Download Nino Zanghì Dipartimento di Fisica dell`Università di Genova, INFN

TWO QUANTUM THEORIES
THAT BELL LIKED
Nino Zanghì
Dipartimento di Fisica dell'Università
di Genova, INFN sezione di Genova, Italy
[email protected]
Summer School on the Foundations of Quantum Mechanics dedicated to
John Bell,
Sesto, July 2014
John Bell, 1989
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sa
Towards An
Exact
Quantum
Mechanics
What is the problem
of
quantum mechanics?
the measurement problem?
Erwin Schrödinger
(1887 – 1961)
the deeper problem
(of which the
measurement
problem is just a
manifestation)...
1
':ili
=
Ei*;u l'*E =s
illtt1l*
t,
::i;= ;
John Stewart Bell
(1928 – 1990)
a
This is the problem
:JJ
,l::tJ
*-J*
All the variants of
Quantum Field Theory
(Cut-offs, Algebraic,
etc.)
\:t/
Non-Relativistic
Quantum Mechanics
Are there solutions of
the the problem?
Yes!
There are many,
indeed!
o
t(t=
o
!
9
V
6>;
-l
?6
(=
-#
!-3-
I
-el9
/o-'
e8;
c-a-
3I=
*lii*
2*. ; s +':'
-p_1,
=alggla'F
3r-t- leF
,"n rut.'=A;;"
*
9
3
Beables
“The terminology, be-able
as against observ-able, is
not designed to frighten
with metaphysic those
dedicated to realphysic. It
is chosen rather to
help in making
explicit some notions
already implicit in,
and basic to, ordinary
quantum theory. For, in
the words of Bohr, 'it is
decisive to recognize that,
however far the phenomena
transcend the scope of
classical physical
explanation, the account of
all evidence must be
e x p re s s e d i n c l a s s i c a l
terms'.”
A
algebra of operators on H
operators as observables
A∈A
Z
macroscopic variables
classical variables
!ψ, Aψ"
< Z >ψ =
!ψ, ψ"
“The concept of 'observable' lends itself to very
precise mathematics when identified with 'selfadjoint operator'. But physically, it is a rather
wooly concept. It is not easy to identify precisely
which physical processes are to be given status
of 'observations' and which are to be relegated
to the limbo between one observation and
another. So it could be hoped that some increase
in precision might be possible by concentration
on the beables, which can be described in
'classical terms', because they are there. The
beables must include the settings of switches and
knobs on experimental equipment, the currents in
coils , and the readings of instruments.
'Observables' must be made,
somehow, out of beables. The theory
of local beables should contain, and
give precise physical meaning to, the
algebra of local observables.”
(ψt , Z)
(ψt , X)
LOCAL BEABLES
EXACT (on all scales)
When von Bortkewitch collected
statistics on the kicking of soldiers to
death by horses, in the Prussian army, in
different years, he found a Poisson
distribution. Now, you don’t go out into
the world looking for the Poisson
distribution, you go out looking for
soldiers and horses and kicks.
BM
universal wave function
Ψ
up to the universal
scale
↑
(ψ, Z)
down to the microscopic
scale
↓
Q
universal wave function
Ψ
up to the universal
scale
↑
(ψ, Z)
down to the microscopic
scale
↓
Q
Q variables: what the
theory is fundamentally
about
for example, particles,
fields, strings, et.
expressing the laws which govern the
behavior of the Q variables in a simple
and natural way.
(Q, Ψ)
universal wave function
Ψ
up to the universal
scale
expressing the laws which govern the
behavior of the Q variables in a simple
and natural way.
↑
(Q, Ψ)
(ψ, Z)
down to the microscopic
scale
↓
Q
Q variables: what the
theory is fundamentally
about
for example, particles,
fields, strings, et.
Q = (X, Y )
X
system
Y
environment
ψ(x) = Ψ(x, Y )
Z = F (Q)
universal wave function
Ψ
up to the universal
scale
expressing the laws which govern the
behavior of the Q variables in a simple
and natural way.
↑
(Q, Ψ)
(ψ, Z)
down to the microscopic
scale
↓
Q
Q variables: what the
theory is fundamentally
about
for example, particles,
fields, strings, et.
Q = (X, Y )
X
system
Y
environment
ψ(x) = Ψ(x, Y )
Z = F (Q)
!ψ, Aψ"
Z −→ A < Z >ψ = !ψ, ψ"
universal wave function
Ψ
up to the universal
scale
expressing the laws which govern the
behavior of the Q variables in a simple
and natural way.
↑
(Q, Ψ)
(ψ, Z)
down to the microscopic
scale
↓
Q = (X, Y )
Q
X
system
Q variables: what the
theory is fundamentally
about
for example, particles,
fields, strings, et.
Y
environment
|Ψ|
2
ψ(x) = Ψ(x, Y )
Z = F (Q)
!ψ, Aψ"
Z −→ A < Z >ψ = !ψ, ψ"
plug the actual configuration Y of the
environment into the second slot of Ψ (x, y)
to obtain a function of x,
ψ(x) = Ψ (x, Y )
(Almost) all implications of BM follow
from this formula
2
PΨ (X ∈ dx | Y ) = |ψ(x)| dx
GRW
ψ !→ ψ
= set {1, . . . , N}. with uniform distribution.
(6)
is chosen at random
in the
Th
Mechanics
1/2 ψ #
T12 Bohmian
T1 +
#Λ
(X
)
1/2
I
1
T1 succinctly su
experiments
that
may
distinguish
this
theory
1We 3shall
#ΛI1 (Xin
ψT1 # space.
ation,
a
stochastic
jump
process
this
process
as
follows.
1 ) Hilbert
1/2
3
chosen
randomly
with
probability
distribution
Λ
(X
)
ψ
I
1
T
the
collapse
X
is
chosen
randomly
with
probability
distribution
1
1
[6].
According
to
the
way
in
which
this
theory
1
ψ
→
!
ψ
=
. uniform distribution
(
Consider
a
quantum
system
described
(in
the
standard
language)
by
an
N-“p
process
as
follows.
T1 in the
T1 + set {1, . . . , N}
I
is
chosen
at
random
with
1/2
1
3
Ghirardi,
Rimini,
and
Weber
sen at
random
in theinstead
set {1, of
. . .Schrödinger’s
, N } with uniform#Λdistribution.
ψT1 # The center
I1 (X1 )
wave
function
follows,
3
3
3
wave
function
ψ
=
ψ(q
,
...,
q
),
q
∈
R
,
i
=
1,
.
.
.
,
N;
for
any
point
x
in
R
(the
3
1/2
2
of
the
collapse
X
is
chosen
randomly
with
probability
distribution
1
N
i
Consider
a
quantum
system
described
(in
the
standard
language)
by
an
N-“
1/2
2. . .
1 probability
ollapse
X
is
chosen
randomly
with
distribution
3.1
GRWm
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
|ψ
,
I
=
i
)
%ψ
|Λ
(x
)ψ
&
dx
#Λ
(x
)
ψ
#
dx
.
(7)
P(X
∈
dx
|ψ
,
I
=
i
)
%ψ
|Λ
(x
)ψ
&
dx
#Λ
(x
)
ψ
#
dxcen
Hilbert
space.
We
shall
succinctly
summarize
T1 1 I11 is1 chosen
i11 1 in
T1 set
1i1. . . ,1iN}
1
1 T1at
T1random
1 the
T1 {1,
1 T1i1distribution.
1 1
T1 The
1.
1 T
1
with
uniform
3
3 sy
the collapse
that,will
be
defined
next),
define
on
the
Hilbert
space
of
the
e of
function
ψ
=
ψ(q
...,
q
),
q
∈
R
,
i
=
1,
.
.
.
,
N;
for
any
point
x
in
R
(the
3.2
GRWf
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. #. 2
3
1
N
i
1/2
1/2
2
of
the
collapse
X
is
chosen
randomly
with
probability
distribution
P(X
∈
dx
|ψ
,
I
=
i
)
%ψ
|Λ
(x
)ψ
&
dx
#Λ
(x
)
ψ
1
1 i (xT11) i1ψT 1# dx
T1 1 . 1
i1 1
T1
P(X
dx
#Λ
(7)
1 ∈ dxoperator
1 |ψT1 , I1 = i1 ) %ψT11|Λi1 (x11 )ψTT11 & 1
collapse
1
1
1 1
he
collapse
that
will
be
next),
define
on
the
Hilbert
space
of
the
sy
(in
the
standard
language)
by defined
anEmpirical
N-“particle”
3.3
Equivalence
Between
GRWm
and
GRWf
2
" −x)
(
Q
1
i
en
the
algorithm
is
iterated:
ψ
evolves
unitarily
until
a
random
tim
1/2
2 T
T
+
−
1
is
iterated:
ψ
evolves
unitarily
until
a
random
time
=
3
2
−7
P(X
dxR1 |ψ(the
, Ii“center”
= i=
dx1 #Λ, i1 (x1σ
) ∼ψ10
.
(
T
T1Λ
1(x)
1 ) %ψT1 |Λi1 (x1 )ψ
T1 & 2σ
T1 # dx
12
1 +1x∈in
e
= 1, . .operator
. , N; for3.any
point
m
apse
2until
3/2
Then is
theaψTalgorithm
is
iterated:
ψ
evolves
unitarily
until
a
random
T
+
he
algorithm
is
iterated:
evolves
unitarily
a
random
time
T
=
1
(2πσ
)
+
2
+∆T
,
where
∆T
random
time
(independent
of
∆T
)
distributed
ac
1
2
2
2
1
"
(
Q
−x)
define
on
the
Hilbert
space
of
the
system
the
4 Primitive
Ontology
1
i
a
random
time
(independent
of
∆T
)
distributed
according
−
2 , is
1
T
+∆T
,
where
∆T
is
a
random
time
(independent
of
∆T1 ) distribut
2
1 algorithm
2 timeΛ
2=
where
∆T
is
a
random
(independent
of
∆T
)
distributed
according
2σ
2the
2
1
(x)
e
,
3.
Then
the
is
iterated:
ψ
evolves
unitarily
until
a
random
time T2
i with
T1 +
"i is the position
exponential
distribution
rate
Nλ,
and
so
on.
2
3/2
where
Q
operator
of
“particle”
i.
Here
σ
is
a
new
constant
of
4.1
Primitive
Ontology
and
Physical
Equivalence
.
.
.
(2πσ
)
2
to
the
exponential
distribution
with
rate
Nλ,
and
so
on.
"
distribution
with
rate
Nλ,
and
so
on.
(Q −x)
xponential
distribution
with
rate
λ,
andtime
so on.
T2 1−7
+∆T
∆T
is the
aNrandom
(independent
of ∆T1 ) distributed accordi
1
2 , where
2is
− ithe
evolution
of
ψ
Schrödinger
evolution
2σ
order
of
10 , m.
e
(4)
4.2
Symmetry
Properties
and
Relativity
.
.
.
.
.
.
.
2
3/2
to
the
exponential
distribution
with
rate
Nλ,
and
so
on.
σre) Q
"words,
her
the
evolution
of the
waveoffunction
is
the
evolu
interrupted
by
collapses
is the
position
operator
of “particle”
Here
σ
is Schrödinger
a isnew
ofe
In
other
words,
the
evolution
thei.wave
function
the constant
Schrödinger
•
•
•
words,iLet
the ψevolution
the
wave
function
is i.e.,
the Schrödinger
evolution
in-. . . . at
initial
wave
function,
the
normalized
wave
function
so
t0 be the of
4.3
Without
Primitive
Ontology
.
.
.
.
.
.
.
.
volution
of
the
wave
function
is
the
Schrödinger
evolution
in−7
d
by
collapses.
When
the
wave
function
is
ψ
a
collapse
with
center
x
a
ticle”
i.
Here
σ
is
a
new
constant
of
nature
of
terrupted
by
collapses.
When
the
wave
function
isthe
ψxSchrödinger
afollowing
collapse
with
center
In
other
words,
the
evolution
of
the
wave
function
is
evolution
er
10
m.
collapses.
When
the
wave
function
is
ψ
a
collapse
with
center
and
label
•
tof
arbitrarily
chosen
as
initial
time.
Then
ψ
evolves
in
the
way:
0
by
collapses.
When
the
wave
function
is
ψand
a collapse
with
center
xlabel
and
lab
When
the
function
is
ψ
a
collapse
with
center
x
and
iwave
occurs
at
rate
te
at rate
Let
ψtterrupted
be
the
initial
wave
function,
i.e.,
the
normalized
wave
function
at
s
5
Differences
between
BM
GRW
0
e., the1.inormalized
wave
function
at
some
time
occurs
at rate
It
evolves
unitarily,
according
to
Schrödinger’s
equation,
until
a
rand
r(x,
i|ψ)
=
λ
%ψ
|
Λ
(x)ψ&
r(x,
i|ψ)
=
λ
%ψ
|
Λ
(x)ψ&
(8)
i
i
−15
−1
5.1 r(x,
Primitive
and Quadratic
Functionals
. .
rbitrarily chosen as initial
time.
in the following
way:
i|ψ) Then
= Ontology
λ %ψψ |evolves
Λi (x)ψ&
λ
∼
10
s
ψ evolves
the
following
way:
r(x, i|ψ) = λ %ψ | Λi (x)ψ&
(
T1 in
=
t
+
∆T
,
so
that
1/2
1/2
0
1
1/2 and
1/2 to Λ (x) . ψ/#Λ
and
when
this
happens,
the
wave
function
changes
5.2
Primitive
Ontology
Equivariance
. .1/2.i(x)
.(8)
. .ψ
s happens,
the
wave
function
changes
to
Λ
(x)
ψ/#Λ
(x)
ψ#.
i
i
i
1/2
r(x,
i|ψ)
=
λ
%ψ
|
Λ
(x)ψ&
i
1/2 ψ/#Λ (x)
1/2
n
this
happens,
the
wave
function
changes
to
Λ
(x)
ψ#.
ψ
=
U
ψ
,
and
when
this
happens,
the
wave
function
changes
to
Λ
(x)
ψ/#Λ
(x)
ψ#.
i
i
•
i
i
T
∆T
t
.
It
evolves
unitarily,
according
to
Schrödinger’s
equation,
until
a
rand
1
1
0
Schrödinger’s
until
random
time any time
if between
time
and
> t0 , n collapses
have occurred
etween time tequation,
and any
time at >
t0 , nt0collapses
have toccurred
at the times
0 Thus,
Thus,to
if emphasize
between
time
t0that
and
any
time
t01/2
>
t0 , collapses
n collapses
have occurred
at “partic
the
tim
1
1/2
,he
if
between
time
t
and
any
time
t
>
t
,
n
have
occurred
at
6
A
Plethora
of
Theories
0
We
wish
here
there
are
no
particles
in
this
theory:
the
word
T
=
t
+
∆T
,
so
that
< t,T11with
< T2centers
< . . .ψ
<
t, inwith
andwave
labels I1 , . . .th
,
< .wave
X1T
, nto
. .<
. ,Λ
X
and centers
labels
IX
.. .. ,. I, nX, nthe
1. . < 0Ttnfunction
0<
changes
(x)
ψ/#Λ
ψ#.
→
1 ,1i.,(x)
tfor
<convenience
T
. .order
. < Tto
t,able
withtocenters
X
Iterminology.
0 ψ
1 < T2 <in
n <
1 , . . . , Xnotation
n and labels
1 , . . . , In , the wa
only
be
use
the
standard
and
=
U
,
(5)
t.0function
6.1
Particles,
Fields,
and
Flashes
. . . . I.1 ,. . .. .., .In., .t
< Tt 2∆T
<
. . < Tnat<time
t, with
centers
X
.∆T
. ,1X
and
labels
t will
be
1
1 be
me
will
1 , .U
n
ψ
=
ψ
,
T
t
1
0
F
t0 and
any attime
> be
t0L,Ft,tnnψcollapses
have
occurred
at
the
times
n
function
time ttwill
L
ψ
t
F
t,t0 0 and Many-Worlds . .
n Functions
oat
particles
intthis
theory:
the6.2
word “particle”
0 t0 is usedWave
Schrödinger
L
ψ
time
will
be
t
ψt,tt 0= 0 in
ψt = thatFnthere are
(9)word “parti(
We
wish
to
emphasize
here
no
particles
this
theory:
the
F
n
F
ψ
=
n Flabels
he<
standard
notation
and terminology.
t and
t, with
centers
X#L
. .0 ψ. t,0 X
I
,
.
.
.
,
I
,
the
wave
#L
ψ
#
# nL
n
1 , t,t
1
n
t
ψ
t,t
0
0
t00 ψt0 # notation and terminology.
t,t#L
0 standard
t,t
for convenience in order to be able
to
use
the
ψ =
t
time location of that collapse. Accordingly, equation (11) gives the joint distribution of
GRWf
the first n flashes, after some initial time t0 . The flashes form the set
F = {(X1 , T1 ), . . . , (Xk , Tk ), . . .}
(with T1 < T2 < . . .).
In Bell’s words:
[...] the GRW jumps (which are part of the wave function, not something
else) are well localized in ordinary space. Indeed each is centered on a particular spacetime point (x, t). So we can propose these events as the basis of
the ‘local beables’ of the theory. These are the mathematical counterparts
in the theory to real events at definite places and times in the real world (as
distinct from the many purely mathematical constructions that occur in the
13
working out of physical theories, as distinct from things which may be real
but not localized, and distinct from the ‘observables’ of other formulations
of quantum mechanics, for which we have no use here). A piece of matter
then is a galaxy of such events. (Bell, 1987a)
That is, Bell’s idea is that GRW can account for objective reality in three-dimensional
GRWm
e m function is given by the following equation:
m(x, t) =
N
�
i=1
mi
�
�
�
2
3
3
3
�
�
d x1 · · · d xN δ (x − xi ) ψt (x1 , . . . , xN ) .
ψt is a wave function as in quantum mechanics, a function on R3
ψt is a GRW process
ng to the usual Schrödinger equation
N
2
�
∂ψ
�
2
i�
=−
∇i ψ + V ψ ,
∂t
2m
i
i=1
denotes the mass of particle i, i = 1, . . . , N .
m function (1) is basically the natural density function in 3-spac
Lorenz
invariance
Relativistic (Lorentz) Invariance
Those paradoxes are simply disposed of by the 1952
theory of Bohm, leaving as the question, the question
of Lorentz invariance. So one of my missions in life is
to get people to see that if they want to talk about the
problems of quantum mechanics – the real problems
of quantum mechanics – they must be talking about
Lorentz invariance.
Bell (1990)
The big question, in my opinion, is which, if either,
of these two precise pictures [GRW and Bohm] can be
redeveloped in a Lorentz invariant way.
Bell (1990)
[N]either of these theories [GRW and Bohm] is Lorentz
invariant, and ... it is pretty clear that no theory in ei-
associated with the GRW evolution and the fact that GRWf, as discussed in Section
5.2, is microscopically equivariant relative to that evolution. In fact, it follows from the
GRW warming that there is, for GRWf, no equivariant association ψ !→ Pψ with ψ a
Schrödinger-evolving wave function.17
What is a precise quantum theory?
8
What is a Quantum Theory without Observers?
To conclude, we delineate the common structure of GRWm, GRWf, and BM:
(i) There is a clear primitive ontology, and it describes matter in space and time.
(ii) There is a state vector ψ in Hilbert space that evolves either unitarily or, at least,
for microscopic systems very probably for a long time approximately unitarily.
(iii) The state vector ψ governs the behavior of the PO by means of (possibly stochastic)
laws.
17
Since the GRWf flash process is non–Markovian, the formulation of the notion of equivariant as(iv) Thegiven
theory
provides
a not
notion
of a typical
history Pofψ should
the POnow
(ofbethe
universe),
fora
sociation
in Section
5.2 is
appropriate
here; instead,
understood
to be
example
by aonprobability
distribution
on the
space
probability
measure
the space Ω of
possible histories
of the
PO of
for all
all possible
times, buthistories;
one whose from
conditional this
probabilities
the future ofthe
anyprobabilistic
time given its past
are as prescribed,
notion for
of typicality
predictions
emerge. here by the formula (33).
" ψ
The association is equivariant if T−t
P = Pψt , with Tτ now the time translation mapping on Ω.
(v) The predicted probability distribution of the macroscopic configuration at time t
determined by the PO (usually) agrees (at least approximately) with that of the
31
quantum formalism.
The features (i)–(v) are common to these three theories, but they are also desiderata,
presumably even necessary conditions, for any satisfactory quantum theory without
quantum probability
P (Â = α|B̂ = β) = | < α|β > |
< ψ|χ >
| < ψ|χ > |
probability amplitude
2
probability
non- Kolmogorovian ?
2
momentum
probability to find the value p of P̂
if the system is (initially)
in the the state ψ
�
| < ψ|p > | = ψ(p)
2
Fourier transform
time of flight measurement of momentum
(Heisenberg, Bohm52, Feynman & Hibbs)
ψ wf at time 0
free evolution
measure X̂ at large time T
1
X̂(T ) = P̂ T + X̂
m
→
mX̂(T ) X
mX̂(T )
P̂ =
+
≈
T
T
T
Bohm
The particle has a well defined position X whose evolution is guided by ψ
�
ψt∗ ∇ψt
(X(t))
Ẋ(t) = Im ∗
m
ψt ψt
P (X(t) = x|ψ, t = 0) = |ψt (x)|2
P (X̂(t) = x|ψ, t = 0) = |ψt (x)|2
mX(T )
T
Thus the asymptotic momentum P =
(T large) is a
RANDOM VARIABLE on the space on initial conditions
with probability distribution
2
�
P (P = p|ψ, t = 0) = |ψ(p)|
spin
σx =
�
| ↑>=
0
1
1
0
�
�
1
0
| < ψ| ↑> |2
| < ψ| ↓> |
2
�
,
σy =
and
�
| ↓>=
0
i
−i
0
�
�
0
1
�
,
σz =
probability spin up along z
probability spin down along z
Classically Non-Describable Two-Valuedness (Pauli)
�
1
0
0
−1
�
ascio di atomi produrrebbe sullo schermo una singola
ra). Quello cheStern
invece Gerlach
si osserva measurement
sono due macchioline
of spin
ra a destra)
chermo
Schermo
Magneti
� ≈ (b + az)σz
HI = µ�σ · B
nella direzione z del campo magnetico, a seinitial
Ψ
=
ψ
⊗
Φ(z)
entazione NORD-SUD dell’atomo.
o vengono preparati con un forno e poi colli-
P̂z
a
2
Ẑ(T ) = Ẑ +
T+
σz T
m
2m
in the space of initial positions with
prob. distribution |Φ(z)|2
Thus the RANDOM VARIABLE
FT (Z(T )) ,
2mz
FT (z) =
aT 2
in the limit of T large has values
+1 with probability < ψ| ↑> |2
−1 with probability < ψ| ↓> |2
FT (z) is the calibration function of the experiment
assignment of num. values to the outcome of the exp.
Morals
• Association between random variable Z
(numerical result of the experiment) and
operators
• Operators compactly express the statistics of
the experiment
Ā =< ψ|Aψ > mean value of Z
< ψ|(A − Ā)2 ψ > variance of Z
< ψ|An ψ > higher moments of Z
• One can completely understand what's going on in
the experiments measuring momentum or spin.
• No need of invoking any putative property of the
electron such as its actual z-component of spin that
is supposed to be revealed in the experiment.
• There is nothing the least bit remarkable about the
nonexistence of this property.
• Measurements are “active.”