Transcript Geen diatitel - Ettore Majorana

Utrecht University
Gerard ’t Hooft,
quant-ph/0604008
Erice, September 6, 2006
1
A few of my slides shown at the beginning of the lecture were accidentally erased.
Here is a summary:
Motivations for believing in a deterministic theory:
0.
1.
2.
3.
4.
Religion
“Quantum cosmology”
Locality in General Relativity and String Theory
Evidence from the Standard Model (admittedly weak).
It can be done.
“Collapse of wave function” makes no sense
QM for a closed, finite space-time makes no sense
String theory only obeys locality principles in perturbation
expansion. But only on-shell wave functions can be defined.
In the S.M., local gauge invariance may point towards the
existence of equivalence classes (see later in lecture)
The math of QM appears to allow for a deterministic
interpretation, if only some “small, technical” problems
could be overcome.
2
3
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.
4
The use of Hilbert Space Techniques as technical
devices for the treatment of the statistics of chaos ...
A “state” of the universe:
í x, ... , p, ..., i, ...,
A simple model universe:
0 0 1
U   1 0 0 
0 1 0


, anything ... ý
í 1ý  í 2ý  í 3ý  í 1ý
  1   2  3 ;
P1   , P2   , P3  
2
1

Diagonalize: U  
e 2i / 3


2
2
2
3


iH
0

e

e  2i / 3 
 23 
5
Emergent quantum mechanics in a deterministic system
d
x (t )  f (x )
dt

ˆ
p  i
x
Hˆ  pˆ  f (x )  g (x )
d
x (t )   i  x (t ) , Hˆ   f ( x )
dt
Hˆ  Hˆ †  i    f  2 Im(g )   0
The
but
Hˆ  0 ??
POSITIVITY
Problem
5a
In any periodic system, the Hamiltonian
can be written as
H p ;
e
 iHT
2

T
2 n
1  H 
n ;
T
5
4
3
2
1
0
-1
-2
-3
-4
-5
n  0,  1,  2, ...
This is the spectrum of a
harmonic oscillator !!
6
In search for a
Lock-in mechanism
7
Lock-in mechanism
8
A key ingredient for an
ontological theory:
Information loss
Introduce equivalence classes
{1},{4}  {2}  {3}
9
With (virtual) black holes, information loss
will be very large! →
Large equivalence classes !
10
Consider a periodic variable:
kets
Beable
d
 ;
dt

  [ 0, 2 ]

H   p   i
  Lz

 m ,
m  0, 1, 2,
3
2
1
0
̶1
̶2
̶3
The quantum harmonic oscillator has
only:
Changeable
H  n ,
m  0, 1, 2,
bras
1
1
This and the nect 2 slides were not shown during the
lecture but may be instructive:
Interactions can take place in two ways.
Consider two (or more) periodic variables.
1:
dqi
 i qi   fi (q );
dt
H int   f i pi
Do perturbation theory in the usual way by
computing
n H int m .
1
2
q2
0 1
2: Write  x  
 for the
1 0
hopping operator.
q1  q1 (0)  1t
q1
a
H  1 p1  2 p2  A ( x 1)  (q1  a)  (q2 )
Use
 x  1;
to derive
H
int
e
 12  i

 i ; e
t2
1  i
x
2

 i x ; e
1  i ( 1)
x
2
x
exp i  H dt  exp(i A (q2 )( x  1) 1
x
t1
 A  12   1
if 
  (q2 )  1
1
)
1
3
However, in both cases,
nH
int
m
will take values over the entire range of values
for n and m .
Positive and negative values for n and m
are mixed !
→ negative energy states cannot be projected
out !
But it can “nearly” be done! suppose we take many slits, and
average:  (q )  f (q )
2
2
Then we can choose to have the desired Fourier coefficients
for
n H int m
But this leads to decoherence !
1
4
Consider two non - interacting systems:
E2
E1  E2
E1
H  1 n1  2 n2
1
5
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 )
1
6
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
En  (n  12 )  p11  p22 
-5
1
7
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
1
8
A key ingredient for an
ontological theory:
Information loss
Introduce equivalence classes
í 1ý,í 4ý  í 2ý  í 3ý
1
9
With (virtual) black holes, information loss
will be very large! →
Large equivalence classes !
2
0
This is one
equivalence class
2
p1 and p2 are beables
(ontological quantities), to be
determined by interactions
with other variables.
x2
Note that x1 is now
equivalent with
x1  2 (k / p1 ) (Mod 2 )
x1
with y1  x1 / p1 and
1
Therefore, we should work
y2  x 2 / p 2 instead of x1 and x 2
In terms of these new variables, we have
p1  p 2  1
2
1
Some important conclusions:
An isolated system will have equivalence classes that show
periodic motion; the period ω will be a “beable”.
The Hamiltonian will be H = (n +½) ω , where n is a changeable:
it generates the evolution.
However, in combination with other systems, n plays the role of
identifying how many of the states are catapulted into
one equivalence class: x1  x1  2 /(2n  1)
If all states are assumed to be non-equivalent, then n = 0 .
The Hamiltonian for the combined system 1 and 2 is then
H 12  (n12  1 2)(1   2 )
The ω are all beables, so there is no problem keeping them
all positive. The only changeable in the Hamiltonian is the
number n for the entire universe, which may be positive (kets)
or negative (bras)
This is how the Hamiltonian “locks in” with a beable !
Positivity problem solved.
2
2
1. 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.
2
3
Gravity
H  (n  1 2) i
Since only the overall n variable is a changeable, whereas the
rest of the Hamiltonian,  i , are beables, our theory will
allow to couple the Hamiltonian to gravity such that the
gravitational field is a beable.
(Note, however, that momentum is still a changeable, and it
couples to gravity at higher orders in 1/c )
Gauge theories
The equivalence classes have so much in common with the
gauge orbits in a local gauge theory, that one might suspect
these actually to be the same, in many cases
( → Future speculation)
2
4
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 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.
We still have a problem: how to realize the information loss
process before the interactions are switched on !
2
5
This and the next slide were not shown during lecture
Claim:
Quantum mechanics is exact, yet
the Classical World exists !!
The equations of motion of the
(classical and [sub-]atomic) world are deterministic,
but involve Planck-scale variables.
The conventional classical e.o.m. are (of course)
inaccurate.
Quantum mechanics as we know it, only refers to the lowenergy domain, where the fast-oscillating Planckian
variables have been averaged over.
Atoms and electrons are not “real”
26
At the Planck scale, we presume that all
laws of physics, describing the evolution.
refer to beables only.
Going from the Planck scale to the atomic
scale, we mix the bables with changeables;
beables and changeables become
indistinguisable !!
MACROSCOPIC variables, such as the
positions of planets, are probably again
beables. They are ontological !
Atomic observables are not, in general, beables.
2
7
G. ’t Hooft
A mathematical theory for
deterministic quantum mechanics
The End
2
8