassical EM-tensor:

O ×Aut(O)

T // O ×Aut(O)

O TE // O.

Since quite often “practice” outpaces “interpretation” 1 wewill develop the conceptual formalism as if it would be an in-terpretation of an “Abstract String Theory”.

The first crucial guess consists in interpreting the informa-tion charge Q = −I yielding the entropy (derivation) S, as(related to?) the BRST charge and think of the ghosts as play-ing a similar role to the role played by neutrinos: carrying the

1E.g. Lorentz and Einstein etc.


“missing” information (from “entropy increase” to a conserva-tion law) and being responsible of generating mass (related toHiggs field?) 2.

Also the “unification between energy-mass and informationshould be hinted by I = F +iM (E = (F,M)) where E = |I| =

(p2+M2)12 . Recall that classical energy E = Z = |Z| (partition

function) is related to probability p = Ei/Z and that entropyS = I(Out) − I(In) is an information flux, perhaps

∫

div(E)...? Also, the thermodynamic equation for energy, with thedistinction between internal and external energy, and work, is:

E − U = W − TS, F = E − U.

It is not clear at this point (nor essential) if the semi-Riemannianformalism is used (metric/norm etc.) of the quaternionic for-malism is used. We suspect that the quaternionic formalism ismore powerful and could provide the pair of above equationsas an analog of Cauchy-Riemann equations (holomorphy in thequaternionic algebraic setup - C/H∗ − algebra?; skew field /supersymmetry?).

Returning to the Einstein-Hawking equivalences (“coinci-dences”), we interprete them as “light-cone gauge constraints”to be dealt with using BRST or the “more geometric” (butequivalent) approach [H], p.130, which is deformation theoryleading to Maurer-Cartan equation etc., as explained in §5):

“proper time′′ : |(E, p)| = M, gauge : |(T, a)| = 0,

“external′′ metric : |d(t, q)| = ds, gauge : |d(S, a)| = 0.

A real EM-tensor T : R4 → R4 (symmetric) has 10 components,quaternionic 1D-bundle morphism restrict the base transforma-tion to a quaternion TE ∼= (E, p).

It is time to list the main ingredients (PCF forms) of theaction, and ... entropy definitely is part of the action!

2To be explored later; in many cases “practice” precedes interpretation(Lorentz and Einstein etc.).

1. ENERGY-MOMENTUM TENSORS: EXTERNAL AND INTERNAL179

The Legendre transform relates the Hamiltonian H = Eand Lagrangian L ([]):

L = Ldt = PCF = pdq − Edt.

The other (curvature) 1-forms are:

dF = PDV − TdS, dM = ΩdJ + adA.

Recall that the “light-cone gauge allows to trade TdS and adA.Also relativistic energy unifies dF and dM .

The above forms satisfy a relation yielding a (preliminary)“Chern-Simons Lagrangian/Action” is:

L = dE − Tr(T ∗dT ).

The topological WZW-term Tr(T ∗dT ) would make possible thepresence of a “information monopole”, or may be interpretingmass as a “kink in spacetime” (instantons etc.).

As mentioned before, a Chern-Simons action yields (thetruncated) Maurer-Cartan equation (vanishing curvature, i.e.local integrability condition; e.g. [Marcus], p.64) as Euler-Lagrange equations, as required by a “deformation theory”interpretation of the equations of dynamics (Master equationDω = 0 etc.):

dS = L = 0 ↔ dω + ω ∧ ω = 0.

In the geometric picture, the energy-momentum tensor corre-sponds to a connection ω (geometric quantization etc.); it istempting to look for a correspondence H = E = F + iM < − >ω ...

The possible alternative incarnation as H2 = 0, related toconformal invariance should be kept in mind (H(z)dz2 = 0?),i.e. “complex or symplectic formalism”? (Mirror symmetry:they are equivalent):

Exercise: Does the corresponding (CS) action S built withthe above PCFs (pieces of the CS-Lagrangian 1-form) S =


∫

“Ldt′′ yield the above EM-tensors?

T =δSδg.

Recall also that dS ∼ dt ≥ 0, “time and entropy point”in the same direction (information goes from the system to theobserver: In→ Out). The proportionality is reminiscent of La-grange multipliers (The correspondence between probabilitiesand energy maximizes entropy).

2. More clues ...

Conformal theories in two dimensions seem to be special.This is consistent with st-duality (generalized Wick rotation),and especially with the dynamical qubit as being the funda-mental building block of the theory.

Once spacetime hypersymmetry is broken, a quaternionicapproach “E = E + p ” (or rather Clifford algebra approach:E, p, t, q and T, a, S, A span 16 = 24 real dimensions!) seems tobe appropriate, (and corresponding with the semi-Riemannianpicture).

Indeed, there are too many dualities between the funda-mental variables:

External InternalMomentum

Current pE = (E, p) qI = (T, a)Internal

CoordinatesExternal

Coordinates qE = (t, q) pI = (S, A)Info

CurrentThe External-Internal duality should correspond to a bi-

complex structure (δ = d∗ etc.).Recall that the (4D) energy-momentum tensor T is the flux

of a (4D) current, expressing a certain duality between “space-time” and internal DOFs:

flux =

∫

∂M

T =< ∂M, T > .

2. MORE CLUES ... 181

The external momentum current (E, p) producing the momen-tum flow TE, are conjugate to the external coordinates (t, q),which seem to be dual to the entropy internal current (S, A)producing the internal flow (energy-momentum tensor TI) rel-ative to the coordinates (T, a) dual to (E, p). Then a “Leibnitzrule” will involve 1-forms (total differentials) like:

d? = dtE + TdS, dqp+ adA.

The 4D-vector valued 1-form is (above notation):

dqE pE + qI dpI = d(qEpE + qIpI) = d(qp),

which is prone for interpretations:

q = (qI, qE), p = (pE,pI),

or better, in view of a Master Equation, should be comparedwith:

d||q||2, ||q||∗ = q · q∗, q∗ = p.

Recall that:

Tr(T ∗EdTE) = pdq−Edt+PdV, Tr(T ∗

I dTI) = adA−TdS+ΩdJ.

It exhibits a “breaking of symmetry”, which, when restored,should be:

T ∗EdTE + T ∗

I dTI = dSE + dF + dM

= (−Edt+ pdq) + (−TdS + PdV ) + (adA+ ΩdJ)

= pdq + adA− (Edt+ TdS)+ (dV P + Ω dJ).Here SE denotes the external action (free energy term?), dF orrather “dU” is the analog internal energy term (work term?)and dM appears associated with the internal DOFs, includingan angular momentum term.

The “correct order” when merging internal and externalDOFs, appears to be:

dS = “LEdt+ LIdS′′ =

(−Edt+ pdq + PdV ) + (−TdS + adA+ ΩdJ) =


−Edt− TdS + pdq + adA+ PdV + ΩdJ3.

Now the graded 1-form (vector and tensor valued):

ω =1

2d||T ||2 = ReTr(T †T ) = T ∗

EdTE + T ∗I dTI = ω1 + ω2

4

is a nice candidate as a WZW-term for a Chern-Simons action,which may generate interesting physics under “dimensional re-duction” ([GIJP]; see later on).

Note the satisfied “light-cone gauge” constraints:

T = ||a||, S = ||A||, ||(E, p)||= m = M0

⇒ TdS = adA, etc.

as well as Lagrangian submanifolds constraints:

p = Eq, ⇔ pdt−Edq = 0.

The “internal analog” is:

a = TdA

dS, ⇔ adS − TdA = 0,

which is satisfied under the “light-cone gauge” conditions fromabove.

2.1. ... and an other possibility. On the other handthe two original 1-forms dF and dM suggest a complex setup(towards a CR-equations, Calabi-Yau etc.):

Re : dF = PdV − TdS, Im : dM = adA+ ΩdJ...?

In view of the duality between I and E DOFs, there should be a“Lagrangian submanifold” constraint providing a basis for thenew principle of equivalence between energy and information:

Edt+ kTdS = 0, (14)

3Corresponds to the internal and external momentum and informa-tion currents together with angular momentum tensors associated to theinternal and external symmetries: translational and rotational.

4Tr(T †T ) should be real somehow ...

4. CURRENTS IN STRING THEORY 183

where k denoted Boltzmann’s constant, having the meaningthat the “energy per interaction of a DOF” Et/kT (quantumjump”) is equivalent to information transfer (flow) 5.

3. Quantizing I/E DOFs

Speaking of quantization, the “internal DOFs” analog ofthe Dirac quantization based on a Poisson bracket a, A = 1(analog of q, p = 1) should be a = 1

i∂∂A . The derivation of the

Schrodinger equation in the “combined picture” (I/E) shouldbe compared with the Bohmian interpretation in terms of apolar representation of the wave function (later on).

3.1. The string field and entropy. A parallel betweenString Theory and Entropy field (entropy as the flux of the in-formation current etc.) might be rewarding. To start with, con-sider a RS as the analog of the corolla: the system’s I/E DOFs“branch” (one string into many, or splitting internal states intoa partition etc.); then follow the “quantization of a string” [H],p.130.

4. Currents in String Theory

The basic ideas of QFT (and alike theories) can be expressesin different languages, as in many other cases, for example ingeometric language (geometric quantization, vector potentialsas connections etc.) or algebraic language (VOAs, TQFTs,A/L-infinity algebras/categories etc.). A bridge is always there:Yoneda Lemma which allows to interpret (visualize) a repre-sentable functor as an irreducible connection 6.

The geometric picture (physicists’ gauge theory) is widespread; to provide a few links with deformation theory, we com-ment on [AW].

5“Quantization of energy/information”.6Grad school observation.


The article starts from the (natural) interpretation of thepartition function as a norm (as noticed earlier §4), except inthe context of gauge theory the finite index set must be re-placed by a line bundle (etc.). Then the energy-momentumtensor (EP-tensor) is the flat connection providing the paralleltransport implemented by the partition function as a section.As expected, the flatness condition (local integrability) is ex-pressed by Maurer-Cartan equation.

The flatness condition is proposed as a fundamental dynam-ical equation (p.77); this is natural, since MC-equation appearsas EL-equation of a CS-action (see §[H]):

CS −Action

FPIuukkkkkkkkkkkkkkkV ariation

((RRRRRRRRRRRRR

Partition function ↔ EP − Tensor.

The result is established by deriving the partition function from(a collection of) EP-tensors. The flux of the generalized currentj(A) associated with the background fields (p.79):

ω =

∫

X

j[A]dA

satisfies MC-equation (deformation equation):

Ω = dω + ω ∧ ω = 0 ↔ D2 = 0,

where D = d + ω, which is the dynamical equation of a CS-action (see §[H]).

Indeed, if Z[A] = Z0e−ωA is the solution of dZ = ωZ (p.78),

then differentiating one obtains the MC-equation for ω.The relation between the “standard definition of the energy

momentum tensor density” (p.81):

d logZ[g] =1

2

∫

T abdgab

and the entropy flux will be considered elsewhere. In the con-text of [AW] (p.81), the (complex) connection is related to the

5. GRAVITY AND ELECTROMAGNETISM: THE REVIVAL OF AN OLD STORY!185

partition function by:

ω = −∂Z, ω = −∂ logZ,

where d = ∂ + ∂ splits.As mentioned before, if d is rather thought of as the per-

turbation (i.e. satisfying MC-equation) in the symmetric defor-mation theory picture, then

d2 = ∂∂ + ∂∂ = [d, ∂] = add(∂),

i.e. we have a double complex modulo homotopy.The condition of being a complex corresponds to d2 = 0,

and in the context of [AW], is equivalent to:

∂ω + ∂ω = 0 (dω = 0)

i.e. with the assumption that ω ∧ ω = 0 (ω is related with Zas above).

As an example consider Quillen’s choice (p.87) given by:

ω0 = Tr(D−1∂D).

The correspondence with the algebraic picture is via the inter-pretation of the Chern class of the line bundle as the centralcharge of the Virasoro algebra (central extension etc.). ¿Fromthe physical point of view, the topological WZW-term imple-ments “topological particles” (monopoles etc.).

5. Gravity and electromagnetism: the revival of anold story!

It seems that gravity and electromagnetism can be unifiedafter all (Einstein, Kaluza-Klein etc. were almost right), ex-cept the external space must be “enlarged” to include internalspace too! In other words, DWT with its duality between ex-ternal and internal space (DOFs; QDR etc.) may be thoughtof as a “super” Kaluza-Klein theory (or rather “hyper” ST,since “spacetime” is replaced by the Feynman Causal struc-ture - paths/transitions etc.). External dynamics and Internal


dynamics (an upgrade of the Standard Model to include en-tropic effect, duality etc.) seem to lead to a pure gauge theory7 with a Chern-Simons action on the “double” of traditionalspace (E, p, S, A etc.). A reduction of the CS-action to exter-nal space, in the traditional setup (e.g. 3+1 to 2+1 dimensions[JNP]) is known to afford solutions which break the symmetryand produce topological kinks [GIJP].

It would be interesting to try to reduce gravitationalChern-Simons terms in the DWT framework (with energy-entropy du-ality (E, S), or rather momentum-entropy 4-current-4-currentinteractions) to see if gravity is produced, incorporating EMand GR. Then accounting for the other interactions (SM andGR) should be ... just hard work :-)

Indeed, as mentioned before, there is an alternative regard-ing the interpretation of the entropy flux: 1) either a 4-current(S, A) with conjugate 4-current (T, a) or 2) S = |A| and T = |a|with a entropy-temperature field F = Alt(A, a).

6. The Quantum Temperature-Entropy Field Theory

In this Maxwell like interpretation (pure gauge theory withCS-action), the temperature T and the gravitational acceler-ation a could be the analog of the electric field (and energy),while the entropy S and “area” A, or rather the entropy cur-rent (divergence of probability amplitude in a corolla etc.) isthought of as the analog of the magnetic field (some kind ofcurvature, therefore suited to model gravity 8).

The entropy tensor as the flux of the information 4-current,probably corresponds to Wick hypersymmetry as an enlargedlocal gauge symmetry, dual to Weyl symmetry:

Hom(Ext× Int,C).

7Are massless two-spinors [Horvathy] ...“infons” (quanta ofinformation)?

8Speculation level “argument” ...

6. THE QUANTUM TEMPERATURE-ENTROPY FIELD THEORY 187

“Ext” represents EDOFs (say a manifold M in ST or a Feyn-man causal structure as a QDR) while “Int” refers to IDOFs(gauge theory principle bundle or Feynman-Kontsevich rulesetc.).

The external local gauge symmetries (translation and rota-tion invariance) of a Lagrangian determine the EP-tensor andangular momentum tensor. A U(1) local symmetry on the tar-get space C yields EM (current, tensor (E,B) etc.). By dualitythe complex phase may be traded between internal and exter-nal spaces, allowing to couple EM and entropy; is the resultingtheory gravity? But space is 3D!?

Due to the I/E-duality, maybe SU(2) “is” the 3D-space (i.e.the homogeneous space of SO(3)) when viewed as external, andthe local gauge symmetry of external space leads to particledynamics (momenta: (E, p) and angular)

“Hom(graphs⊗ SU(2),C) ∼= Hom(graphs, SU(2)).

As internal, it enables the U(1)-gauge symmetry (gravity/entropyas a pure U(1)-gauge topological theory).

Looking around, one finds such attempts to modify Einsteinequations by including CS-actions [JP, J](WZW-topologicalterms), but again “a la” Kaluza-Klein. It would be instruc-tive to try to “transmute” these attempts in the context ofI/E-duality (hyper-spacetime-energy symmetry, including s/t-duality).

The formula yielding the “total entropy” change (per “unitof time”):

“d/dtH ′′ : ∆S = −I(t+) + I(t−), I(p) =

∫ p

0ln zdz.

should probably be thought of as “Gauss Theorem”, includ-ing a unit information charge I(t−) (source term) besides theinformation flux term I(t+).


6.1. Is the merger of EM and GM possible? So,u = (E, p) is the 4-momentum current and (a, A) may be in-terpreted as an analog of the electromagnetic field (E,B), withentropy S = |A| associated to the information potential andT = |a| as a “curvature” (magnetic field: Hodge dual of EM-tensor/curvature F [R], p.18). The energy density of EM is

1/2(E2 +B2) =1

4F 2 −F 2

0 (∼ g + Ric)

tr(F ∧ F =1

2F 2 = B2 −E2.

The analog expression of T and S is not “OK”; may be T andS represent rotated coordinates:

TS =1√2(e+ ib)

1√2(e− ib).

The electric field is the momentum conjugate the vector po-tential A, and the scalar potential A0 appears as a Lagrangemultiplier, like β (or kT ; Hmmm ...).

As a spin 1 bosonic U(1) gauge theory with fixed spacetimeM , one gets electromagnetism (coupled with a particle withcharge). But quantum information (amplitude of probability)couples with everything ... Gravity, as an entropic effect, couldmean information as a conformal structure with action deter-mined by von Newman entropy (see §8.1) with a metric is agauge field. The electrons and ions plasma confinement CS-model of loc. cit. p.27 might be start up point (a wild guess:model for the particles and vacancies of the QDR; it supports“stable knot like solitons” loc. cit. p.33).

Now allow the topology of M to vary (graphs, RS/Galoiscovers etc.) together with the complex structure. If the “targetC is fixed”, the entropy should be invariant under local “con-formal gauge”, as well as “topological gauge” transformations(H(π) or quantum information potential S(RS), including aWZW-term), possibly introducing mass as a topological kink(central charge?).

6. THE QUANTUM TEMPERATURE-ENTROPY FIELD THEORY 189

Now the metric defines a conformal class (complex struc-ture / RS), but if SU(2) “is” space (pointwise, or rather vertex-wise), then no metric is necessary. It is rather misleading (me-chanicist point of view) and the key structures are the (dynam-ical) qubit and its symmetries SU(2) together with the causalnetwork (transmute Calabi-Yau formalism).

6.2. Variation of the action: manifold too! The dual-ity in spacetime (Stokes Th.) plays a crucial role, especially inconnection with dimensional reduction of WZW-terms [GIJP](j∗dG may count in lower dimensions; dG exact term in theLagrangian). What new phenomena does I/E-duality bring?If action is a pairing (integral) but spacetime is not fixed, noteven topologically (NOP/subsystems change due to a changein the scale in the QDR), then a variation of the action (un-der fixed boundary conditions) involves internal and externalterms:

δS = δ < M, L >=< dintM,L > + < M, δL > .

In the Feynman process framework the role of “spacetime” Mis taken by a colored graph Γ. It is worth investigating the non-classical term corresponding to the variation of the path itselfas a quantum correction (e.g. Jacobi fields in semi-Riemanniangeometry [O’Neill] etc.). 9.

An exact term, under duality, is not just a boundary term:

< M, dG >=< dintM + ∂M,G >=

∫

dintM+

∫

∂MG.

Exploring the current literature [JP], maybe EM and GR canbe unified, not as Einstein tried as a “pure” Kaluza-Klein the-ory, but in the context of DWT with its new principle of equiv-alence between internal and external DOFs (hyper-space-timesymmetry).

9A categorical interface to Field Theory as Feynman processes is over-due by now :-)


7. Bohmian mechanics: a “hidden” message

Broadly speaking, information (∼ energy!) can be createdand destroyed, and the probability (↔ energy) is just the “mod-ulus” of quantum information (the wave function!):

ψ = Ee−iS .

The 4-momentum flow of the wave function (classical energyand momentum) represents half of the picture; the other halfshould probably be the 4-entropy flow, after a closer inspec-tion of the Schrodinger’s equation written in polar coordinates(Bohmian mechanics [Gosson], p.29, [HCM], p.15). Then thequantum potential Q is conceivably an entropic term: TS.

Recall that the real part of the Schrodinger equation is:

∂S

dt+

1

2m(∇S)2 + V +Q = 0

with an associated conservation energy:

E =1

2mp2 + V +Q,

under the correspondence:

E = −∂S∂t, p = ∇S.

The quantum potential is given by ([HCM], p.15):

Q = − 1

2m

∆R

R.

The imaginary part is

∂P

∂t+ ∇ · (Pv) = 0,

where R2 = P is identified as the probability.This last equation is a conservation equation, with P the

probability density and Pv the probability current:

div(P, Pv) = 0,

8. HEAT TRANSFER AND ENTROPY FLUX 191

i.e. there are no “probability sources” 10. The relation betweenthe quantum potential and our probability current lnp is underinvestigation:

∆(lnp) =∆p

p− (∇p)2

|p|2 .

We should recall that the ‘’correct” state function is TS, notS: Ψ = Ee−iTS.

That energy is a “modulus” is suggested by the Boltzmanncorrespondence with the probability picture, as well as the rela-tivistic version (E2 = p2 +m2). In order to implement the gen-eralized Wick rotation (imaginary time, s/t-duality, euclidian-statistics mechanics versions etc.) perhaps one has to com-plexify the action (energy and entropy as modulus and phase)as a meromorphic function (on Calabi-Yau manifolds, perhaps...) such that the Cauchy-Riemann equations would play anessential role.

8. Heat transfer and entropy flux

Conservation of internal energy gives the classical heat equa-tion:

d

dt[

∫

edV ] = heat flux+ heat generated(sources).

Heat/temperature are the 4th component of the information4-current (the “density of information”). It is not clear if theI/O-flux through a corolla should be understood as a In/Outflux or as consisting of two terms, an information charge andan outward information flux:

∆H = −Q(t+) +Q(t−) = flux + produced entropy.

On the other hand conservation of probability:

p =

n∑

i=1

pi

10No “floating daemons” to measure/ask questions :-)


may be viewed as a conservation of informational charge, to-wards a “string theory” picture of the above equation: thecorolla represents “Faraday’s lines of forces” on a puncturedRiemann sphere with one positive charge and n negative charges(in equilibrium), and Shannon entropy is the information po-tential of the configuration.

As another (possibly helpful) analogy, consider a particle(system) breaking up into n particles (subsystems) (“mass” mis really our previous ν):

m =

n∑

1

mi.

The non-relativistic Lagrangians (internal and external) are(could be):

LE = mq2/2− 0, LI = 0− T, dS = mv2/2dt− TdS,

i.e. no external potential energy and heat has no kinetic term.Then the action is:

S =

∫

mv2/2−∫

TdS = kinetic energy+entropic potential11.

To relate with Shannon’s entropy, further represent entropyas a surface term:

Q(p) =

∫ p

0ln rdr =

∫ p

0

∫ r

1

dA

|A|and think that normalized probability p runs from [0, 2π] andr is a “polar radius”, while W (A) = 1

A is an information po-tential. Then Shannon’s entropy should be expressed in termsof this potential (analogy with gravitational potential):

Q(pi) =

∫ ∫

Σi

dA

|A|

∆H =

∫

S2

W (|A|)dA, W (|A|) =1

|A| .

11Or rather lnΨ in view of Bohmian mechanics ...?

8. HEAT TRANSFER AND ENTROPY FLUX 193

The above interpretation may be of use in connection with theABE-model for the measurement problem. The conservationof energy and information:

d/dt(total) = I/O− flux+ I/O− sources

involves a deterministic and unitary Schrodinger flow of ampli-tude of probability, as carrier of energy and information/entropy,as well as source terms (sinks: info “leaking” out of the A →B quantum interaction/communication) due to the observer’schange in entropy (Eve), and related to the “collapse of thewave function” 12.

The Hamilton-Jacobi formalism of the above internal dy-namics will be considered later on. In a sense Fourier’s lawφ = −∇U for the heat transfer implies that “heat/entropyhas no inertia” (U ∼ E ∼ kT ), which should be related withthe equivalence between the gravitational and inertial mass:mGR = Mext + Mint = Mext. The internal analog of Fourierlaw should relate “acceleration” (info current) and entropy po-tential a = −nablaW (|A|).

8.1. Is it you, String Theory? The equipartition prob-ability distribution, as a colored corolla, is reminiscent of theGalois cover f(z) = zn. It is conceivable that a potential the-ory analysis (ln z =

∫

1r , H =

∫

ln z etc.) might reveal en-tropy as the flux of the information current produced by a“Coulombian” quantum information potential with standardGreen function

G(A→ B) = 1/r, ∆G = δ

and dynamics (information flow) governed by Schrodinger dif-fusion equation (or rather relativistic version). The metric is aconvenient “devise”, and the above equation could be read the

12The other n − 1 possible outcomes branch into ... other paralleluniverses!?


other way around:

“|A− B|′′ = G(A→ B)−1.

The primary concept (besides Feynman paths, the category)would be transition amplitude “table” (in an axiomatic ap-proach: the “S-matrix”; see also §6.1).

Information theory based on the combinatorics of labeledconfigurations is in fact Galois theory (extensions/bundles andtheir symmetries). This sheds light on the role of (pointed)Riemann surfaces for String Theory. After all the WZW-termof the CS-action is a winding number ([JNP], p.1):

CS(A) ∼∫

tr(A)3 = W (g), A = g−1g (FP − ghost)

detecting the “jump distance” between the topological branches13. The fundamental structure is the bifield with its antipodalmap. On one side

∫

r = r2/2 “yields” the kinetic term (and itsinverse, the propagator etc.), while on the other side

∫

1/r =ln r yields a current etc.

As an “inverse scattering problem”, the process modeledby f(z) = zn has a partition function as a section (see §4).Measuring the wave function Ψ (also a section), “localizes” thebranch, collapsing the wave function.

The more general problem leads to meromorphic functions,Galois covers, local systems and flat connections, Fuchsian sys-tems, KZ-equations etc.

An analysis of the potential theory of the amplitudes ofprobability will be started in the context of the Bohmian inter-pretation of QM.

13The role of the Faddeev-Popov ghost will be investigated later on.

CHAPTER 13

The Bohmian mechanics interpretation

The Bohr’s rule:

probability = |amplitude|2

becomes clear as a requirement that the L1-norm of a prob-ability distribution (partition function in statistics formalism)equals the L2-norm of the complex formulation in QM. Prob-abilities can only decrease “with time” (succession of cause-effect), yet the quantum reality shows that information can benot only destroyed, but also created (interference: destructivebut also constructive!).

So, the wave function Ψ = ReiS (density of amplitude ofprobability), thought of as the quantum information potential,satisfies the Schrodinger equation and the probability density(distribution) associated to it is P = R2. Assume Ψ is a su-perposition of basic measurement states corresponding to someGalois cover of a meromorphic function f . Using a metric isa classical way to specify the potential of to events to be cor-related (interact) in some way, but ultimately the conformalstructure is the primary structure (more than topology yet lessthan geometry: information theory :-).

1. The quantum dice

‘Up or down” is the basic question in the quantum world.God plays quantum dice after all, whether we like it or not.The quantum dice is the “unit quantum configuration space”,which is the projective space of the qubit C⊕C: the Riemann

195

196 13. THE BOHMIAN MECHANICS INTERPRETATION

sphere (it has no corners, faces etc.; it is the perfect shape, asthe Greeks would agree).

A coordinate system amounts to an axis; then “Up or Down”is the basic quantum question (Shannon/von-Newman sense).

Conservation of the number of configurations (or energyetc.) under partitioning ν =

∑

i νi (actual space) correspondsto the normality condition for classical probabilities

∑

i pi = 1.This is analysis (L1-norm picture etc.). In the correspond-ing geometric picture of amplitudes, the conservation equationis Pythagoras’ theorem. More precisely, The possible tran-sitions (paths/histories) between two “effective” state spacesHom(A,B) are orthogonal (measurement basis), and the lengths/weights are the amplitudes. The description is a diagonal rep-resentation relative to an observable/observer.

The “actual size model” is probably the pointed RS picturedecorated with vertex operators. Then a String Theory with abackground independent (Shannon-von Newman) entropy CS-action (WZW-term) could be the answer of a massless bosonicpure gauge field: gravity. The Kontsevich graphs (and ho-mology) [K97] in the light of the Feynman graphs cohomol-ogy interpretation of Ionescu [Ion03, Ion04-1] will be calledthe Kontsevich-Ionescu QDR. In the context of the appropriatepairing (Hochschild DGLA or A∞-algebra etc.) can be realizedas a double complex, and quasi isomorphic as total complexwith the RS-complex (relative some differential; moduli spaceas the cohomology).

So RS are essentially configurations of quantum information(“quantum questions”:interactions/communications). Know-ing the “real” probabilities amounts to knowing the factoriza-tion Pi = ZiZi, where Zi belongs to some quantum dice (Hopffiled is better then “bifield”). Still, why can’t we “see” the

2. ENTROPY AND STRING ACTION 197

phase? If it’s just a mathematical model to implement cre-ation of quantum information (and destruction; everything isperiodic at some scale ... 1), what is its meaning? 2

RS exhibit a rigidity which comes together with the max-imum principle for analytic functions (observables), which aredetermined by their values on the boundary. This maybe thoughtof as the perfect encoding-decoding scheme for quantum com-munications (interactions). In some sense a complex structureis such a scheme, and the entropy of a RS measures the quan-tum information capacity of the quantum interaction channel. As mentioned before, a background independent action istherefore the entropy of a RS which, if there is any justice atall, should be Polyakov’s action: the area of the world sheetunder an embedding. And this is the “hidden massage” of theblack holes Area Theorem, that entropy “is” area. Our ap-proach shows that I-entropy 3 is the flux of the informationcurrent. It should be related with the von-Newman entropy(later on: f ′(z), Jacobian etc.)

2. Entropy and String action

Probability/entropy reflects a partitioning of the internalspace and time reflects a partitioning of the external space 4.We used to think that these two “realms” are totally distinct;

1The “life cycle”; “All universe is in an atom” and life, i.e. quantuminformation, starts with a roll of dice :-)

2In some sense (periodicity: “birth and death”), the quantum phase isa proper quantum time (proper time is a trademark of relativity :-) of thelocal quantum process. Each qubit/q-dice is an “elementary” Q-processorwith its own “clock”; network enough of them, and “ta-ta”: The Universe!

3It is not Shannon and not automatically von Newman; of course, it’sIonescu’s :-)

4We have a “superspace situation” here: external, “visible” space, isassociated with our classical understanding of reality, and internal, “hid-den” space, is associated with our quantum description of reality.


but eventually it is a matter of scale (resolution: zoom in andout).

I/E-duality goes beyond the generalized Wick rotation andtrades I and E DOFs; at the level of QI this is “just” a complexstructure. A quantum interaction/communication is modeledby a RS and the complex structure is a shadow of the I/E-duality.

Since we do not believe in coincidences, we state the follow-ing as a research goal worth studying.

Conjecture 2.1. The background independent String Ac-tion is S = Entropy(Riemann Surface), representing the quan-tum information capacity of the quantum communication chan-nel.

The Euler-Lagrange (Maurer-Cartan) equations maximizeentropy, i.e. the Quantum Information Capacity of the Quan-tum Channel (interaction/communication).

In some sense it is the “perfect action” Nature could thingoff: maximizing the interaction / communication efficiency.The duality between I/E includes both traditional maximiza-tion principles: minimum action principle in mechanics andmaximum entropy in thermodynamics.

Glancing at [Gosson] p.30:

HΨ = H +QΨ

confirms that the action (as wished for before) should be of theform

S = Lextdt + Lintds.

The corresponding Hamiltonian should allow a “merger” of theheat equation with Schrodinger’s equation (relativistic form), i.e. merging dynamics and thermodynamics first, and only thenquantize the theory.

2. ENTROPY AND STRING ACTION 199

A corollary regarding the measurement paradox is that al-though information “diffuses” internally and externally, the in-formation flow is conserved:

∆Total QI = flux through RS + charges at ∂RS

and deterministic “in between measurements” (rigidity: I/O-boundary determined). The measurement produces a “rota-tion” from external to internal space.

The simplest idea for implementing the I/E-duality is tomerge energies (Hamiltonians), i.e. the external and internal:

H = E + iU

Looking at [HCM] p.15 (the use of polar coordinates) sug-gests that S as a flux, is a current on the RS, say the complexplane; the “nodes” [Gosson], p.30, are the poles of informa-tion charges (punctured RS), with an associated Green function(the Hamiltonian?). If there is just a pole at 0, locally:

G(0, z) = lnΨ = ln |z − 0|+ iSloc

while globally dG = d|z|/|z|+ iα defines a closed “angle form”α.

So the Hamiltonian dynamics equation of our complex Hamil-tonian is analyticity, the wave function is the quantum potential(the internal dynamics has no kinetic term: zero mass field),and the real part (classical Hamiltonian equations) are Cauchy-Riemann equations (harmonic function with sources).

Externally, time corresponds to the “real” Hamiltonian flow:external energy. Internally, we have the entropy / quantumphase and internal energy/temperature. The entropy is an“imaginary time-direction” corresponding to the imaginary Hamil-tonian: internal energy, which comes from a quantum potentialonly (no kinetic part). An important difference between gaugetheory and DWT, is that in gauge theory the distinction ex-ternal space - internal space is definitive (although allowingtwisting) as a principle bundle. The internal symmetries andexternal symmetries (via the homogeneous space) are separated


(No go theorem: cannot “merge” Poincare and gauge groups).In DWT, with its QDR with duality approach, the hyper-symmetry between I/E-spaces allow for a “rotation” betweenexternal motion (time) and internal motion (entropy/phase).In this way the Wick rotation, imaginary temperature, s/t-duality etc. “tricks” become mandatory for understanding re-ality.

Reality is not, and will never be what it seems, since ourunderstanding has its own ... dynamics!

3. Information current revisited (II)

Whether we work in a “Hamilton-Jacobi gauge” (p and Econjugates: p = ∇E) or “Lagrangian gauge” (p and q con-jugates: p = mq) should be irrelevant, due to the algebraic-geometric correspondence: Space = Spec(Observables) (states-observables duality; Pontrjagin duality etc.):

E pt q

Then we “amend” the internal/external correspondence statedearlier with an “analytic Hamiltonian” in mind:

H : H → C5.

Internal energy U ∼ T (temperature) is I/E-dual to E anda = Mv an internal momentum is I/E-dual to p. Together j =(U, V ) is the 2D-entropy current (I/E-dual to 4D-momentum;is dH a pairing!?). The corresponding coordinates are (τ, r), i.e.the proper local time (the quantum phase) and (essentially) theprobability density. U, v should be another pair of conformalcoordinates, so why not identifying (another gauge ...) U ∼T ∼ τ (related with time via Wick rotation τ = it?): z = reiτ ∼(U, V ). Recall that “the ST practice” is to use independentcomplex variables z and z, which reduce on shell: z = x +

5WZW-like: Tr(H−1dH = Trd lnH ... is von Newman entropy?

4. AGAIN, WHY IS ENERGY MASS? 201

iy then z = x − iy (PCF form reduces to Lagrangian). TheShannon entropy H(P ) as the flux of probability

In the HJ-gauge a = ∇T (heat / internal energy current)and U and V are harmonic conjugates.

Also, if energy is complex H = E + iT (imaginary tem-perature ∼ internal energy U), we should have correspondingcomplex coordinates: the complex time is z = t + is, wheres is entropy (related); a polarization should restore our con-viction that “time” and “entropy” have similar arrows (TdScorresponds to Edt).

4. Again, why is energy mass?

Energy is mass and TS is an internal energy. Is there anobvious relation?

E2 −M20 = p2 ↔ (E −M0)(E +M0) = p2

Are mass and entropy directly related M0 = iT (or M0 = iU)?

Edt+ Tds = pdq ?

What about mass, as a coupling constant? Does mass coupleexternal and internal energy?

H = He + iMHi?

What “natural” maps from quaternions z, z to the complex fieldare there? (Use Edt− pdq and Tds ...):

H(z, z)dz = ...?

In other words, what is the “obvious” complex Hamiltonianflow?

We should look at what a Calabi-Yau manifold is, since ifa polarization (Lagrange multiplier/manifold etc.) relates timet and entropy s and harmonicity relates the real and complexcomponents of the Hamiltonian, then, out of 8 dimensions only6 are left; is it a coincidence? (Ditto!)

Regarding the number of dimensions of external space (whichis essentially irrelevant, as it is by now acknowledged), once the


manifold approach is abandoned (framework of the QDR) mir-ror symmetry should appear as an I/E-duality (the algebraicKontsevich’s version).

5. Bohmian mechanics’ Schrodinger equation

Since (x, y) and (r, φ) are coordinates conformal related, wewill interprete first Schrodinger’s equation (~ = 1):

i∂Ψ

∂t= −1/2m∆Ψ + (u+ iv)Ψ

with a complex potentialU = u+iv as a CR-equation, assumingthat the real and imaginary parts of Ψ = X + iY satisfy asystem of coupled heat equations.

Recall that the external coordinates are (t, q) with q =(x, y, z) and the internal coordinates are (s, a) 6 (previouslydenoted as S and A, with conjugate variables (T, a), since thePCF’s are Edt − pdq and TdS − adA; or should it be (S, T )since S = A and T = a ...).

Then, taking the real and imaginary parts and assumingheat equation with coupled sources are satisfied:

∂X

∂s= (1/2m∇− u)X + vY ⇒ ∂Y

∂t=∂X

∂s,

∂Y

∂s= (1/2m∇− u)Y − vX ⇒ ∂Y

∂s= −∂X

∂tso CR-equations are satisfied. The need for more independentcoordinates, reflecting the extention of observables from exter-nal to external and internal:

Space = Spectrum(Algebra of Observables)

is clear: (t, q) (H) and (s, a) (internal Hom dual to C; Hodgediamond and PCF to Lagrange reduction etc.) are consistentwith a generalized Wick rotation (a more symmetric notationis used):

Ψ(t, t, q, q) ∈ C

66 dimensions!

5. BOHMIAN MECHANICS’ SCHRODINGER EQUATION 203

Yet to understand the significance of the Schrodinger equationin the spirit of the Bohmian mechanics, we must revert to polarcoordinates (R, φ) (S ↔ φ and A ↔ R), i.e. Ψ = Reiφ, wherethe probability is P = R2 and the local proper time (entropy/“information arrow”) is φ.

We will try to rewrite the imaginary and real parts involvingthe Green function (potential; see [Conway], p.275):

lnΨ = lnR+ iφ.

A Green function is associated to a RS and a one distinguishedpoint (see Dirichlet problem etc.). To suggest the connectionwith the entropy flow and entropy determined by the boundaryvalues, we will call such a (genus 0) RS (Dirichlet region) anRS-corolla.

The real part of Schrodinger’s equation for Ψ(R,Φ) (polarcoordinates) is (equivalent to [Gosson], p.29; [HCM], p.15):

∂φ

∂t+ 1/2m(∇φ)2 + U − 1/2m (∆R)/R = 0. (15)

Substituting ln Ψ and using:

∆ lnR = ∆R/R− (∇R)2/R2, (∇ lnR)2 = (∇R)2/R2

and therefore:

∆R/R = ∆ lnR+ (∇ lnR)2,

we obtain

∂φ

∂t− 1/2m ∆ lnR+ 1/2m ((∇Φ)2 − (∇ lnR)2) = −U.

(16)

Since

(∇φ)2 − (∇R)2 = Re(lnΨ)2

we obtain:

∂φ

∂t− 1/2m(∆ lnR) + 1/2m Re(∇ lnΨ)2 + U = 0

(17)


Introducing the complex operator (parabolic case/ non-relativistic):

HC = 1/2m ∆ + i∂t,

then the equation is:

Re[HC ln Ψ + 1/2m (∇ lnΨ)2] = U.

(probably) yields a Klein-Gordon equation:

lnΨ +m(D ln Ψ)2 = U (18)

for the (infinitesimal) current A = ln Φ corresponding to theinformation flux (in view of our flux interpretation of entropyQ(p) =

∫ p0 ln zdz).

Since we must have = D2 (at least at the level of a corre-sponding Dirac equation), then the above “Klein-Gordon equa-tion” 18 is “just” a Master Equation with Matter-InformationSources 7 (U = u+ iv to include internal and external sources:information and energy-matter).

Since the above “issue” is important, namely trying to in-terprete Schrodinger’s equation from the physical point of viewwe need to revert to the alternative conformal coordinates hav-ing a physical meaning (internal entropy and proper time), a“fresh start” might help.

6. Interpreting the new “Klein-Gordon equation”

Although we will start with Schrodinger’s equation, we aimat a KG-equation (and ultimately a Dirac/ spinorial version,perhaps with a “stop” at a quaternionic form).

In contrast with the Bohmian approach which focuses onprobability (half of the “visible” picture), we focus on the quan-tum information carries by the wave function, and are lookingfor a Green function (2-point function and the correspondingequation):

G(z, z0) = ln |z − z0| + g(z0, z)

7Non-homogeneous Maurer-Cartan equation is an algebraic substitutefor the Einstein’s equation “Average Curvature=Energy-Matter Tensor”.

6. INTERPRETING THE NEW “KLEIN-GORDON EQUATION” 205

When z0 = 0, then our candidate (the infinitesimal generatorof the wave function) is (the 2-point function):

G(z) = lnΨ = ln |z|+ iΦ = lnR+ iΦ.

Therefore the real part of the Schrodinger equation 15 in termsof the infinitesimal generator G(z) is Equation 17.

In view of a “Maxwellian interpretation” of entropy/gravity,instead of the Bohmian Hamiltonian H +QΨ [Gosson], .p 30,let us try the Hamiltonian which besides a scalar potential Uincludes a vector potential A (loc. cit. p.17):

H = 1/2m(p− A)2 + U.

We have in mind A = ∇S (conjugate to entropy), towards agauge theory framework where information/entropy is a con-nection (topological / WZW-term etc.) and the curvature isrelated to mass in the spirit of GR.

Then the real part of the deformed Schrodinger equation(see H loc. cit. p.18) by including a quantum vector potential,is our above equation 16:

−∂φ∂t

+ 1/2m (A2 − (∇Φ)2) = U + ρ. (19)

where the Laplacian term for lnR corresponds to the distribu-tion of information charges:

1

2m∆ lnR = ρ.

It is tempting to rewrite it in terms of a probability potential:

∆ lnP = mρ.

The above “KG-equation” 18 can now be interpreted 8 as in-cluding a vector potential corresponding to a connection D =

8... with “confidence” :-)


D + imA 9

D2G = U . (20)

A Dirac-like equation should be “around”; a quaternionic form(before tackling a full Clifford algebra version) should involvethe “obvious” operator:

H = ∂t + h · ∇, h = (I, J, K)

where the pure quaternionic basis elements I, J, K act as acomplex structure in the plane C:

· : H → End(C).

It is still an incomplete form without a generalized Wick rota-tion, so a complexified version should be considered:

∂t 7→ ∂τ = ∂t + i∂T , ∇q 7→ ∇f = ∇q + i∇a.

Indeed G“=”E + iS under the probability-energy correspon-dence (normal distributions etc.). Note that the “natural” sym-metries between the internal and external variables and theirconjugates used above corresponds to the Hodge diamond of aCalabi-Yau manifold (...?).

To conclude, we state the “result” of the above discussion,as a goal for the future implementation (see Annex A).

Theorem 6.1. The Quantum Digital Gravity dynamics equa-tion (20) is the Master Equation (non-homogeneous Maurer-Cartan) satisfied by the quantum information potential (2-pointfunction):

D2G = Uwhere U represents the sources: energy-matter and information.

9To “pack” (∇ · ∇)G + (∇G) · (∇G) one may use a coproduct trick:

< ∇ · ∇, G > + < ∇ ∗ ∇, G >= (∇ ⋆ ∇)(G)

with ∗ the “convolution” dual to ⊗ and ⋆ = · + ∗; ... too “convoluted” :-)

8. PROBABILITIES AND AMPLITUDES 207

Proof. If the background free String Theory action is notenough, use some Chern-Simons modification of the Hilbert-Einstein action [JP, J], but on a foamy causal structure [Rovelli-1]as a QDR :-)

7. Entropy entanglement and Riemann surfaces

Asking “The Oracle” (i.e. the Web, or course) “What is theentropy of a Riemann surface”, we found that the link betweenentropy and RS is not new . That a RS is a merger of the inputsystems not just intuitively (level of pictures), and it entailsentanglement, is confirmed by the point of view of [LB] thatan interaction of pure states is a merger, the system remaining“pure”, even if the parts become mixtures (in the “effectivelanguage” of Hilbert spaces).

This confirms that a Riemann surface carries entropy, asnoticed earlier as the Riemann-corolla analogy, and leading toan interpretation of the string action.

8. Probabilities and amplitudes

The crucial issue remains: what is the relation betweenquantum information (and therefore entropy) and time. Thevarious dualities (Wick, s/t, I/E) require a complexified de-scription even at the level of spacetime (twistor program?).More precisely, the I/E-duality should be implemented as aHermitean pairing, and “hiding” EDOFs by duality into IDOFsshould correspond to a complex rotation; e.g. iq, ip become in-ternal coordinates etc.

Time is special; it is an external causal correlation. TheI/E-DOFs model, we picture internal DOFs as the IO-quantumcomputation with the IO (input/preparation and output/theresult or observation) as the external DOFs as modeled/associatedwith the observer. A crude diagram of the “universe” (the two


parts - Feynman or just the Q-computation loop 10 etc.) issuited for a “convolution framework”:

C∆→ I∗n ⊗ In

Ext⊗Int→ O∗n+1 ⊗ On+1

<,>→ C

graded by “external time” (a “Markov-Feynman” chain of pair-ings: I/Os, of the system and observer).

Here Ext and Int denote the quantum computing processes(“hardware and software”) of the corresponding system or ob-server. The model tends to favor (is appropriate for) processesexhibiting a heavy information processing (“leaving systems”)rather then “mechanical processes” (motion/change of inter-action capability with a “fixed processing frequency” internalclock - de Broglie pilot wave!). That the internal time of leavingorganisms is naturally modeled in a flexible way, is well known.

It appears that time especially, is indeed only a coordinateof the model, used for labeling purposes in a quantum com-munication (interaction). It reflects causal correlations (rela-tive to the observer), while the other correlations appear tothe observer as non-local/spacial; one may chose to “collapse”the corresponding non-locality and represent it by adjunctionas internal (imaginary indeed!). The relativistic unification ofspace and time is by now misleading (era of quantum interac-tion/communication modeling).

The need for a complete complexification of QM is now ap-parent (beyond Wick rotation etc.). If we inspect the classicalpicture versus the quantum picture:

CM QMProbabilities R Amplitudes C

Observables Real numbers Real eigenvaluesTime Energy/mass ≥ 0 Phase

10The QC-loop as a black-box processor plays a crucial role in manytheories, including Loop Quantum Gravity.


we see that the observables are kept real in QM althoughthe complex wave function contains the “real description” of thesystem. The symmetry when considering Bohr’s rule P = Z∗Z(and probability distributions, random variables etc.) as op-posed to “mechanical observables” 11 is restored if one allowsthe Quantum Observables to have complex eigenvalues (Bohm-Aharonov effect-like question: is the complex quantum po-tential (vector potential) “real”? Yes, it is part of the “realmodel”).

CM QM

Type External Internal

Statistics Probabilities R Amplitudes C

Observables Real eigenvalues Complex eigenvalues

Energy/massO∗O

Quantum InformationI/O

Then the “classical description” and “quantum description” areuniformly separated.

Classically P = Z∗Z and external space/paths correspondsto mixtures and classical logic (path can be concatenated andstacked in parallel: sequential and parallel computing). Theinternal space is “quantum” pure states, superposition withinterference etc. The classical (external) observables are thereal part of Q-observables C = Q∗Q etc.

But “How do we factorize classical observables (“quantize”)?”,this is the question.

With a totally complex framework, there are lots of possi-bilities: energy and temperature can be mixed into E+iT , timeand entropy related by a “Lagrange multiplier” t = is etc. Or

11Random variable and wave functions.


perhaps a Wick rotation relates energy time-flow and entropy-information flow (holomorphic flow) in a “Dirac quantization”paradigm:

(t, i∂t) (T, i∂S).

to implement the Heisenberg CCR as required by the categor-ical view of transitions (see quantum jumps: q and p = ∂q donot commute). Whether the Dirac spinors or quaternions areused for the implementation, remains to be seen.

The external metric (no time variable) describes externalcorrelations (angle form, PCF etc.). Time is an informationprocessing coordinate; it’s “user dependent”. The quantum in-formation flow is rather a holomorphic flow (2 real dimensions).

But these require rethinking the basics (relativity, Diracquantization etc.). And if the external causal structure is 3Dafter all (SU(2) really), but as part of a double ... we shouldstill try to understand the message encoded in the Calabi-Yau3-folds. 12

A look at [D2] (what a coincidence; there has to be fate ...)confirms the above “impression”: for a Calabi-Yau three (com-plex dimensions) fold, the quantum pre-potentials (or rather thewhole sum) seems to play the role of the infinitesimal currentof quantum information (our G = lnΨ; loc. cit. p.164):

lnZ ∼∑

g≥0

g2g−2s Fg(q).

The conifold (diffeomorphic with T ∗S3) could be related withthe quantum qubit with its three external internal and threeinternal coordinates, since the associated Riemann surface (loc.cit. p. 159): q = x2 + y2 is reminiscent of the Bohr’s rule.The connection between matrix integrals and ribbon graphsexhibits q as a “partition function” with qi/q = Ni/N = pias asymptotic probabilities of a corresponding distribution ofprobabilities.

12“Reality” is not quite what it seems ...


Now the idea of complex eigenvalues appears too (p.165):“To really make sense of these ... consider a generalized ma-trix integral, in which one does not integrate over Hermiteanmatrices, but over a particular contour in the space of complexmatrices. 13”.

The Riemann surface of p.166 is in our opinion a QC-routine, composed of two RS-corollas “B∗A” (see A → B pro-cesses in our Part II). It also suggests a flow of quantum in-formation entangling the In states, and also reminiscent of a“three-slit experiment”. The first quantum pre-potential isprobably just a Green function:

F1(q) = −1

2ln detA− 1

12D.

So, what is the entropy (as a flux) of such a RS-corolla? (andmany other questions: PCF is ydx, what it means to “select” aspace-time coordinate? Is it equivalent to split the I/E DOFs?etc.).

13I.e. “full” quantum computing!

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APPENDIX A

VIReQuEST: a V.I. Virtual Institute

The Virtual Institute for Quantum Entropy and Space-Timehad initially the role of a non-standard, permanent, grant pro-posal.

The Virtual Institute can play the role of an interface be-tween sponsors of all categories and the Focused Research Groupcoordinating the implementation and development of the Dig-ital World Theory under an “open source” research strategy(Linux project worked, why not DWT?).

Depending on the funding available (sabbatical, NSF grants,WWW support etc.) a technical version of DWT (ver. 2.0) isscheduled for the summer 2007 (on a web site near you; relativethe clicking distance :-) 1).

For more details please visit our web site http://www.virequest.comand the DWT dedicated site http://www.thedigitalworldtheory.com/.

Comments regarding this “uncensored” version and otherkind of support 2 are most welcome!

1. Mathematical-physics and top-down design

To justify the possibility for a larger team to actually de-signing the theory, not just “discovering” it, we will sketch aparallel with the design process of expert systems.

1A bookmark away: http:www.theDigitalWorldTheory.com.

2http://www.virequest.com/VIReQuest Sponsors.htm

219

220 A. VIREQUEST: A V.I. VIRTUAL INSTITUTE

Physics theories are implemented in mathematical language,to allow precise computations / processing both for computersand human mind.

The design of physical theories (“the product”) by the the-oretical physicist (designer) with help from the mathematician(the implementation specialist) is supposed to be used by theexperimentalist (“the beneficiary”).

The application analyst acts as a mediator between the ben-eficiary and executant (designer and implementation special-ist): collects info from the actual site (phenomenon), benefit-ting from how the beneficiary perceives the phenomenon (Thebeneficiary does not always know what they want in terms ofproduct capabilities ...). The application analyst after the col-lection of data phase is over elaborates a projet description.The executant (team) develops the project description into animplementation specification (“blue prints”); implementationspecialists (“programmers”) build the system (theory) accord-ing to the implementation specifications. Other phases follow(testing, back to the drawing board etc.).

A systemic approach in developing physical theories couldbe a step further in achieving several improvements: portabil-ity, longevity, better performance (prediction).

The interface between user and product (physics interface)should be “simple”, not reflecting the internal complexity of theimplementation (“buttons” and “switches”, not all the wiresvisible!). The user’s interface should be device independent.

The implementation of the user’s interface is normally notdone by the designer of the user’s (physics) interface (e.g. letmathematicians chose the right language/technology etc.). Theimplementation should just make the interface work as speci-fied.

The two levels alluded above (interface/implementation)constitute only one stratum in a complex application (e.g. ST,QFT, QG etc.). Care for “less convection” within this “strati-fied complexity” should be exercised.

2. DWT V.1 IMPLEMENTATION GOALS 221

All the above are more or less common (instinctive) prac-tice; yet a deliberate effort towards enforcing a better method-ology should pay off quickly.

The main goal of VIReQuEST (the mediator), is to “prove”that such a methodology can be applied to mathematics andphysics, when it comes to building a better theory, and thatsuch a methodology is more efficient.

The “Platonic way” is not the only way; mathematics canbe discovered/uncovered etc., but it can also be designed ac-cording to the “application’s needs”. Physics (the application)is driven by observation (conforming to reality as the ultimate“beneficiary”), but the design of a theory should conform tothe designer’s expectations and standards.

2. DWT v.1 implementation goals

A few more specific goals will be sketched bellow.

2.1. The QDR and Entropy. To convert the connectionbetween energy and entropy, an old story indeed, into a funda-mental new equivalence principle, we will tentatively claim therelation between Heisenberg CCR and entropy at the level ofthe QDR.

The QDR, say in its graph (co)homology incarnation, en-ables the creation and annihilation operators a = p + iq anda∗ = p−iq to be extended from the internal (Fock) space to theexternal space (DOFs), as the graph homology and cohomologydifferentials. A superspace structure on the QDR should allowto implement Heisenberg CCR and the Hamiltonian (Lapla-cian) ([LP], p.3 3):

Resolution : a, a† = 1 (HCCR ↔ Entropy quanta)(21)

3Forget about some constants/parameters for now ...

222 A. VIREQUEST: A V.I. VIRTUAL INSTITUTE

Laplacian : [a, a†] = H (Hamiltonian↔ Energy quanta)(22)

The entropy is related to the number of micro-states, S =k lnW , which in turn appears as the size of the symmetrygroup of the object. In the combinatorial picture, for example:W = n!/

∏

ni!, is the size of the bundle |Aut(E)| correspondingto a given partition, etc. We will think of the “total informa-tion” (rather then the average represented by H), as represent-ing a measure of the symmetry at hand: NH = ln |Aut(E)|.4

Then, creating a new “edge” (a particle-antiparticle pair)in a “space-time graph” Γ of degree N (NOP: N → N + 1),and then annihilating an edge (N + 1 → N ) will produce avariation of the symmetry groups which is different from theother scenario, when first the annihilation operator is applied(N → N − 1) and then the creation operator (N − 1 → N ).After all, the physicists say that the entropy is carried by thefield ...

This should be consistent with the role of the entropy, withsymmetry as its measure, as part of the action principle :

K(A,B) =

∫

Γ∈Hom(A,B)

eiS(Γ)

|Aut(Γ)| =

∫

Γ∈Hom(A,B)

e(NH+iS)(Γ).

The superspace approach for the QDR is also consistentwith the demand 5 of doubling the parameters (internal/external:hyperdynamics). But the “holly grail” remains the theorie’sability to explain (generate) mass (and gravity etc.) as an en-tropic effect.

4Not surprising, if recalling that a standard mechanics for generatingmass involves breaking the symmetry ...

5!? “research on demand” ... hm ... why not?

2. DWT V.1 IMPLEMENTATION GOALS 223

2.2. Blow up the Time! 6 It is not space in need of“extra-dimensions”, but rather “time”. In order to accommo-date the information flow, with its quanta the qubit, the in-ternal description of causality, i.e. sequential versus parallelquantum computing, probably needs three dimensions repre-senting the internal symmetries of the information unit: SU(2).This would allow to implement a full set of Feynman-Wick ro-tations: one particles’ time rotated into space, and further intoits anti-particles time via conjugation of the qubit’s symmetries(the quantum dot), or better using a generalized complex struc-ture allowing to look for a precise meaning for the fundamentalparticle-wave duality.

From a local space-time structure x, y, z, x, y, z (like Dar-beaux coordinates in the symplectic case), special relativityshould emerge as result of collapsing the three space-like or-thogonal dimensions, where time t2 = x2 + y2 + z2 representsthe classical external parameter labeling quantum events fromthe point of view of the observer.

The resulting super-resolution together with the I/E-dualityand the generalized complex structure, would provide enoughstructure to account for the classical symmetries: conformalsymmetry, st-duality (Feynman-Wick rotations) and super-sym-metry.

On the other hand, at the conceptual level, the classicalfundamental principles would be represented in a unified frame-work: Heisenberg’s uncertainty relations and particle-wave du-ality, space and time without the 3-1 assymetry. The new mat-ter (energy) and information (entropy) unification should beable to tame gravity, one way (as a forth interaction) or theother (as an “entropic effect”).

The (implementation) goals sketched above will be investi-gated as part of the DWT ver.2 7.

6... a small print “Evrika”!7... coming up the summer of 2007, on a web site near you :-)

Documents

ISBN: xxx-xx-xxxx-xmy.ilstu.edu/~lmiones/DWTv1.pdfAppendix A. VIReQuEST: a V.I. Virtual Institute 219 1. Mathematical-physics and top-down design 219 2. DWT v.1 implementation goals