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Utrecht University
Can
be
at the Planck scale?
Gerard ’t Hooft
Santiago de Compostela
December 15 2008
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
Determinism
Omar Khayyam
(1048-1131)
in his robā‘īyāt :
“And the first Morning of creation wrote /
What the Last Dawn of Reckoning shall read.”
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.
M. Born’s probability
interpretation
E. Schrödinger’s
wave equation
Paul Dirac’s
state vectors
D. Bohm’s Pilot waves
EPR paradox
t  0:
[x
k
x( A)  x( B ) ;
, p ( j ) ]  i  ij 
l
(i )
[x
k
( A)
x
k
p( A)  p( B )  0
kl
, p ( A)  p ( B ) ]  0
l
( B)
l
J.S. Bell
Particles (1) and (2) are “entangled”.
Let’s talk about spin rather
than position and momentum
S ( A )  S ( B )  S ( total )  0
if you measure spin A, you know spin B, and they commute
A1
z
B1
A:
x
B2
A2
1

z  
;

 1
B : 1 
1
2
0 1 
x  

1
0


1 1 
1 1 ;  2 


1
2
 1 1
 1 1


A1
B1
A2
B2
1

A: z  
;

 1
1 1 
1
B : 1  2 
;

1 1
0 1 
x  

1
0


 1
1
2  2 
1
 x  z  1  2  0 ;
 x 1 
1
2
; z 2  
1
2
 z 1   x 1   z  2   x  2  2 2
1
1
 z 1   x 1   z  2   x  2  2 2
A1
B1
A2
B2
However, all σ only take values
±1 , and you can’t have all four
of these entries contribute +1, so
this number should be between
―2 and 2.
Bell’s inequality:
 z 1   x 1   z  2   x  2
contradicts QM
 2
α and β are entangled. P cannot depend on B , and
Q cannot depend on A → Bell’s inequality → contradiction!
t =0
And yet no useful signal can be sent from B to P
or A to Q.
A new variety of the same idea: the
Conway – Kochen Free Will Theorem
Consider two entangled massive spin 1
particles, with total spin S = 0 :
[ Sa(i ) , Sb( j ) ]  i i j  abc Sc(i )
S (i )2   Sa( i )2  2
a
(S
(1)
a
S ) 0
(2)
a
In case of spin 2 :
1

 12



2
2
Sz   0  , Sx  
1

 1 2
1
[ S x2 , S z2 ]  [ S y2 , S z2 ]  [ S x2 , S y2 ]  0
2
y
2
1
S S S 2 
2
x



1 
2
1
2
z
2
2
2
S
,
S
,
S
 x y z 
(1,1, 0) or (1, 0,1) or (0,1,1)
0
1
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 orthogonal
triples of axes have exactly the (1,1,0) combination.
1
Source
S (1)  S (2)  0
Conclude:
Free Will Theorem:
If observers on the two different
sites have the free will to choose
which axes to pick, the spin values
of the two particles cannot be
pre-determined.
No “hidden variables ”
2
But is there “Free Will” ???
What is Free Will ??
Our present models of
Naturepoints
are quantum mechanical.
Starting
Does that prove that Nature itself is quantum mechanical?
Suppose one assumed a ToE that literally determines
all events in the universe:
“Theory of
Everything”
determinism
We imagine three scales in physics:
Pico: The Planck scale: 10-33 cm
Micro: The microscopic, or atomic, scale: 10-8 cm
Macro: The macroscopic scale (people, planets):
> 1 cm
At the Planck scale (pico) there are strictly
deterministic laws.
At the atomic scale (micro) everything seems chaotic.
What we call atoms and molecules, are just minimal
deviations from the statistical average
(deviations of only a few bits of information
on many billions!
At the macroscopic scale, these deviations seem
to adopt classical behavior (“classic limit”)
The use of Hilbert Space Techniques as technical
devices for the treatment of the statistics of chaos ...
A “state” of the universe:
TOP DOWN
í x , ... , p, ..., i, ...,
A simple model universe:
, anything ... ý
í 1ý  í 2ý  í 3ý  í 1ý
BOTTOM
UP
   1“Beable”

 2 
 0 0 1
U   1 0 0 
 0 1 0


Diagonalize:
3;
P1   , P2   , P3  
1

U 
e 2i / 3

“Changeable” 
2
2


iH

e

e  2i / 3 
2
2
3
0
 23 
How could a local, deterministic dynamical system
obtain a quantum field theory at large distance scales?
Example: a Cellular Automaton (CA)
t
f ( x, t ) 
f (1,1)
f (0, 0)
f (2, 0)
N
;
x  t  even
Cauchy line
x
t
Cauchy line
f (1,1)
f (0, 0)
x
f (2, 0)
The evolution rule is of the type:
ft  2, x  ft , x 
N
F  ft 1, x 1 , ft 1, x 1 
This is time reversible:
ft , x  ft  2, x
N
F  ft 1, x 1 , ft 1, x 1 
On the Cauchy surface at time t :
 (t )  ..., f 0,t , f1,t 1 , f 2,t , f3,t 1 , f 4,t ,...
At even time t :
 (t  1)  ... A0 A2 A4 ...  (t
Ax f x ,t  f x,t  2  f x,t 
N

A  (t
F ( f x 1,t 1 , f x 1,t 1 )
At odd time t :
 (t  1)  ... B1 B3 B5 ...  (t
Bx f x ,t  f x,t  2  f x,t 
N

B  (t
F ( f x 1,t 1 , f x 1,t 1 )
 (t  1)  A  (t )
At even time t :
Ax f x ,t  f x,t  2  f x,t 
Then the next step :
N
F ( f x 1,t 1 , f x 1,t 1 )
 (t  2)  B  (t  1)
Bx f x ,t  f x ,t  2  f x ,t 
N
F ( f x 1,t 1 , f x 1,t 1 )
[ Ax , Ax ' ]  0 ; [ Bx , Bx ' ]  0 ;
[ Ax , Bx ' ]  0
e
e
iH 2
e
| x  x'| 1
 (t  2)  e2iH  eiH eiH ;
Now write :
iH1
only if
| x  x ' |  0, 2, 4,...
1
i (... A 0  A 2  A 4 ...)
e
i (... B 1  B 3  B 5 ...)
;
;
 iA x
 Ax
 iB x
 Bx
e
e
2
 (t  2)  e
e
e
iH1
iH 2
e
2iH
i (... A 0  A 2  A 4 ...)
e
i (... B 1  B 3  B 5 ...)
e
;
;
iH1 iH 2
e
 iA x
 Ax
 iB x
 Bx
e
e
;
Thus we find the Hamiltonian: H from
Campbell-Baker-Hausdorff :
eQ  e R eS  Q  R  S  12 [ R, S ]
 121 [ R,[ R, S ]]  121 [[ R, S ], S ]  ...
R  iH1 , S  iH 2 ,
Q  2iH 
2H  H1  H 2  12 i[ H1 , H 2 ]  ...   H xn ;
R  iH1 , S  iH 2 ,
Q  2iH 
2H  H1  H 2  i[ H1 , H 2 ]  ...   H ;
n
x
1
2
All terms obey:
[ H xn , H xn'' ]  0
if
| x  x'|  n  n'
If CBH would converge, then we would have
H   H ( x) ; [H ( x), H ( x ')]  0 if | x  x ' |  
x
Note that H ( x ) is a finite-dimensional hermitean
matrix, hence has a lower bound. Therefore, H ( x )
has a lowest energy eigenvalue: the “vacuum”.
The above may seem to be a beautiful approach
towards obtaining a local quantum field theory
from a deterministic, local CA.
However, CBA does not converge, especially not
when sandwiched between states with energy
difference O (2 )
One could conjecture that our theory needs to
be accurate only at energy scales << Planck length.
Rotating B is not possible without affecting vacuum B ;
to without
Bell’s inequalities
???
Rotating What
A is happened
not possible
affecting vacuum
A;
vacuum A and vacuum B will affect the entangled
particles α and β .
The “Quantum non-locality” may merely be a property of
what we define to be the vacuum !
“vacuum” A
“vacuum” B
It is essential to distinguish:
The use of quantum statistics (Hilbert space)
while assuming
deterministic underlying laws.
This is not a contradiction !
If there is info-loss, our formalism will not
change much, provided that we introduce
í 1ý,í 4ý  í 2ý  í 3ý
Two (weakly) coupled degrees of freedom
The (perturbed) oscillator has discretized
stable orbits. This is what causes quantization.
The equivalence classes have to be very large
these info - equivalence classes are very
reminiscent of local gauge equivalence classes.
It could be that that’s what
gauge equivalence classes are
H V/G
Two states could be gauge-equivalent if the
information distinguishing them gets lost.
This might also be true for the coordinate
transformations
Emergent general relativity
Consider a periodic system:
3
2
1
kets
a harmonic
oscillator !!
E=0
q(t  T )  q(t )
e
 iHT
q  q
E  2 n / T
―1
―2
―3
bras
A simple model
generating the following quantum theory for an
N dimensional vector space
of states:
d
 iH  ;
dt
 H11
H  
 H N 1
H1N 


H NN 
2 (continuous) degrees of freedom, φ and ω :
d (t )
  (t ) ,   [0, 2 ) ;
dt
d  (t )
   f ( ) f '( ); f ( )  det( H   )
dt
d (t )
  (t ) ,   [0, 2 ) ;
dt
d  (t )
   f ( ) f '( ); f ( )  det( H   )
dt
f '( )
  ein(  t )  n ( )
i t
1  e
 ( ),
  (1 ,..., N )
f ( )
d
dt
In this model, the energy
ω is a beable.
stable
fixed points
Quite generally, contradictions between
QM 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, in
particular their phases.
But the wave functions are man-made utensils
that are not ontological, just as
probability distributions.
A “classical” measuring device cannot be rotated
without affecting the wave functions of the objects
measured.
One of the questionable elements in
the usual discussions concerning Bell’s
inequalities, 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 change
the present without also changing the past.
Changing the past might well affect the correlation
functions of the physical degrees of freedom in
the present –
the phases of the wave functions, may well be
modified by the observer’s “change of mind”.
Do we have a
FREE WILL , that does not even
affect the phases?
Using this concept, physicists “prove” that deterministic theories
for QM are impossible.
The existence of this “free will” seems to be indisputable.
Citations:
R. Tumulka:
weKochen:
have to abandon
[Conway’s]
four incompatible
Conway,
free will isone
justofthat
the experimenter
can
premises.
It seems
thatany
anyone
theory
the freedom
assumption
freely
choosetotome
make
of aviolating
small number
of
invokes
a conspiracy
as unsatisfactory
...
observations
...and
thisshould
failurebe
[ofregarded
QM] to predict
is a merit rather
than a defect, since these results involve free decisions that
We should
require ahas
physical
the universe
not yettheory
made.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).
Bassi,
Ghirardi:
Needlessiftosomehow
say, the the
[theinitial
free-will
assumption]
A theory
seems
unsatisfactory
conditions
of the
mustare
beso
true,
thus B that
is free
to measure
along
any in
triple
of
universe
contrived
EPR
pairs always
know
advance
which
directions.
... experimenters will choose.
magnetic
fields the
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 to
enhance classical symmetries:
The displacement operator:
U { f x }  { f x1} ;
xU  U ( x  1)
Eigenstates:
U p, r  e i p p, r ;
0  p  2
Fractional displacement operator:
U (a)  e ia p
This is an extension of translation symmetry
Other continuous
Symmetries 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
momentum space
Rotation symmetry
Renormalization Group: how does one derive
large distance correlation features knowing the
small distance behavior?
K. Wilson
momentum space
Unsolved
problems:
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 magnificent
mathematical 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 a
mathematical problem.
As if:
We know the solution, but what EXACTLY
was the problem ?
Maar hoe kan dit klassieke gedrag nu afwijken
van de Bell-ongelijkheden ?
De Bell-ongelijkheden zijn afgeleid uit de
veronderstelling dat onze huidige beschrijving
een “werkelijkheid” betreft.
Maar de lege ruimte, het “vacuüm”, correspondeert
niet met één werkelijkheid.
Het vacuüm is en lineaire superpositie van
werkelijkheden, die wij in onze beschrijving ervan
hebben ingevoerd.
Wat doet dat met de Bell-ongelijkheden ?
Dit is hoe de Bell-ongelijkheden omzeild kunnen
worden
Echter, we hebben hier nog geen harde wiskundige
formulering van
Een grote moeilijkheid is de precieze wiskundige
definitie van het vacuüm als toestand met
exact 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 modellen
lastig te behandelen.