Gerard ’t HooftGerard ’t Hooft

Santiago de CompostelaDecember 15 2008

Utrecht University

Can

be

at the Planck scale?

The Quantum Discussion The probability interpretation Pilot waves The ontological interpretation EPR paradox Bell inequality Conway – Kochen “Free Will theorem” Why all these results are important Why all these results might be wrong Local hidden variables

A model that seems to work

in his robā‘īyāt :

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

Determinism

Omar Khayyam (1048-1131)

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.

Paul Dirac’sstate vectors

E. Schrödinger’swave equation

M. Born’s probabilityinterpretation

D. Bohm’s Pilot waves

EPR paradox

( ) ( ) ( ) ( )

( ) ( )

0 : ; 0

[ , ]

A B A B

k l kli j ij

t x x p p

x p i

( ) ( ) ( ) ( )[ , ] 0k k l lA B A Bx x p p

J.S. Bell

Particles (1) and (2) are “entangled”. Let’s talk about spin rather

than position and momentum

A1

A2

B1

B2

1 11 22 2

1 0 1;

1 1 0

1 1 1 1;

1 1 1:

:

1

z x

B

A

( ) ( ) ( total )0

A BS S S

if you measure spin A, you know spin B, and they commute

z

x

A1

A2

B1

B2

1 11 22 2

1 0 1;

1 1 0

1 1 1 1;

1 1 1:

:

1

z x

B

A

1 2

1 11 22 2

0 ;

;

x z

x z

1 1 2 2 2 2z x z x

1 1 2 2 2 2z x z x

A1

A2

B1

B2

However, all σ only take values±1 , and you can’t have all fourof these entries contribute +1, sothis number should be between―2 and 2.Bell’s inequality:

1 1 2 2 2z x z x

contradicts QM

t = 0

α and β are entangled. P cannot depend on B , andQ cannot depend on A → Bell’s inequality → contradiction!

And yet no useful signal can be sent from B to P or A to Q.

A new variety of the same idea: theConway – Kochen Free Will Theorem

Consider two entangled massive spin 1particles, with total spin S = 0 :

( ) ( ) ( )

( )2 ( )2

(1) (2)

[ , ]

2

( ) 0

i j i j ia b abc c

i ia

a

a a

S S i S

S S

S S

2 2 2 2 2 2

2 2 2

2 2 2

or or

[ , ] [ , ] [ , ] 0

2

, ,

(1,1,0) (1,0,1) (0,1,1)

x z y z x y

x y z

x y z

S S S S S S

S S S

S S S

In case of spin 2 :1 1

2 2

2 2

1 12 2

1

0 , 1

1z xS S

1

10

The 4 cubes of Conway & Kochen

It is impossible to attach 0’s and 1’s to all axes at the positions of the dots, such that all orthogonaltriples of axes have exactly the (1,1,0) combination.

Source

(1) (2) 0S S

1 2

Conclude:Free Will Theorem:

If observers on the two differentsites have the free will to choose

which axes to pick, the spin valuesof the two particles cannot be

pre-determined. No “hidden variables ”

But is there “Free Will” ???

What is Free Will ??

Starting points

Suppose one assumed 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?

We imagine three scales in physics:

Pico: The Planck scale: 10-33 cmMicro: The microscopic, or atomic, scale: 10-8 cmMacro: The macroscopic scale (people, planets):

> 1 cm

At the Planck scale (pico) there are strictlydeterministic laws.

At the atomic scale (micro) everything seems chaotic.What we call atoms and molecules, are just minimaldeviations from the statistical average(deviations of only a few bits of information

on many billions!

At the macroscopic scale, these deviations seemto adopt classical behavior (“classic limit”)

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”

Cauchy line

How could a local, deterministic dynamical systemobtain a quantum field theory at large distance scales?

Example: a Cellular Automaton (CA)

x

t even( , ) ;Nf x t x t

(0,0)f (2,0)f

(1,1)f

x

t

(0,0)f (2,0)f

(1,1)f Cauchy line

The evolution rule is of the type:

2, , 1, 1 1, 1,Nt x t x t x t xf f F f f

This is time reversible:

, 2, 1, 1 1, 1,Nt x t x t x t xf f F f f

On the Cauchy surface at time t :

0, 1, 1 2, 3, 1 4,( ) ..., , , , , ,...t t t t tt f f f f f

At even time t :

0 2 4

, , 2 , 1, 1 1, 1

( 1) ... ... ( (

( , )Nx x t x t x t x t x t

t A A A t A t

A f f f F f f

At odd time t :

1 3 5

, , 2 , 1, 1 1, 1

( 1) ... ... ( (

( , )Nx x t x t x t x t x t

t B B B t B t

B f f f F f f

, , 2 , 1, 1 1, 1

( 1) ( )

( , )Nx x t x t x t x t x t

t A t

A f f f F f f

, , 2 , 1, 1 1, 1

( 2) ( 1)

( , )Nx x t x t x t x t x t

t B t

B f f f F f f

At even time t :

Then the next step :

' '

'

[ , ] 0 ; [ , ] 0 ; | ' | 0, 2, 4,...

[ , ] 0 | ' | 1x x x x

x x

A A B B x x

A B x x

only if

1 2

0 2 41

1 3 52

2

(... ...)

(... ...)

( 2) ;

;

;

x

x

x

x

iH iHiH

i iiH

i iiH

t e e e

e e e A

e e e B

A A A A

B B B B

Now write :

1 2

0 2 41

1 3 52

2

(... ...)

(... ...)

( 2) ;

;

;

x

x

x

x

iH iHiH

i iiH

i iiH

t e e e

e e e A

e e e B

A A A A

B B B B

Thus we find the Hamiltonian: H from Campbell-Baker-Hausdorff :

12

1 112 12

[ , ]

[ ,[ , ]] [[ , ], ] ...

Q R Se e e Q R S R S

R R S R S S

1 2

11 2 1 22

, , 2

2 [ , ] ... ;nx

R iH S iH Q iH

H H H i H H H

Note that is a finite-dimensional hermitean matrix, hence has a lower bound. Therefore,has a lowest energy eigenvalue: the “vacuum”.

( )xH( )xH

1 2

11 2 1 22

, , 2

2 [ , ] ... ;nx

R iH S iH Q iH

H H H i H H H

All terms obey:

'' if[ , ] 0 | ' | 'n n

x xH H x x n n

If CBH would converge, then we would have

( ) ; [ ( ), ( ')] 0 if | ' |x

H x x x x x H H H

The above may seem to be a beautiful approachtowards obtaining a local quantum field theory from a deterministic, local CA.

However, CBA does not converge, especially notwhen sandwiched between states with energydifference (2 )O

One could conjecture that our theory needs tobe accurate only at energy scales << Planck length.

What happened to Bell’s inequalities ???

“vacuum” B“vacuum” A

Rotating B is not possible without affecting vacuum B ;Rotating A is not possible without affecting vacuum A ;vacuum A and vacuum B will affect the entangledparticles α and β . The “Quantum non-locality” may merely be a property ofwhat we define to be the vacuum !

It is essential to distinguish:

The use of quantum statistics (Hilbert space)

while assuming

deterministic underlying laws.

This is not a contradiction !

í 1ý,í 4ý í 2ý í 3ý

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

Two (weakly) coupled degrees of freedom

The (perturbed) oscillator has discretizedstable orbits. This is what causes quantization.

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

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

a harmonic oscillator !!

kets

bras

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

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.

One of the questionable elements 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

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

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

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 ?

Maar hoe kan dit klassieke gedrag nu afwijkenvan de Bell-ongelijkheden ?

De Bell-ongelijkheden zijn afgeleid uit deveronderstelling dat onze huidige beschrijvingeen “werkelijkheid” betreft.

Maar de lege ruimte, het “vacuüm”, correspondeertniet met één werkelijkheid.

Het vacuüm is en lineaire superpositie vanwerkelijkheden, die wij in onze beschrijving ervanhebben ingevoerd.

Wat doet dat met de Bell-ongelijkheden ?

Dit is hoe de Bell-ongelijkheden omzeild kunnen worden

Echter, we hebben hier nog geen harde wiskundigeformulering van

Een grote moeilijkheid is de precieze wiskundigedefinitie van het vacuüm als toestand metexact omschreven correlatiefuncties.

In de QM is het vacuüm de toestand met de laagste energie mogelijk. Dat is een verstrengelde toestand.

Maar juist het energiebegrip is in deze modellenlastig te behandelen.