Gerard Gerard ’’t Hooftt Hooft

Tilburg,Keynote Address

Utrecht University

the

The heuristic theory: local determinism, info-lossand quantization (discretization)

Quantum statistics: QM as a tool.

Quantum states;The deterministic Hamiltonian

The significance of the info-loss assumption:equivalence classes

Energy and Poincaré cycles

Free Will, Bell’s inequalities; the observability of non-commuting operators

Symmetries, local gauge-invariance

QM vs determinism: opposing religions?

Newton’s laws: not fully deterministic

in his robā‘īyāt :

“And the first Morning of creation wrote /What the Last Dawn of Reckoning shall read.”

Determinism

Omar Khayyam (1048-1131)

Starting points

We assume a ToE that literally determines all events

in the universe: determinism

“Theory of

Everything”

Our present models of Nature are quantum mechanical.Does that prove that Nature itself is quantum mechanical?

Newton’s laws: not fully deterministic:

t=0: x = 1.23456789012345

t=1: x = 2.41638507294163

t=2: x = 4.62810325476981

prototypeexample of a‘chaotic’ mapping:

How would a fully deterministic theory look?

t=0: x = 1.23456789012345

t=1: x = 1.02345678901234

t=2: x = 1.00234567890123

There is information loss.

Information loss can also explain

Information is not conserved

This is anecessary

assumption

Two (weakly) coupled degrees of freedom

One might imagine that there are equations of Nature that can only be solved in a statistical sense. Quantum Mechanics appears to be a magnificentmathematical scheme to do such calculations.

Example of such a system: the ISING MODEL

L. Onsager,B. Kaufman

1949

In short: QM appears to be the solution of amathematical problem. As if:

We know the solution, but what EXACTLY was the problem ?

TOP DOWN

BOTTOM UP

iH

i

i e

e

eU

3/2

3/2

1

The use of Hilbert Space Techniques as technical

devices for the treatment of the statistics of chaos ...

, ... , , ..., , ..., , anything ... x p i

í ýA “state” of the universe:

A simple model universe: í 1ý í 2ý í 3ý í 1ý

Diagonalize:

0 0 1

1 0 0

0 1 0

U2 2 2

1 2 3, , P P P

;321

23

23

0

“Beable”

“Changeable”

Quantum States

d( )

di

i

qf q

t

( , ) ( ) ( )i i iH p q p f q g q

d[ ( , ), ] [ , ] ( )

d

[ ]

ii j i j

i

qi H p q q i p q f q

t

f q

With this Hamiltonian, the quantum system

is identical to the classical system.

í 1ý,í 4ý í 2ý í 3ý

If there is info-loss, this formalism will notchange much, provided that we introduce

Consider a periodic system:

( ) ( )

2 /

iHT

q t T q t

e q q

E n T

E = 0

1

2

3

― 1

― 2

― 3

T

½T

¹/₃T

a harmonic oscillator !!

For a system that is completely isolated fromthe rest of the universe, and which is in anenergy eigenstate, is theperiodicity of its Poincaré cycle !

The quantum phase appears to be the position of this state in its Poincaré cycle.

2 /T E

The equivalence classes have to be very large

these info - equivalence classes are veryreminiscent of local gauge equivalence classes.It could be that that’s what gauge equivalence classes are

Two states could be gauge-equivalent if theinformation distinguishing them gets lost.

This might also be true for the coordinatetransformations

Emergent general relativity

/V GH

Other continuousSymmetries such as: rotation, translation, Lorentz, local gauge inv., coordinate reparametrization invariance, may emerge together with QM ...

They may be exact locally, but not a property of the underlying ToE, and not be a property of the boundary conditions of the universe

Rotation symmetry

momentum space

Renormalization Group: how does one derivelarge distance correlation features knowing the small distance behavior?

K. Wilson

momentum space

Unsolvedproblems:

Flatness problem,Hierarchy problem

A simple model

generating the following quantum theory for an N dimensional vector space of states:

2 (continuous) degrees of freedom, φ and ω :

11 1

1

;N

N NN

H Hd

iH Hdt

H H

( )( ) , [0,2 ) ;

( )( ) '( ) ; ( ) det( )

d tt

dtd t

f f f Hdt

( )( ) , [0,2 ) ;

( )( ) '( ) ; ( ) det( )

d tt

dtd t

f f f Hdt

( )f

'( )f

d

dt

stable fixed points

( ) ( )nin te

In this model, the energyω is a beable.

1

1

( ) ,

( ,..., )N

i te

Bell inequalities

And what about the

?

electron

Measuring device

vacuum

John S. Bell

Quite generally, contradictions betweenQM and determinism arise when it is assumed that an observer

may choose between non-commuting operators,to measure whatever (s)he wishes to measure,

without affecting the wave functions, inparticular their phases.

But the wave functions are man-made utensilsthat are not ontological, just as probability distributions.

A “classical” measuring device cannot be rotatedwithout affecting the wave functions of the objectsmeasured.

The most questionable element inthe usual discussions concerning Bell’sinequalities, is the assumption of

Propose to replace it with

Free Will :

“Any observer can freely choose which feature of a system he/she wishes to measure or observe.”

Is that so, in a deterministic theory ?

In a deterministic theory, one cannot changethe present without also changing the past.

Changing the past might well affect the correlationfunctions of the physical degrees of freedom inthe present – the phases of the wave functions, may well bemodified by the observer’s “change of mind”.

R. Tumulka: we have to abandon one of [Conway’s] four incompatible premises. It seems to me that any theory violating the freedom assumption invokes a conspiracy and should be regarded as unsatisfactory ...

We should require a physical theory to be non-conspirational, which means here that it can cope with arbitrary choices of the experimenters, as if they had free will (no matter whether or not there exists ``genuine" free will). A theory seems unsatisfactory if somehow the initial conditions of the universe are so contrived that EPR pairs always know in advance which magnetic fields the experimenters will choose.

Do we have a FREE WILL , that does not even affect the phases?

Citations:

Using this concept, physicists “prove” that deterministic theories for QM are impossible.

The existence of this “free will” seems to be indisputable.

Bassi, Ghirardi: Needless to say, the [the free-will assumption] must be true, thus B is free to measure along any triple of directions. ...

Conway, Kochen: free will is just that the experimenter can freely choose to make any one of a small number of observations ... this failure [of QM] to predict is a merit rather than a defect, since these results involve free decisions thatthe universe has not yet made.

General conclusions

At the Planck scale, Quantum Mechanics is not wrong, but its interpretation may have to be revised, not only for philosophical reasons, but also to enable us to construct more concise theories, recovering e.g. locality (which appears to have been lost in string theory).

The “random numbers”, inherent in the usual statistical interpretation of the wave functions, may well find their origins at the Planck scale, so that, there, we have an ontological (deterministic) mechanics

For this to work, this deterministic system must feature information loss at a vast scale

Holography: any isolated system, with fixed boundary, if left by itself for long enough time, will go into a limit cycle, with a very short period.

Energy is defined to be the inverse of that period: E = hν

Lock-in mechanism

In search for a

Lock-in mechanism

The vacuum state must be aChaotic solution

Just as in Conway’s Game of Life ...

stationary at large distance scales

“Free will” islimited by laws of physics

The ultimate religion:( Moslem ?? )

“The will of God is absolute ...

1. Any live cell with fewer than two neighbours dies, as if by loneliness.

2. Any live cell with more than three neighbours dies, as if by overcrowding.

3. Any live cell with two or three neighbours lives, unchanged, to the next generation.

4. Any dead cell with exactly three neighbours comes to life.

This allows us to introduce quantum symmetries

Example of a quantum symmetry:A 1+1 dimensional space-time lattice

with only even sites: x + t = even.

“Law of nature”:

, 1, 1 , 2, 1, 11,x t x t x t x t x t x →

↑ t

Classically, this has asymmetry:

But in quantum language, we have:

3 1, , 1, 1,,x t x t x t x t

1 1 1 1, 1 1, 1, , 1x t x t x t x t x t odd

1 0 1

1 0

x →

↑ t

;x x x

x tt t t

even

x

t

1,x t

x

t

1 1, 1, 1x t x t

1 11, 1 , 2x t x t

What about rotations and translations?

One easy way to use quantum operators toenhance classical symmetries:

The displacement operator:

1{ } { } ; ( 1)x xU f f xU U x

Eigenstates:, , ; 0 2i pU p r e p r p

Fractional displacement operator:

( ) ia pU a e

This is an extension of translation symmetry

Consider two non - interacting periodic systems:

1E

2E 1 2E E

1 1 2 2H n n

The allowed states have “kets ” with

1 11 2 1 1 2 2 1,22 2( ) ( ) , 0H E E n n n

and “bras ” with

1 11 2 1 1 2 2 1,22 2( ) ( ) , 1H E E n n n

1 2 1 2 1 20 | | | |E E E E E E

12E t

11 1 2 2 1 2 1 2 1 2 1 22 ( )( ) ( )( )E t E t E E t t E E t t

Now, and

So we also have: 1 2 1 2( ) ( )t t t t

The combined system is expected again to behave asa periodic unit, so, its energy spectrum must be some combination of series of integers:

1 2E E

0

5

4

3

2

1

-1

-2

-3

-4

-5

11 1 2 22( )nE n p p

For everythat are odd andrelative primes, wehave a series:

1 2,p p

11 1 2 22( )nE n p p

11

2

T

22

2

T

1 25, 3p p The case

121 1 2 2

2T

p p

This is the periodicity of the equivalence class:

1 1 2 2 nst odC (M 2 )p x p x

1x

2x

22

In the energy eigenstates, the equivalence classes coincidewith the points of constant phaseof the wave function.

Limit cycles

We now turn this around. Assume that our deterministic system will eventually end in a limit cyclewith period . Then is the energy (already long before the limit cycle was reached).

2 /TT

If is indeed the period of the equivalence class, then this must also be the period of the limit cycle of the system !

1 1 2 22 /( )T p p

The phase of the wave function tells us where in the limit cycle we will be.