Utrecht UniversityUtrecht University

Gerard ’t HooftGerard ’t Hooft

Gell-Mann ColloquiumGell-Mann ColloquiumSingapore, February 24, 2010Singapore, February 24, 2010

prototype: (any number of space dimensions)prototype: (any number of space dimensions)

x

t even x t

The evolution law:The evolution law:

variables: variables: ( , ) NF x t Z

1 1 x x x

1

1

t

t

t

Margolus ruleMargolus rule

1 1 x x x

1

1

t

t

tmod

( , 1) (

( 1,

, 1

){ }

)

( )

N

F x t F x t

F x tQ

(when (when x x + + t t is odd) is odd)

Alternatingly, the sites at Alternatingly, the sites at even even t t and the ones at and the ones atoddodd tt are updated: are updated:

( , 2) U t t A B

even odd( ) ; ( )

x xA A x B B x

even odd( ) ; ( )

x xA A x B B x

AA and and BB are are operators. operators. Write them asWrite them as

( ) [{ ( 1)}]( ) ;

1 mod

iP x Q F x

iP

A

F F

e

e

x

N is the is the permutation operatorpermutation operator for the variable for the variable FF

iPe

equal-time commutators:equal-time commutators:

ia x ib xA x B x

a x t a x t b x t b x t

e e

a x t b x t x x

( ) ( )( ) ; ( ) ;

( , ) , ( ', ) 0; ( , ) , ( ', ) 0

( , ) , ( ', ) 0 if ' 1

even

odd

( )

( )

;

( , 2) ;

.

x

x

i a x

i b x

A

U t t A

B e

B

e

Write:Write: i HU e 2What is What is HH ? ?

Use Baker-Campbell-Hausdorff:Use Baker-Campbell-Hausdorff:

P Q R

R P Q P Q P P Q P Q Q

P P Q Q

e e e1 1 12 12 12

124

,

[ , ] [ ,[ , ]] [[ , ], ]

[[ ,[ , ]], ]

1 1 12 12 12

124

[ , ] [ ,[ , ]] [[ , ], ]

[[ ,[ , ]

,

]],

P Q R

R P Q P Q P P Q P Q Q

P P Q

e e e

Q

1 12 4

'124

', ''

( )

( ) ( ) ( ) ( ), ( ' )

( ) ( ), ( ' ), ( '' )

x

x

x x

H x

x a x b x i a x b x

a x b x a x b x

H

H

Faster convergence is reached if we limit Faster convergence is reached if we limit ourselves to the conjugation class of ourselves to the conjugation class of H H ::

P Q P Q F R Fe e e e e Where F is chosen such thatWhere F is chosen such thatWrite repeated commutators, for instanceWrite repeated commutators, for instance

as:as: , to , to findfind

( , ) ( , )R P Q R P Q

3[ ,[ ,[ ,[ , ]]]]Q P P P Q QP Q

4 2 2 2 43160480 8

2 21 112 960

9

( 51 76 33 44 )

2 ( 8 )

( , )

Q P QPQP Q P PQ P Q PQ

R P QPQ Q P Q PQ

P Q

O

appears to be a perfectly local, boundedappears to be a perfectly local, boundedquantum operatorquantum operator, similar to the Hamilton density, similar to the Hamilton densityoperator of a QFT.operator of a QFT.

H x( )

H Hx x x x( ), ( ' ) 0 if '

similarly: stays similarly: stays outside the “light cone”: information does notoutside the “light cone”: information does notspread faster than velocity spread faster than velocity v v =1==1=cc

H Hx x x xif( ), ( ' ) 0 '

as an operator, is as an operator, is (practically) (practically) boundedbounded(from below and above), so (from below and above), so HH should have a should have alowest eigenstate. This is the vacuum state oflowest eigenstate. This is the vacuum state ofthe cellular automaton.the cellular automaton.

H x( )

only if one only if one may may

terminateterminate the the BCH series BCH series

But does the Baker-Campbell-HausdorffBut does the Baker-Campbell-Hausdorffexpansion converge ?expansion converge ?

One can argue that divergence occurs when twoOne can argue that divergence occurs when twoenergy eigenvalues of energy eigenvalues of HH are considered that are are considered that are apart.apart. t2

““Planck energy” ?

Planck energy” ?

But does the Baker-Campbell-HausdorffBut does the Baker-Campbell-Hausdorffexpansion converge ?expansion converge ?

Qu: Qu: time translation invariance only strictly holdstime translation invariance only strictly holdsfor time tranlations over integral multiples of for time tranlations over integral multiples of ΔΔt t ,,the lattice time unit. Is conservation of energythe lattice time unit. Is conservation of energyviolated by multiples of ? violated by multiples of ? t

A1: A1: yes, if you introduce a classical perturbation:yes, if you introduce a classical perturbation:allow the cellular automaton to be perturbed:allow the cellular automaton to be perturbed:

Then, Then, acts with the beat of the lattice clock.acts with the beat of the lattice clock.It only respects energy conservation modulo It only respects energy conservation modulo ..

H H Hx x x( ) ( ) ( )H

t

A2: A2: no, if you expand the complete Hamiltonianno, if you expand the complete HamiltonianH H into a linearlized part and an interaction into a linearlized part and an interactionpiecepiece . The total energy, defined by . The total energy, defined by

is is exactlyexactly conserved. conserved.

H0Hint

H H0 int

Can one resum the BCH series ?Can one resum the BCH series ?

0

1 2

1

0

0

1 2 12

1 212 12 1 2 12( )

0

120

( ) ( )( )

diagonalize at

; ( ) ;

( 0) ; ( ) :

( )

( )( )

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

'

'1

n nni

n

nn

n

t t

E E

ixC ixCiC i A iB de e e dxe C e A

d

C B C

dE C E

d

i E EA i B E E A

e

i B C C C A

C

C

1 2| | 2E E

Converges only if at all Converges only if at all tt

This distinction may be of crucial importanceThis distinction may be of crucial importancefor the following discussion:for the following discussion:

beable:

changeable:

B

C

superimposable:C

t = 0

αα and and ββ are entangled. are entangled. PP cannot depend on cannot depend on BB , and , andQQ cannot depend on cannot depend on A A → Bell’s inequality → Bell’s inequality → contradiction!→ contradiction!

And yet no And yet no useful signaluseful signal can be sent from can be sent from BB to to PP or or AA to to QQ..

It is essential to realize that Bell’s It is essential to realize that Bell’s inequalities refer to the inequalities refer to the statesstates a system a system is in, whereas our “hidden variables” is in, whereas our “hidden variables” are a theory for their are a theory for their dynamicsdynamics..

We can always assume our system to We can always assume our system to be in a state violating Bell’s inequalities, be in a state violating Bell’s inequalities, and evolve it backwards in time, to and evolve it backwards in time, to conclude thatconclude thatthe initial state must have been a the initial state must have been a thoroughly entangled one. The Universethoroughly entangled one. The Universe

must have started out must have started out as a highly entangled state … as a highly entangled state …

or rather, or rather, our understanding of it, our understanding of it,

But so what ?But so what ?

Our world is not quantum mechanical, Our world is not quantum mechanical, but only our perception of it …but only our perception of it …

G. ‘t H, arXiv:0909.3426;G. ‘t H, arXiv:0909.3426;P.Jizba, H. Kleinert, F.Scardigli, arXiv:012.2253,P.Jizba, H. Kleinert, F.Scardigli, arXiv:012.2253,

And others … And others …

The Cellular AutomatonThe Cellular AutomatonPrototypePrototypeIts evolution operatorIts evolution operatorHamilton formalismHamilton formalismConvergence problemConvergence problemQM and GRQM and GRConclusionConclusion

1 2

1 2

1 2

1 )

12 12 120

(

(

1)

' 'i E E

ixE ixE

i E E

edx e e AC C