Transcript Geen diatitel

Utrecht University
the
Gerard
’t Hooft
Tilburg,
Keynote Address
QM vs determinism: opposing religions?
Newton’s laws: not fully deterministic
The heuristic theory: local determinism, info-loss
and quantization (discretization)
Quantum statistics:
QM as a tool.
Quantum states;
The deterministic Hamiltonian
The significance of the info-loss assumption:
equivalence classes
Energy and Poincaré cycles
Free Will, Bell’s inequalities;
the observability
of non-commuting operators
Symmetries, local gauge-invariance
Determinism
Omar Khayyam
(1048-1131)
in his robā‘īyāt :
“And the first Morning of creation wrote /
What the Last Dawn of Reckoning shall read.”
Our present models of
Naturepoints
are quantum mechanical.
Starting
Does that prove that Nature itself is quantum mechanical?
We assume a ToE that literally determines all events
in the universe:
“Theory of
Everything”
determinism
Newton’s laws: not fully deterministic:
prototype
example of a
‘chaotic’ mapping:
t=0: x = 1.23456789012345
t=1: x = 2.41638507294163
t=2: x = 4.62810325476981
How would a fully deterministic theory look?
t=0: x = 1.23456789012345
t=1: x = 1.02345678901234
t=2: x = 1.00234567890123
There is information loss.
Information loss can also explain
Information is not conserved
This is a
necessary
assumption
Two (weakly) coupled degrees of freedom
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 ?
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  3 ;
0 0 1
U   1 0 0 
0 1 0


Diagonalize:
P1   , P2   , P3  
1

U 
e 2i / 3

“Changeable” 
2
2


iH

e

e  2i / 3 
2
2
3
0
 23 
Quantum States
d qi
 fi (q )
dt
H ( p, q)  pi fi (q )  gi (q )
d qi
  i [ H ( p, q ), qi ]   i[ p j , qi ] f j (q ) 
dt
 fi [q ]
If there is info-loss, this formalism will not
change much, provided that we introduce
í 1ý,í 4ý  í 2ý  í 3ý
Consider a periodic system:
3
¹/₃T
2
½T
1
T
E=0
q (t  T )  q (t )
e
 iHT
q  q
E  2 n / T
―1
―2
―3
a harmonic
oscillator !!
For a system that is completely isolated from
the rest of the universe, and which is in an
energy eigenstate,
is the
T  2 / E
periodicity of its Poincaré cycle !
The quantum phase  appears to be the
position of this state in its Poincaré cycle.
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
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
A simple model
generating the following quantum theory for an
N dimensional vector space
of states:
d
 iH  ;
dt
 H11
H  
 H N 1
H1 N 


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
And what about the
Bell inequalities
John S. Bell
electron
vacuum
Measuring device
?
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.
The most questionable element 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
In search for a
Lock-in mechanism
Lock-in mechanism
The vacuum state must be a
Chaotic solution
Just as in Conway’s Game of Life ...
stationary at large distance scales
“Free will” is
limited by
laws of physics
The ultimate religion:
( Moslem ?? )
“The will of God is absolute ...
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.
This allows us to introduce quantum symmetries
Example of a quantum symmetry:
A 1+1 dimensional space-time lattice
with only even sites: x + t = even.
↑
t
x →
 x,t  1,
“Law of nature”:
 x1,t 1   x,t x2,t  x1,t 1
↑
t
Classically,
this has a
symmetry:
x →
 x  x  x
  
 ;  x t
 t   t t 
even
But in quantum language, we have:
 x,t   ,
 x1,t  
3
x,t

1
x,t 1


1
x1,t

1
x 1,t
1
x,t 1
1
x 1,t
0 1 
 

1
0


1
 x t
odd
t
1
 x,t
x
t
 
1
x,t
1
x1,t 1
x

1
x1,t 1

1
x,t 2
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
Consider two non - interacting periodic systems:
E2
E1  E2
E1
H  1 n1  2 n2
The allowed states have “kets ” with
H  E1  E2  1 (n1  12 )   2 (n 2  12 ) ,
n1,2  0
and “bras ” with
H  E1  E2  1 (n1  12 )   2 (n 2  12 ) ,
E1  E2  0
Now,

| E1  E2 |  | E1  E2 |
 E   t  12
E1  t1  E2  t2 
n1,2  1
1
2
and
 (E1  E2 )(t1  t2 )  (E1  E2 )(t1  t2 )
So we also have:
 (t1  t2 )   (t1  t2 )
The combined system is expected again to behave as
a periodic unit, so, its energy spectrum must be some
combination of series of integers:
E1  E2
5
4
3
2
1
For every p1 , p2
that are odd and
relative primes, we
have a series:
0
-1
-2
-3
-4
-5
En  (n  12 )  p11  p22 
En  (n  12 )  p11  p22 
2
2
2 
T2
The case
p1  5, p2  3
2
This is the periodicity of
the equivalence class:
x2
p1 x1  p2 x2  Cnst (Mod 2 )
x1
2
T12 
p11  p22
2
1 
T1
In the energy eigenstates, the equivalence classes coincide
with the points of constant phase
of the wave function.
Limit cycles
If T  2 /( p1 1  p 2  2 ) is indeed
the period of the equivalence class,
then this must also be the period
of the limit cycle of the system !
We now turn this around. Assume
that our deterministic system will eventually end in a limit cycle
with period T . Then 2 /T is the energy (already long before
the limit cycle was reached).
The phase of the wave function tells us where in the
limit cycle we will be.