Download Document

Utrecht University
and
Gerard ’t Hooft
Isaac Newton Institute, December 15, 2004
Deterministic Quantum
Mechanics
Conventional Quantum
Mechanics
The rules for physical calculations are identical
There is a preferred
basis
All choices of basis
are equivalent
Locality can only be
understood in this
basis
Locality applies to
commutators outside
the light cone
Ontological
equivalence classes
Gauge equivalence
classes of states
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


Diagonalize:
, anything ... ý
í 1ý  í 2ý  í 3ý  í 1ý
  1   2  3 ;
P1   , P2   , P3  
2
1

U 
e 2i / 3


2
2


iH

e

e  2i / 3 
æ0
çç
çç
H ® ç - 2p / 3
çç
çç
+ 2p /
è
ö÷
÷
÷
÷
÷
÷
÷
÷
÷
3 ø÷
÷
(An atom in a magnetic field)
An operator that is diagonal in the primordial basis, is
a BEABLE .
In the original basis:
1



O  2  ;


3


[O (t) , O '(t')]  0
all O , all t , t '
 1



1


Other operators such as H, or: 
1


are CHANGEABLES
Deterministic evolution of
continuous degrees of freedom:
d i
i
q (t )  f (q )
dt
H   pi f (q )
i
i
pi 

i
qi
but, … this H is not bounded from below !
The harmonic oscillator
Theorem: its Hilbert Space is that of
a particle moving along a circle
H
?
Our assignment: Find the true beables of our world!
Beables can be identified for:
An atom in a magnetic field
Second quantized MASSLESS,
NON-INTERACTING “neutrinos”
Free scalar bosons
Free Maxwell photons
Beables for the first quantized neutrino:
r r
H = s ×p,
s i s j = i eijk s k + dij I
O (t)  { pˆ , pˆ   , pˆ  x}t ,
pi
where pˆ i   , ( pˆ x  0 )
p
r
r
r
r
r
r
x (t ) = x (0) + s t ; pˆ ×x (t ) = pˆ ×x (0) + pˆ ×s t
r
r
pˆ (t ) = pˆ (0) ,
pˆ ×s (t ) = pˆ ×s (0)


[ pˆ  x , pˆ i ]  ipˆ   pˆ i  0
p

pˆ  
 1
But, single “neutrinos” have

 
H    p , H 0 !!!

Dirac’s second quantization:
0
But a strict discussion
requires a cut-off for every
orientation of p̂ :
H
}
}
empty
full
p̂
But, how do we introduce mass?
How do we introduce interactions?
How do the “flat membranes” behave in
curved space-time ?
A key ingredient for an
ontological theory:
Information loss
Introduce equivalence classes
í 1ý,í 4ý  í 2ý  í 3ý
Neutrinos aren’t sheets ...
They are equivalence classes
There is an ontological
position x , as well as
an orientation p̂ for the
momentum.
The velocity in the p̂ direction
is c , but there is “random”, or
“Brownian” motion in the transverse
direction.
Note:
v> c
p̂
Two coupled degrees of freedom
Does dissipation help to produce a lower
bound to the Hamiltonian ?
Consider first the harmonic oscillator:
The deterministic case:
.
x = y ,
.
y = -x .
write
[x, px ] = i ,
[y, py ] = i
H  ypx  xpy
H  14 ( p x  y ) 2  14 ( p x  y ) 2  14 ( p y  x) 2  14 ( p y  x) 2 
 ( P1  Q1 )  ( P2  Q2 )  H1  H 2
1
2
Q1 
2
x  py
2
2
, P1 
2
1
2
px  y
2
2
,
Q2 
y  px
2
, P2 
py  x
2
H  H1  H 2 ,
[ H1 , H 2 ]  0
Two independent QUANTUM
harmonic oscillators!
We now impose a constraint
(caused by information loss?)
H2  
1
2
  ,
Q1  x 2 ,
y  px ,
P1  p x
x   py
2y 2 ,
H  x2  y 2 .
Important to note:
The Hamiltonian nearly coincides with the
Classical conserved quantity x 2  y 2
This oscillator has two conserved quantities:
(D = D † )
D º xpx + ypy - i
;
r 2 º x2 + y2
[H , D ] = 0 ;
Write
;
H 1,2 =
1 (1 H
4 r
± r )2 +
[H , r ] = 0 .
1 (D
4r 2
+ i )2
Alternatively, one may simply remove the last
part, and write H 2 = 0
®
H = r2
Or, more generally:
H = a r2 ,
where a
is a conserved quant it y.
Then, the operator D is no longer needed.
Compare the Hamiltonian for a (static)
black hole.
H  H I  H II We only “see” universe # I.
Information to and from universe II is lost.
We may indeed impose the constraint:
H II
is equivalent to " 12  ". H  H I only.
Let H be the Hamiltonian and U
ontological energy function.
Projecting onto states
 with H
only happen if there is information loss.
be an
 U 
But, even in a harmonic oscillator, this lock-in is
difficult to realize in a model.
The “classical quantization” of energy:
  ein ;    t
H   n 
U  kn
k 
can
This way, one can also get into grips with
the anharmonic oscillator.
Since H must obey
2 n
H 

T
where T is the period of the (classical) motion,
we get that only special orbits are allowed.
Here, information loss sets in. The special orbits
are the stable limit cycles!
If T is not independent of r , then the
allowed values En of H are not equidistant, as
in a genuine anharmonic oscillator.
The perturbed oscillator has discretized
stable orbits. This is what causes quantization.
A deterministic “universe” may show
POINCARÉ CYCLES:
Equivalence classes form pure cycles:
Gen. Relativity: time is a gauge parameter !
H E ,
Heisenberg Picture:
E  fixed  0
  fixed.
Dim(  ) = # different Poincaré cycles
For black holes, the equivalence classes are very large!
The black hole as an information processing machine
One bit of
information
on every
0. 724  10 - 65 cm2
These states
are also
equivalence
classes.
The ontological
states are in
the bulk !!
The cellular automaton
Suppose:
★ a theory of ubiquitous fluctuating variables
★ not resembling particles, or fields ...
Suppose:
★ that what we call particles and fields are
actually complicated statistical features of said
theory ...
One would expect
★ statistical features very much as in QM
(although more probably resembling Brownian
motion etc.
★ Attempts to explain the observations in
ontological terms would also fail, unless we’d
hit upon exactly the right theory ...
dobbelgod