Utrecht University

Gerard ’t Hooft

and

Isaac Newton Institute, December 15, 2004

Conventional QuantumMechanics

Deterministic QuantumMechanics

All choices of basisare equivalent

There is a preferred

basis

The rules for physical calculations are identical

Locality applies tocommutators

outsidethe light cone

Locality can only be

understood in this basis

Gauge equivalence classes of states

Ontological equivalence classes

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 11 0 00 1 0

U2 2 2

1 2 3, , P P P

;321

iH

i

i ee

eU

3/2

3/2

1

02 / 3

2 / 3H p

p

æ ö÷ç ÷ç ÷ç ÷ç ÷® -ç ÷÷ç ÷ç ÷ç ÷+ ÷çè ø

(An atom in a magnetic field)

An operator that is diagonal in the primordial basis, is a BEABLE .

1[ ] 0

2 ; all all , '

3t t

(t) , '(t')

, O O O O

11

1

Other operators such as H, or:

are CHANGEABLES

In the original basis:

Deterministic evolution ofcontinuous degrees of freedom:

( ) ( )i id q t f qdt

( )ii

i

H p f q iip qi

but, … this H is not bounded from below !

The harmonic oscillatorTheorem: its Hilbert Space is that ofa particle moving along a circle

H

?

Our assignment: Find the true beables of our world!

Beables can be identified for:

An atom in a magnetic field

Second quantized MASSLESS, NON-INTERACTING “neutrinos”

Free scalar bosons

Free Maxwell photons

Beables for the first quantized neutrino:

, i j ijk k ijH p i Is s s e s d= × = +rr

ˆ ˆ ˆ{ , , } ,tp p p x (t)O

) 0ˆ ( , ˆ where xi

i pppp

ˆ ˆ ˆ( ) (0) ; ( ) (0) ˆ ˆ ˆ ˆ( ) (0) , ( ) (0)x t x t p x t p x p tp t p p t p

s ss s

= + × = × + ×= × = ×

r r r rr rr r

0 ˆ ˆ]ˆ ,ˆ[ ii p

ppipxp

1ˆp

But, single “neutrinos” have

!!! 0 ,

HpH

Dirac’s second quantization:H

0}empty

}fullBut a strict discussionrequires a cut-off for every orientation of :p̂

But, how do we introduce mass?How do we introduce interactions?How do the “flat membranes” behave in curved space-time ?

p̂

A key ingredient for an ontological theory: Information loss

Introduce equivalence classes

í 1ý,í 4ý í 2ý í 3ý

Neutrinos aren’t sheets ...They are equivalence classes

p̂There is an ontological position x , as well asan orientation for themomentum.

p̂

The velocity in the directionis c , but there is “random”, or“Brownian” motion in the transversedirection.

p̂

Note: v c>

Two coupled degrees of freedom

Does dissipation help to produce a lower bound to the Hamiltonian ?

Consider first the harmonic oscillator:The deterministic case: write

. . , . [ , ] , [ , ]x yx y y x x p i y p i= = - = =

yx xpypH

212

22

2212

12

121

2412

412

412

41

)()(

)()()()(

HHQPQP

xpxpypypH yyxx

22222121 , , , xp

PpyQypPpx

Q yxxy

0 ] , [ , 2121 HHHHH

. H , 2 2P , 2

, , 22

11

21

2

yxypxQ

pxpyH

x

yx

Important to note: The Hamiltonian nearly coincides with theClassical conserved quantity 22 yx

We now impose a constraint (caused by information loss?)

Two independent QUANTUM harmonic oscillators!

This oscillator has two conserved quantities:†

2 2 2 ; ( )

; [ , ] 0 ; [ , ] 0 .x yD xp yp i D D

x y H D Hr r

º + - =º + = =

22 21 1 11,2 4 4( ) ( )H H D ir rr= ± + +Write

Alternatively, one may simply remove the last part, and write 2

2 0 H H r= ® =

Or, more generally:2 , where is a conserved quantity.H ra a=

Then, the operator D is no longer needed.

Compare the Hamiltonian for a (static)black hole.

III HHH We only “see” universe # I.

Information to and from universe II is lost.We may indeed impose the constraint:

12 is equivalent to " ". only.II IH H H

But, even in a harmonic oscillator, this lock-in isdifficult to realize in a model.

Projecting onto states with can only happen if there is information loss.

H U

The “classical quantization” of energy:

;ine tH n

U k n

k

Let H be the Hamiltonian and U be an ontological energy function.

This way, one can also get into grips with the anharmonic oscillator.

Since H must obey T

nH 2

where T is the period of the (classical) motion,we get that only special orbits are allowed.

Here, information loss sets in. The special orbitsare the stable limit cycles!

If T is not independent of , then the allowed values of H are not equidistant, as in a genuine anharmonic oscillator.

rnE

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

A deterministic “universe” may showPOINCARÉ CYCLES:

Equivalence classes form pure cycles:

Gen. Relativity: time is a gauge parameter !

Dim( ) = # different Poincaré cycles

0 fixed , EEH

Heisenberg Picture: fixed.

For black holes, the equivalence classes are very large!

The black hole as an information processing machine

65 2

One bit ofinformationon every

cm0 724 10 -.

These statesare alsoequivalence

classes.The ontologicalstates are inthe bulk !!

The cellular automaton

Suppose: ★ a theory of ubiquitous fluctuating variables ★ not resembling particles, or fields ... Suppose: ★ that what we call particles and fields are actually complicated statistical features of said theory ...

One would expect ★ statistical features very much as in QM

(although more probably resembling Brownian motion etc.

★ Attempts to explain the observations in ontological terms would also fail, unless

we’d hit upon exactly the right theory ...

dobbelgod