贝尔-隐变量1
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REVIKAVS OF MOT)F.
RN
PHYSI(".
s
VOt.
UM
f.
',
.
"8,
NUMBER
3
JUT,
V 1966
On
1
~'e
.
'.
.
ron.
em
o1:
"
.
.
.ic.c.
en
Varia
o.
es
in,
Quantum
iV.
;ec.
~anics*
JOHN
S.
BELL)
Stanford
linear Accelerator
Center, Stanford
Unioersity,
Stanford,
California
The demonstrations
of von Neumann
and
others,
that
quantum
mechanics does not
permit
a hidden variable
inter-
pretation,
are reconsidered.
It is shown that their
essential axioms are
unreasonable.
It is
urged
that
in
further
examination
of this
problem
an
interesting
axiom
would
be
that
mutually
distant
systems
are
independent of one
another.
I.
INTRODUCTION
To know the
quantum
mechanical
state
of a
system
implies,
in
general,
only
statistical
restrictions
on
the
results
of
measurements. It
seems
interesting
to
ask
if
this statistical
element
be
thought
of as
arising,
as
in classical statistical
mechanics, because
the states
in
question
are
averages
over better de6ned states for
which
individually
the results
would
be
quite
deter-
rnined.
These
hypothetical
"dispersion
free"
states
would
be
specified
not
only
by
the
quantum
mechanical
state vector
but also
by
additional
"hidden
variables"—
"hidden"
because
if
states with
prescribed values
of
these variables could
actually
be
prepared,
quantum
mechanics
would be
observably inadequate.
Whether this
question
is indeed
interesting has been
the
subject
of debate.
'
'
The
present
paper
does not
contribute to that
debate. It is addressed to
those
who
do
find the
question
interesting,
and more
particularly
to those
among
them who believe
that'
"the
question
concerning
the
existence of such hidden variables
re-
ceived
an
early
and
rather decisive answer
in
the
form
of von
Neumann's
proof
on the mathematical
impos-
sibility
of
such variables in
quantum theory.
"
An
at-
tempt
will
be made to
clarify
what von
Neumann and
his successors
actually
demonstrated. This will
cover,
as
well
as von
Neumann's
treatment,
the
recent version
of the
argument
by
Jauch
and
Piron,
'
and the
stronger
*
Work
supported
by
U.S. Atomic
Energy
Commission.
/Permanent
address:
CERN,
Geneva.
'The
following
works contain
discussions
of and references
on the hidden variable
problem:
L. de
Broglie,
Physicien
et
Pensemr
(Albin Michel,
Paris,
1953);
W.
Heisenberg,
in Xiels
Bohr
and the Development
of
Physics,
W.
Pauli,
Ed. (McGraw-Hill
Book Co.
,
Inc.
,
New
York,
and
Pergamon
Press,
Ltd.
,
London,
1955);
Observation and
Interpretation,
S.
Korner,
Ed.
(Academic
Press Inc.
,
New
York,
and Butterworths Scientific Publ.
,
Ltd.
,
London,
1957);
N.
R.
Hansen,
The
Concept
of
the Positron
(Cam-
bridge University
Press,
Cambridge, England,
1963).
See
also
the
various works
by
D. Bohm
cited
later,
and Bell and
Nauen-
berg.
For the
view
that
the
possibility
of
hidden variables has
little
interest,
see
especially
the
contributions
of Rosenfeld to the
first and third of these
references,
of Pauli to the
first,
the
article
of
Heisenberg,
and
many
passages
in Hansen.
A.
Einstein, Philosopher Scientist, P.
A.
Schilp,
Ed.
(Library
of
Living Philosophers, Evanston,
Ill.
,
1949).
Einstein's
"Auto-
biographical
Notes"
and "Reply to
Critics"
suggest
that--the
hidden variable
problem
has
some interest.
'
J.
M.
Jauch
and
C.
Piron,
Helv.
Phys.
Acta
36,
827
(1963).
44
result
consequent
on the
work
of
Gleason.
'
It will
be
urged
that these
analyses
leave the
real
question
un-
touched. In fact it will
be
seen that these demonstra-
tions
require
from the
hypothetical
dispersion
free
states,
not
only
that
appropriate
ensembles thereof
should
have
all
measurable
properties
of
quantum
mechanical
states,
but certain
other
properties
as
well.
These additional
demands
appear
reasonable when
re-
sults
of
measurement are
loosely
identified with
prop-
erties
of isolated
systems.
They
are
seen to
be
quite
unreasonable when
one
remembers with
Bohrs
"the
impossibility
of
any
sharp
distinction between
the
behavior
of atomic
objects
and the interaction with
the
measuring
instruments
which serve to de6ne the
conditions
under
which the
phenomena
appear.
"
The realization that
von
Neumann's
proof
is
of
limited
relevance
has been
gaining
ground
since the
1952
work
of
Bohm.
'
However,
it
is
far from universal.
Moreover,
the writer
has not found in the literature
any
adequate analysis
of what went
wrong.
~
Like
all
authors
of noncommissioned
reviews,
he
thinks that
he can
restate the
position
with
such
clarity
and sirn.
-
plicity
that all
previous
discussions
will
be
eclipsed.
II.
ASSUMPTIONS,
AND
A
SIMPLE EXAMPLE
The authors
of
the
demonstrations
to
be reviewed
were concerned to
assume
as
little
as
possible
about
quantum
mechanics.
This is valuable for some
purposes,
but not
for
ours.
We
are interested
only
in the possi-
bility
of hidden variables in
ordinary quantum
me-
4
A. M.
Gleason,
J.
Math.
Bz
Mech.
6,
885
(1957).
I
am much
indebted
to Professor
Jauch
for
drawing
my
attention to
this
work.
'N.
Bohr,
in
Ref. 2.
'
D.
Bohm,
Phys.
Rev.
SS,
166,
180
(1952).
7
In
particular
the
analysis
of
Bohme
seems
to
lack
clarity,
or
else
accuracy.
He
fully
emphasizes
the role
of
the
experimental
arrangement.
However,
it seems to
be
implied (Ref.
6, p.
187)
that
the
circumvention
of
the theorem
reqlires
the association
of
hidden variables with the
apparatus
as well as
with the
system
observed. The
scheme
of
Sec. II
is
a counter
example
to
this.
Moreover, it will
be
seen in Sec. III
that
if the essential
additivity
assumption
of
von Neumann were
granted,
hidden variables
wherever located
would
not
avail.
Bohm's
further
remarks in
Ref.
16
(p.
95)
and Ref.
17
(p.
358)
are also
unconvincing.
Other
critiques
of the
theorem
are
cited,
and some of
them
rebutted,
by
Albertson
PJ.
Albertson,
Am.
J.
Phys. 29,
478
(1961
)
].
7
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