Gell-Mann Colloquium Singapore, February 24, 2010

Τεχνίτη Νοημοσύνη και Ρομποτική

1 Δεκ 2013 (πριν από 4 χρόνια και 5 μήνες)

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Utrecht University

Gerard ’t Hooft

Gell
-
Mann Colloquium

Singapore, February 24, 2010

prototype: (any number of space dimensions)

x
t
even
 
x t
The evolution law:

variables:

(,)

N
Fxt
Z
1 1
 
x x x
1
1

t
t
t
Margolus

rule

1 1
 
x x x
1
1

t
t
t
mod
(,1) (
( 1,
,1
)
{ }
)
( )
 

N
Fxt Fxt
Fx t
Q
(when

x

+
t

is odd)

Alternatingly
, the sites at

even

t

and the ones at

odd

t

are updated:

(,2)
  
Utt AB
even odd
( );( )
 
 
 
x x
A Ax B Bx
even odd
( );( )
 
 
 
x x
A Ax B Bx
A

and
B

are
operators.
Write them as

( ) [{ ( 1)}]
( );
1 mod
 

 
iPxQ Fx
iP
A
F F
e
e
x
N

is the

permutation operator

for the variable
F

iP
e
equal
-
time
commutators
:

 
 
 
  
iax ibx
Ax Bx
axt axt bxt bxt
e e
axt bxt x x
( ) ( )
( );( );
(,),(',) 0;(,),(',) 0
(,),(',) 0 if'1
even
odd
( )
( )
;
(,2);
.
x
x
i ax
i bx
A
Utt A
B
e
B
e

  

Write:

iH
U
e
2
What is
H

?

Use Baker
-
Campbell
-
Hausdorff
:

    
 
P Q R
R P Q PQ PPQ PQQ
PPQ Q
ee e
1 1 1
2 12 12
1
24
,
[,] [,[,]] [[,],]
[[,[,]],]
1 1 1
2 12 12
1
24
[,] [,[,]] [[,],]
[[,[,]
,
]],
P Q R
R P Q PQ PPQ PQQ
PPQ
ee e
Q
    

1 1
2 4
'
1
24
',''
( )
( ) ( ) ( ) ( ),(')
( ) ( ),('),('')
x
x
x x
H x
x ax bx i ax bx
ax bx ax bx

 
  
 
 
 
  
 
 

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

PQ PQ F R F
e e eee
  

Where F is chosen such that

Write repeated
commutators
, for instance

as:

, to find

(,) (,)
RPQ RP Q
 
3
[,[,[,[,]]]]
QPPPQ QPQ
4 2 2 2 4
3
1
60480 8
2 2
1 1
12 960
9
( 51 76 33 44 )
2 ( 8 )
(,)
Q P QPQP QP PQP Q PQ
R P QPQ Q P Q PQ
PQ
    
    

O

appears to be a perfectly local, bounded

quantum operator
, similar to the Hamilton density

operator of a QFT.

H
x
( )

   
H H
x x x x
( ),(') 0 if '
similarly: stays

outside the “light cone”: information does not

spread faster than velocity
v
=1=
c

 
H H
x x x x
if
( ),(') 0'

as an operator, is
(practically)
bounded

(from below and above), so
H

should have a

lowest
eigenstate
. This is the vacuum state of

the cellular automaton.

H
x
( )
only if one
may
terminate

the BCH series

But does the Baker
-
Campbell
-
Hausdorff

expansion converge ?

One can argue that divergence occurs when two

energy
eigenvalues

of
H

are considered that are

apart.

t
2
But does the Baker
-
Campbell
-
Hausdorff

expansion converge ?

Qu
:
time translation invariance only strictly holds

for time
tranlations

over integral multiples of
Δ
t
,

the lattice time unit. Is conservation of energy

violated by multiples of ?

t
A1:
yes, if you introduce a classical perturbation:

allow the cellular automaton to be perturbed:

Then,

acts with the beat of the lattice clock.

It only respects energy conservation modulo

.

 
H H H
x x x
( ) ( ) ( )

H

t
A2:
no, if you expand the complete Hamiltonian

H

into a
linearlized

part and an interaction

piece

. The total energy, defined by

is
exactly

conserved.

H
0
H
int

H H
0 int
Can one
resum

the BCH series ?

0
1 2
1
0
0
1 2 12
1 2
12 12 1 2 12
( )
0
12
0
( ) ( )
( )
diagonalize at
;( );
( 0);( ):
( )
( )
( )
[,[,...,[,]]...]
'
'
1
n n
n
i
n
n
n
n
t t
E E
ixC ixC
iC i A iB
d
e e e dxe C e A
d
C B C
d
E C E
d
iE E
A iB E E A
e
iBCC CA
C
C
 
 

  

 
 

  

1 2
| | 2
E E

 
Converges only if at all
t

This distinction may be of crucial importance

for the following discussion:

beable:
changeable:
B
C
 
 
 

 
 
 
 
 
 
 

 
 
 
 

superimposable:
C
 
 
 

 
 
 
 
   
   
   
   

t
= 0

α

and
β

are entangled.
P

cannot depend on
B

, and

Q

cannot depend on
A
→ Bell’s inequality

And yet no
useful signal

can be sent from
B

to
P

or
A

to
Q
.

It is essential to realize that Bell’s inequalities
refer to the
states

a system is in, whereas
our “hidden variables” are a theory for their
dynamics
.

We can always assume our system to be in
a state violating Bell’s inequalities, and evolve
it backwards in time, to conclude that

the initial state must have been a thoroughly
entangled one. The Universe

must have started out as
a highly entangled state …

or rather, our
understanding of it,

But so what ?

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

G. ‘t H, arXiv:0909.3426;

P.Jizba
, H. Kleinert,
F.Scardigli
, arXiv:012.2253,

And others …

The Cellular Automaton

Prototype

Its evolution operator

Hamilton formalism

Convergence problem

QM and GR

Conclusion

1 2
1 2
1 2
1
)
12 12 12
0
(
(
1
)
''
iEE
ixE ixE
iE E
e
dxe e A
C C

 