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
→ contradiction!
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
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