John S. Bell
The Foundations of
Quantum Mechanics

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John S. Bell

The Foundations of
Quantum Mechanics
Editors

RI. Bell
CERN

K. Gottfried
Cornell University

M, Veltman
University of Michigan, Ann Arbor

vpy r l d Scientific

rngapore New Jersey London Hong Kong
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JOHN S. BELL ON THE FOUNDATIONS OF QUANTUM MECHANICS
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Publisher’s Note
The material in this volume first appeared as Section 3 of

Quantum Mechanics, High Energy Physics and Accelerators
(World Scientific, 1995). It has been reprinted owing to
demand from the physics community. Once again, World
Scientific would like to thank the publishers of the various
books and journals for their permission to reproduce the
articles found in Quantum Mechanics, High Energy Physics
and Accelerators.

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vii

Contents
1. On the Problem of Hidden Variables in Quantum Mechanics
Rev, Mod. Phys. 38 (1966) 447-452 ....................................................................................

1

2. On the Einstein Podolsky Rosen Paradox
Physics 1 (1964) 195-200 ...................................................................................................

7

3. The Moral Aspect of Quantum Mechanics
with M . Nauenberg
Preludes in Theoretical Physics - in Honor of V: E Weisskopf,

eds. A. De-Shalit, H. Feshbach and L. Van Hove (North-Holland,
Amsterdam, 1966), pp. 279-286 .........................................................................................

13

4. Introduction to the Hidden-Variable Question
Foundations of Quantum Mechanics - Proc. Int. Sch. of Physics
‘Enrico Fermi, ’ course IL, ed. B.d’Espagnat
(Academic, New York, 1971), pp. 171-181 ........................................................................

22

5. The Measurement Theory of Everett and de Broglie’s Pilot Wave
Quantum Mechanics, Determinism, Causalig, and Particles,
eds. M. Flato et al. (Reidel, Dordrecht, 1976), pp. 11-17 ...................................................

33

6. Subject andobject
The Physicist’s Conception of Nature, ed. J . Mehra
(Reidel, Dordrecht, 1973), pp. 687-690 ..............................................................................

40

7. On Wave Packet Reduction in the Coleman-Hepp Model
Helv. Phys. Acta 48 (1975) 93-98 .......................................................................................

44

8. The Theory of Local Beables

Epistemological Lett. 9 (1976); Dialectica 39 (1985) 86-96

50

..............................................

9. How to Teach Special Relativity
Prog. Sci. Culture 1 (1976) ..................................................................................................

61

10. Einstein-Podolsky-Rosen Experiments
Proc. Symp.on Frontier Problems in High Energy Physics
fin Honour of Gilberto Bernardini on His 70th Birthday),
Pisa, June 1976, pp. 33-45 ..................................................................................................

74

11. Free Variables and Local Causality
Epistemological Left. 15 (1977); Dialectica 39 (1985) 103-106

84

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...


Vlll

12. Atomic-Cascade Photons and Quantum-Mechanical Nonlocality
Invited talk at Conf. European Group for Atomic Spectroscopy,
Orsay-Paris, 10-13 Jul. 1979; CommentsAtom. Mol. Phys. 9 (1980) 121-126

.................

88

13. de Broglie-Bohm, Delayed-Choice, Double-Slit Experiment,
and Density Matrix
Int. J. Quantum Chem.: Quantum Chem. Symp. 14 (1980) 155-159 ..................................

94

14. Quantum Mechanics for Cosrnologists
Quantum Gravity 2, eds. C. Isham, R. Penrose and D. Sciama
(Oxford University Press, 1981), pp. 611-637 ....................................................................

99

15. Bertlmann’s Socks and the Nature of Reality
Journal de Physique, Colloque C2, Suppl. 3 (1981) 41-62

................................................

126

16. On the Impossible Pilot Wave

Found. Phys. 12 (1982) 989-999 ........................................................................................

148

17. Beables for Quantum Field Theory
CERN-TH.4035184 (1984); Quantum Implications, ed. B , Hiley
(Routledge and Kegan Paul, 1987), p. 227 ..........................................................................

159

18. EPR Correlations and EPW Distributions
New Techniques and Ideas in Quantum Measurement Theory (21-24 Jan. 1986),
ed. D. M. Greenberger; Ann. N.Y. Acad. Sci. 480 (1986) 263 .............................................

167

19. Are There Quantum Jumps?
Schrijdinger: Centenary o f a Polymath (Cambridge University Press, 1987) .................... 172
20. Six Possible Worlds of Quantum Mechanics
Proc. Nobel Symp. 65: Possible Worlds in Humanities, Arts and Sciences

(Stockholm, 11-15 Aug. 1986), ed. S . AllCn (Walter de Gruyter, 1989), pp. 359-373 ....... 193

21. Against ‘Measurement’
Phys. World 3 (1990) 33-40 ................................................................................................

208

22. La Nouvelle Cuisine
Between Science and Technology, eds. A. Sarlemijn and

P. Kroes (Elsevier/North-Holland, 1990), pp. 97-1 15 .........................................................

21 6

23. In Memory of Ceorge Francis FitzGerald
Lecture given at Trinity College, Dublin, on the 100th anniversary of the
FitzGerald contraction. Published in Phys. World -5 (1992) 31-35.
Abridged version written by Denis Weaire, Trinity College, Dublin ..................................

235

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1

O n the Problem of Hidden Variables in Quantum
Mechanics*
JOHN S. BELLt
Stunford Linear AccJcrclkir Cenkr, Sbnford Ulnicclzit>~,Slunfmf, Calijornio

The demonstrations of von Neumann and others, that quantum mechanics does not permit a hidden variable interpretation. are.
reconsidend.It is shown that their essentialaxioms 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.

result consequent on the work of Gleason.‘ It will be
urged that these analyses leave the real question unTo know the quantum mechanical state of a system touched. In fact it will be seen that these demonstraimplies, in general, only statistical restrictions on the tions require from the hypothetical dispersion free
results of measurements. It seems interesting to ask states, not only that appropriate ensembles thereof
if this statistical element be thought of as arising, as should have all measurable properties of quantum
in c k i c a l statistical mechanics, because the states in mechanical states, but certain other properties as well.

question are averages over better defined states for These additional demands appear reasonable when rewhich individually the results would be quite deter- sults of measurement are loosely identified with p r o p
mined. These hypothetical “dispersion free” states erties of isolated systems. They are seen to be quite
would be specified not only by the quantum mechanical unreasonable when one remembers with Bohr‘ “the
state \-ector but also by additional “hidden variables’’impossibility of any sharp distinction between the
“hidden” because if states with prescribed values of behavior of atomic objects and the interaction with
these variables could actually be prepared, quantum the measuring instruments which serve to define the
mechanics would be observably inadequate.
conditions under which the phenomena appear.,’
UXePher this question is indeed interesting has been
The realization that von Neumann’s proof is of
the subject of debate202 The present paper does not limited relevance has been gaining ground since the
contribute to that debate. It is addressed to those who 1952 work of Bohm.* However, it is far from universal.
do tind the question interesting, and more particularly Moreover, the writer has not found in the literature
to those among them who believe that’ “the question any adequate analysis of what went wrong.’ Like all
conceiring the existence of such hidden variables re- authors of noncommissioned reviews, he thinks thet
ceived an early and rather decisive answer in the form he can restate the position with such clarity and simof von Xeumann’s proof on the mathematical i m p plicity that all previous discussions will be eclipsed.
sibility of such variables in quantum theory.” An attempt d be made to clarify what von Neumann and If. ASSUMPTIONS, AND A SIMPLE EXAMPLE
his successors actually demonstrated. This will cover, as
The authors of the demonstrations to be reviewed
well as von Neumnn’s treatment, the recent version
were
concerned to assume as little as possible about
of the argument by Jauch and Piron,’ and the stronger
quantum mechanics. This is valuable for some purposes,
but not for ours, We are interested only in the p s i * \Yak supported by U.S. Atomic Energy Commission.
bility of hidden variables in’ordinary quantum met P e m n t address: CERN, Geneva.
I. INTRODUCTION

“I’k ioUowing works contain dircuasions of and referrnocs
on the hidden variable robkm: L. de Broplie, Physkkn d

A. M.Gleason, J. Math. & Meih. 6, 885 (1957). I am mu+
P r n s m fAlbin Michcl, %ark, 1953); W. Heisenberg, in h
’
e indebted to Professor Jauch for &wing my attentloo to thts
Bokr d l k c D d o w M o P h y h , W.Pauli, Ed. ( M c G ~ w - work.
~
Book Ca,Inc., New Yor , and Pergamon Press, Ltd., London,
5N. Bohr, in Ref. 2.
1955) ; CXmwtim and Zn&p&Nrm, S. K(lrner, Ed. (Academic
” D . Bohm, Phyg Rev. 85,166, 180 (1952).
Pres
New York and Butterworths Scientific Publ., Ltd..
’ I n particular the analysis of Bohm’ seems to lack ckrity,
London, 1957) ;N. R.kansen, Tks Cmept of the Posifrm (kor else accuracy. He fully emphasized the role of the experimcotnl
bridge Vniversity Pren, Cambridge, England, 1963). See 8lso arrangement. However, it seem to be implied (Ref. 6, p. 187)
the vuiars works by D. Bohm dted later, and Bell and Naoeo- that the circumvention of the themem requires the apsmbtioo
berg.’ For the view that the p d i i l i t y of hidden variables has of hidden variables with the sppamtus as well as with the system
little interest, see eqepedrlly the amtnbutions of Rosenfeld to
observed. The scheme of Sec II h a counter exampk to this.
fust .nd third of these references, of Pauli to the first, the ubde Moreover, it will be seen io Sec III that if the essential ulditnity
assumption of voo Neumann rere granted, hidden wirbks
of Hebcnkrg, and many pasapger io Hansen.
* A. Einstein, Phihopher Sncnlist, P. A sdrilp, Ed. (Libruy wherever located would not avail Bohm’s further re&
in
of Living Philosophers, Evanstoa, Ill., 1949). Emstem’s “Autm Ref. 16 (p. 95) and Ref. 17 (p. 358) are also uocooviruing.
biographical Notes” and “Reply to Critics” suggest that tbe Other critiques of the theorem .R cited, and some d them
rebutted, by Albertson 0.Albcrtson, Am. J. Phya 29, 478
h$ko vuiable problem has some interest.
f. BL fauch and C. Piron, Hclv. Phys. Acta 36, 827 (I=).
(1961) 3


‘

h

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4cd

REntrs 02 bfODElE-4 ~ Y S I C S*

JaLY

1966

chanicsand will use freely all the usual notions. Thereby

the demonstrations will be substantially shortened.
A quantum mechanical “system” is supposed to
have ‘fObservablesJ*represented by Hermitisn operators in a complex linear vector space. Every “measurement” of an observable yields one of the eigenvalues
of the corresponding operator. Observables with commuting operators can be measured simultaneously? A
quantum mechanical “state” is represented by a vector
in the linear state space. For a state vector $ the statistical expectation value of an observable with operator
0 is the normalized inner product ($, O$)/(#, #).
The question at issue is whether the quantum mechanical states can be regarded as ensembles of states
further specified by additional variables, such that
given values of these variables together with the state
vector determine precisely the results of individual

measurements. These hypothetical well-specified states
are said to be “ d i r s i o n free.”
In the following discussion it will be useful to keep
in mind as a simple example a system with a twodimensional state space. Consider for definiteness a
spin -4 partide without translational motion. A
quantum mechanical state is represented by a twocomponent state vector, or spinor, $, The observables
are represented by 2 X 2 Hermitian matrices
a+@*d,

(-1)

where a is a real number, Q a real vector, and d has for
components the Pauli matrices; a is understood to multiply the unit matrix. Measurement of such an observable yields one of the eigenvalues.

with relative probabilities that can be inferred from
the expectation value

For this system a hidden variable scheme can be sup
plied as folIoss: The dispersion free states are specified
as well
by a reai number X, in the interval -$as the spinor $. To describe how X determines which
eigenvalue the measurement gives, we note that by a
rotation of coordinates $ can be brought to the form

‘Recent ppen on the measurement process in quantum
mechanics, with further references, are: E P. Wigner, Am.
Phys. 31,6 (19611; A Shimony, ibid. 31,755 (1963); J. M.Jaud;
HeIv. Phyn Act. 37, 293 (1964); B. d‘Espagnat, Concspliocu
& fa physqua rmekmporainc (Hermann & Cie., Paris, 1965);

J. S. Bell and XL. Smenherg, in Pi$&
in Thcoreficd Physics,
In Bmm ef Y. WyrissPopf (North-Hollnnd Publishing Company,
Amsterdam, 1wj6).

Let @=, & &, be the components of 6 in the new coordinate system. Then measurement of a + @ d on the
state specified by $ and X results with certainty in the
eigenvalue

where

X=@.

ifB.#O

=@.

if 8,=0,B.#O

=@,

if@.=O,

and &=O

and
signX=+l
=--1

ifX>O

ifx
The quantum mechanical state speciiied by $ is obtained by uniform averaging over X. This gives the
expectation value

@+Q4
=Lala+~
@ I ~ ~ ~ ~ ( x I @ Is +i g~ nI ~@~ .=I a) + ~ .
as required.

It should be stressed that no physical significance is
attributed here to the parameter A and that no pretence
is made of giving a complete reinterpretation of quantum mechanics. The sole aim is to show that a t the level
considered by von Neumanu such a reinterpretation
is not excluded. A complete theory would require for
example an account of the behavior of the hidden variables during the measurement process itself. With or
without hidden variables the analysis of the measurement process presents peculiar difKculties,b and we
enter upon it no more than is strictly necessary for our
very l i i t e d purpose.

III. VON NEUMAXH
Consider now the proof of von Neumanns that dispersion free states, and so hidden variables, are hpossible. His essential assumption@is: Any reat lineor
ctnnbimfidn uj any two BGnnifinn optrators represents
an abservoblc, and the sm:e f i wcomZriMtwn of &xFe,!~J. von Seumann, Matkcmafiscka G~andl~gm
d a QWS*
nrcchunik (JuliuSpringer-Verlag, Berlin, 1932) {En lish transl.:
Princeton Cniversity Press, Princeton N.J., M.& AU page
numbers quoted are those of the Eng& edition. The problem
is posed in the preface, and on p. 209. The formal proof occupies

essentially pp. 305324 and is foUowed by several ages of cornmentary. A self-contained exposition of the proof
been p=sented by J. -Ubertson (see Ref. 7).
‘“This is contained in von Neumann’s B’ {p. 311), 1 (p. 3 1 3 ) ~
and I1 (p. 314).

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3
Jom S.BELL H i d d a Vwk& in Quantum Mtxbnics

the e~wtotionvalue oj the combination.
This is true for quantum mechanical states; it is required by von Neumann of the hypothetical dispersion
free states also. Zn the two-dimensional example of
Sec. 11, the expectation value must then be a linear
function of U and 9. But for a dispersion free state
(which has no statistical character) the expectation
value of an observable must equal one of its eigenvalues.
The eigenvalues (2) are certainly not h e a r in @.Therefore, dispersion free states are impossible. If the state
space has more dimensions, we can always consider a
two-dimensional subspace; therefore, the demonstration
is quite general.
The essential assumption a n be criticized as follows,
A t first sight the required additivity of expectation
values seems very reasonable, and it is rather the nonadditivity of allowd values (eigenvalues) which requires explanation. Of course the explanation is well
known: A measurement of a sum of noncommuting
observables cannot be made by combining trivially the
results of separate observations on the two terms-it
requires a quite distinct experiment. For example the
measurement of c* for a magnetic particle might be

made with a suitably oriented Stem Gerlach magnet.
The measurement of U,,would require a different orientation, and of (u,+u~) a third and different orientation.
But this explanation of the nonadditivity of allowed
values also establisbes the nontriviality of tbe additivity
of expectation values. The latter is a quite peculiar
property of quantum mechanical states, not to be expected a piori. There is no reason to demand it individually of the hypothetical dispersion free states,
whose function it is to reproduce the mcasurabZe peculiarities of quantum mechanics when aoerogcd ouw.
In the trivial example of Sec. I1 the dispersion free
states (specified X) have additive expectation values
only for commuting operators. Nevertheless, they give
logically consistent and precise predictions for the results of all possible tnessurements, which when averaged
over X are fully equivalent to the quantum mechanical
predictions. In fsct, for this trivial example, the hidden
variable question M posed informally by von Neumann’l
in his book is answered in the affirmative.
Thus the formal proof of von Neumann does not
justify his infonnal concltwionu: “It is therefore not,
as is often assumed, a question of reinterpretation of
quantum rnedmi-the
present system of quantum
mechanics would have to be objectively false in order
that another description of the elementary process than
the statistical one be possible.” It was not the objective
measurabie predictions of quantum mechanics which
ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation
between the results of incompatible measurements
tion values i s

Reference 9, p. 209.
Reference 9, p. 325.

449

either of which might be made on a given occasion but
only one of which can in fact be made.

IV. JAUCH AND PIRON
A new version of the argument has been given by
Jauch and Piron.’ Like von Neumann they are interested in generalized forms of quantum mechanics
and do not a s s m c the usual connection of quantum
mechanical exptctation values ~ i t hs k t e vectors and
operators. We assume the latter and shorten the argument, for we are concerned here only with possible
interpretations of ordinary quantum mechanics.
Consider only observables represented by projection
operators. The eigenvalues of projection operators are
0 and 1. Their expectation values are equal to the probabilities that 1 rather than 0 is the result of measurement. For any two projection operators, o and b, a third
(anb) is defined as the projection on to the intersection
of the corresponding subspaces. The essential axioms
of Jauch and Piron are the following:
(A) Expectation values of commuting projection
operators are additive.
(B) If, for some state and two projections a and b,
(a)=

( b ) = 1,

then for that state
(anb)= 1.

Jauch and Piron are led to this last axiom (4” in

their numbering) by an analogy with the calculus of
propositions in ordinary logic. The projections are to
some extent analogous to logical propositions, with the
allowed value 1 corresponding to “truth” and 0 to
“falsehood,” and the construction (anb) to ( a “and” b)
In logic we have, of course, if a is true and B is true then
(a and b) is true. The axiom has this same structure.
Now we can quickly rule out dispersion free states
by considering a Zdimensional subspace. In that the
projection operators are the zero, the unit operator,
and those of the form

3+3&*4
where d is a unit vector. In a dispersion free state the
expectation value of an operator must be one of its
eigenvalues, 0 or 1 for projections. Since from A
(3+4&.d)+

(4- $a*d)=1,

we have that for a dispersion free state either
(*+)&*d)-1

or (i-#i?-d)=l.

Let a and 6 be any noncollinear unit vectors and
a= + i t & * d ,

b e t * # & d,

with the signs chosen so that {a)= (b)=l. Then B
requires
(anS)= 1.

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650

REVIEWS OF

But with a and

MODELW PHYSICS

*

JULY

1966

noncollinear, one readily sees that

onb= 0,
so that

If $1 and J.l are another orthogonal basis for the
subspace spanned by some vectors 14 and 4, then

from (4)

>+(P(#*)

(dla)=O.

So there can be no dispersion free states.
The objection to this is the same as before. We are not
dealing in B with logical propositions, but with measurements involving, for exampie, differently oriented magnets. The axiom holds for guantum mechanical states.13
But i t is a quite peculiar property of them, in no way
a necessity of thought. Only the quantum mechanical
averages over the dispersion free states need reproduce
this property, as in the example of Sec. 11.

V. GLEASON
The remarkable mathematical work of Gleason' was
not explicitly addressed to the hidden variable problem.
It was directed to reducing the axiomatic basis of
quantum mechanics. However, as it apparently enables
von Neumam's result to be obtained without objectionable assumptions about noncommuting operators, we
must clearly consider i t The relevant corollary of
Gleason's work is that., if the dimensionality of the
state space is greater than t x o , the additivity requirement for espectation d u e s of commuting operators
cannot be met by dispersion free states. This will now
be proved, and then its significance discussed. It should
be stressed that Gleason obtained more than this, by
a lengthier argument, but this is all that is essential
here.
It su5ces to consider projection operators. Let P ( 9 )
be the projector on to the Hilbert space vector 9, i.e.,

acting on any vector Q

P(*)$= (*.*)-1(+, $)*.
If a set

or

c ( P ( W>

>=1-

(P(i61)

*1,W

>+(P(+l))= (P(W)+( P ( W >-

(P(+l)

Since $1 may be any combination of @I and %, we have:
(B) If for a given state

( P ( W )= (P(**)
)=0
for some pair of orthogonal vectors, then
( P (&+Bchl1) =0
for all a and 8.
(A) and (B) will now be used repeatedly to establish
the following. Let 9 and be some vectors such that
for a given state

(E'(+) >=1,
(5)

+

(E'(+) >-o.
Then 9 and $ cannot be arbitrarily close; in fact

1 *-#I>*

I $ I.

(6)

(7)

To see this let us normalize $ and write @ in the form
9=$+ d',

+

where +' is orthogonal to and normalized and t is a
real number. Let 4'' be a normalized vector orthogonal
to both $ and J.'
(it is here that we need three dimensions a t least) and so to ch. By (A) and ( S ) ,
(P(+') )=4

( E ' ( ! u )=O.

Then by (B) and ( 6 ) ,

are complete and orthogonal,

(P(*+.r-'4'')

)=O,

where y is any real number, and also by (B) ,

(P(- 4'+r4/') >=0.

Since the E'(+.() commute, by hypothesis then

c
s

(P(+i) )= 1.

(4)

S i n e the expectation value of a projector is nonnegative (eacb measurement yields one of the allowed values
0 oi ij, ariu m c e any two oritlogonai vectors can be
regarded as members of a complete set, we have:
( A ) If with some vector 9,( P ( + ) ) = lfor a given
state, then for that state C,P($))=Ofor any $ orthogonal on

The vector arguments in the last two formulas are
orthogonal; so we may add them, again using (B) :

(P(tL+e(-Y+Y-*)+")

Sow if t is less than

In the two-dimensionaliazx

U

> =( b ) = 1 'ior sonie q u a n t u m

mechanical state) is pwsilde only ii the twoprojertorsare idrnticnl
&==&.I'hcrl d n h = d = b a n d , , d V ! = ( a ) = ( b ) = l .

3, there are real y such that

e(y+r-') = f l .

Therefore,

+.

U

) =0.

(P($+$") )= (P(+-+") )-0.
The vectors $&$" are orthogonal; adding them and
ngain using ( R ) ,
(Y!J.) ) = 0.

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5
JOHN

This contradicts the assumption (5). Therefore,
c>f,

as announced in (7).
Consider now the possibility of dispersion free states.
For such states each projector has expectation value
either 0 or 1. It is clear from (4)that both values must
occur, and since there are no other values possible,
there must be arbitrarily close pairs J.,QI with different
expectation values 0 and 1, respectively. But we saw
above such pairs could not be arbitrarily close. Therefore, there are no dispersion free states.
That so much follows from such apparently innocent
assumptions leads us to question their innocence. Are
the requirements imposed, which are satisfied by
quantum mechanical states, reasonable requirements
on the dispersion free states? Indeed they are not.
Consider the statement (B). The operator P(a@l+B@l)
commutes with P(a1) and &‘(a%)
only if either OL or 8
is zero. Thus in general measurement of P(a@1+@+2)
requires a quite distinct experimental arrangement.
We can therefore reject (B) on the grounds already
used: it reIates in a nontrivial way the results of experiments which cannot be performed simultaneously;
the dispersion free states need not have this property,
it will su5ce if the quantum mechanical averages over
them do. How did it come about that (B) was a consequence of assumptions in which only commuting

operators were explicitly mentioned? The danger in
fact was not in the explicit but in the implicit assump
tions. It was tacitly assumed that measurement of an
observable must yield the same value independently
of what other measurements may be made simultaneously. Thus as well as P(+z) say, one might measure
either P(@*)01 P(rt.i),w k r e @%and $1 are orthogonal
to +a but not to one another. These different possibilities
require different experimental arrangements; there is
no a jwiori reason to believe that the results for P(@a)
should be the same. The result of an observation may
reasonably depend not only on the state of the system
(including hidden variables) but also on the complete
disposition of the apparatus; see again the quotation
from Bohr at the end of Sec. I.
To illustrate these remarks, we construct a very
artificial but simple hidden variable decomposition.
If we regard all observables as functions of commuting
projectors, it will suffice to consider measurements
be the set of projectors
of the latter. Let PI,Pt,
measured by a given apparatus, and for a given quantum mechanical state let their expectation values be
XI, &XI,
Xr-X,,
* * * . As hidden variable we take a
real number Oon a state with given X yields the value 1 for P,,if
h,,-land zero otherwise. The quantum mechanical state is obtained by uniform averaging over
X. There is no contradiction with Gleason’s corollary,
because the result for a given P,,depends also on the

s. BELL

3kfdm I’or&s

bt Quantum 3fakarriu

151

choice of the others. Of course it would be silly to let
the result be affected by a mere permutation of the
other P’s, so we specify that the same order is taken
(however defined) when the P’s are in fact the same
set. Reflection d l deepen the initial impression of
artificiality here. However, the example suffices to
show that the implicit assumption of the impossibility
proof was essential to its conclusion. A more serious
hidden variable decomposition will be 3ken up in
Sec. VI.“

VI. LOCALITY AND SEPARABILXTY
Up till now we have been resisting arbitrary demands
upon the hypothetical dispersion free states. However,
as well as reproducing quantum mechanics on averaging, there are features which can reasonably be desired
in a hidden variable scheme. The hidden variables
should surely have some spacial significance and should
evolve in time according to prescribed law. These are
prejudices, but it is just this possibility of interpolating
some (preferably causal) spacetime picture, between
preparation of and measurements on states, that makes

the quest for hidden variables interesting to the unsophisticated.? The ideas of space, time, and causality
are not prominent in the kind of discussion we have
been considering above. To the writer’s knowledge the
most successful attempt in that direction is the 1952
scheme of Bohm for elementary wave mechanics. By
way of conclusion, this will be sketched briefly, and
a curious feature of it stressed.
Consider for example a system of two spin -3 particles. The quantum mechanical state is represented by
a wave function,
tLii(r1, r2) ,
where iand j are spin indices which will be suppressed.
This is governed by the Schrodinger equation,
*/at = -i(

- (#/a?)
- (l~*/&2~)
+V (rl- rr)

where V is the interpartide potential. For simplicity
we have taken neutral particles with magnetic moments, and an external magnetic field H has been dlowed to represent spin analyzing magnets. The hidden
variables are then two vectors XI and X,, which give
directly the results of p i t i o n measurements. Other
measurements are reduced ultimately to position measurements.” For esample, measurement of a spin component means observing whether the particle emerges
with an upward or downward deflection from a Stern“The simplest eumple for illustratin the discussion of Sec V
would then be a particle of spin 1, postdating a sufficient variety

of spin+xternal-&M interactions to permit arbitrary complete
Bets of spin states to be spacialiy separated.
“There are cleul,- enough measurements to be interesting
that can be made in this way. \Ye will not consider whether there

are others.

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452

R ~ v m w sox MODERNPnrsrcs

-

JULY

1966

GeAch magnet. The variables XIand X, are supposed
to be distributed in configuration space with the probability density,
P(XI,&) =

c I h ( X 1 , XZ) I’,
*I

appropriate to the quantum mechanical state. Consistently, with this XIand X*are supposed to vary with
time according to
&/a=~(Xt
X P ~m +u*(Xi, X~)(a/aXJ+AL
*j

dXJdt=p(XI,

X P Im

+,,*(XI, X~(~J/~X&(XI,
Xd.
II

(9)

The curious feature is that the trajectory equations
(9) for the bidden variables have in general a grossly
nonlocal character. If the wave function is factorable
before the analyzing fields become effective (the particles being far apart),

$,,(XI,X,) =*dX1)xJ(X2),
this factorability will be preserved. Equations (8) then
reduce to

& / ~ = [*s*(xI)@t(Xl))-’
~

analyzing fields acting on 2-however remote these
may be from particle 1. So in this theory an e.splicit
causal mechanism exists whereby the disposition of
one piece of apparatus affects the results obtained
with a distant piece. In fact the Einstein-PodoWryRosen paradox is resolved in the way which Einstein
would have liked least (Ref.2, p. 85).
-More generally, the hidden variable account of a
given system becomes entirely different when we remember that it has undoubtedly interacted with numerous other systems in the past and that the total
wave function will certainly not be factorable. The

same effect complicates the hidden variable account
of the theory of measurement, when it is desired to
include part of the “apparatus” in the system.
B o b of course was well aware*Jb18of these features
of his scheme, and has given them much attention.
However, it must be stressed that: to the present
writer’s knowledge, there is no proof that any hidden
variable account of quantum mechanics must have this
extraordinary character.” It would therefore be interesting, perhaps,’ to pursue some further “impossibility proofs,” replacing the arbitrary axioms objected
to above by some condition of locality, or of separability
of distant systems.

I

ACKNOWLEDGMENTS

The first ideas of this paper were conceived in 1952.
I warmly thank Dr. F. Mandl for intensive discussion
I
at that time. I am indebted to many others since then.
latterly, and very especially, to Professor J. 11.
x Im C x i * ( X , ) ( ~ / ~ X , ) X ( X , ) and
.
j
Jauch.
The Schrijdinger equation (8) also separates, and the
D. B o b , Causalily and C h n u in Modem Physics (D. Van
trajectories of XIand X I are determined separately by Nostrand
Co.; Inc., Prhceton, N.J.,1957).
equations involving H(XJ and H ( X 1 ) , respectively.

1’ D. Bohm, in QIlanfum Thcury, D. R. Bates, Ed. (Academic
However, in general, the wave function is not factorable. Press Inc., New York, 1962).
“ D . Bohm and Y. Aharonov, Phys. Rev. 108, 1070 (1957”.
Tbe trajectory of 1 then depends in a complicated way
Is Since the completion of this
r such a proof has been found
on the trajectory and wave function of 2, and so on the U. S Bell, Physics 1, 195 (t9&3?

addl=CC x ~ * ( x ~ ) x ~ ( x J Z I

Reprinted with p e d s s i o n frornRev. ofhfod.Phys., Vol. 38, No. 3, July 1966,
pp. 447-452. Copyright 1966 The American Physical Society.

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7
Reprinted from:
Physics Vol. 1, No. 3, pp. 195-280, 1964

physics Publishing Co.

Printed in the United States

ON THE EINSTEIN PODOLSKY ROSEN PARADOX*
J . s. BELL?
Department of Physics, University of Wisconsin, Madison, Wisconsin
(Received 4 November 1964)

I. Introduction

T H E paradox of Einstein, Podolsky and Rosen [I] w a s advanced as a n argument that quantum mechanics
could not be a complete theory but should be supplemented by additional variables. T h e s e additional varia b l e s were t o restore t o the theory causality and locality [2]. In this note that idea will b e formulated
mathematicaIly and shown t o be incompatible with the statistical predictions of quantum mechanics. It i s
the requirement of locality, or more precisely that the result of a measurement on one system b e unaffected
by operations on a distant system with which it h a s interacted in the past, that creates the essential difficulty. There have been attempts f31 to show that even without such a separability or locality requirement no "hidden variable" interpretation of quantum mechanics is possible. These attempts have been
examined elsewhere [4] and found wanting. Moreover, a hidden variable interpretation of elementary quantum theory 1.51 h a s been explicitly constructed. That particular interpretation h a s indeed a grossly nonlocal structure. T h i s is characteristic, according to the result t o b e proved here, of any such theory which
reproduces exactly the quantum mechanical predictions.

It. Formulation
With the example advocated by Bohm and Aharonov [6],the EPR argument i s the following. Consider
a pair of spin one-half particles formed somehow in the singlet spin s t a t e and moving freely in opposite
directions. Measurements can b e made, s a y by Stern-Gerlach magnets, on selected components of the
-3, where 3 is some unit vector, yields the value
spins
and a2. If measurement of the component
+ 1 then, according to quantum mechanics, measurement of Z2.d must yield the value -1 and vice versa.
Now we make the hypothesis [2], and it seems one a t l e a s t worth considering, that if the two measurements are made at p l a c e s remote from one another the orientation of one magnet does not i n f h e n c e the
result obtained with the other. Since we can predict in advance the result of measuring any chosen cornponent of 3, , by previously measuring the same component of G , , i t follows that the result of any such
measurement must actually be predetermined. Since the initial quantum mechanical wave function d o e s not
determine the result of an individual measurement, this predetermination implies the possibility of a more
complete specification of the state.
L e t this more complete specification b e effected by means of parameters A . It is a matter of indiffere n c e in the following whether A denotes a single variable or a s e t , or even a s e t of functions, and whether
the variables are discrete or continuous. However, we write a s if h were a singIe continuous tarameter.
T h e result A of measuring G , d i s then determined by 3 and A, and the result B of measuring U * -g in the
and A, and
same instance is determined by

-

*Work supported in part by the U.S. Atomic Energy Commission

'0x1l e a v e of absence from S L A C and CERN

195

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8
Vol. 1.

J . S. BELL

196

NO. 3

A(:, A) = t1, B ( & A) = k1.

(1)

T h e vital assumption [2] is that t h e result B for particle 2 does not depend on the s e t t i n g t, of the magnet
for particle 1, nor A on
If p(2) '," the probability distribution of A then the expectation value of the product of the two com-b
ponents ol.a and 0 2 * gi s

B.

P ( z , x ) =&Ap(A)A(S,h)B(&A)
T h i s should equal the quantum mechanical expectation value, which for t h e singlet s t a t e is
-B

< o1

- -a* - .

G2-

X >=

-B

-a.

X.

(3)

But it will be shown that this i s not possible.
Some might prefer a formulation in which the hidden variables fall into two s e t s , with A dependent on
one and B on the other; this possibility i s contained in the above, s i n c e A s t a n d s for any number of varia b l e s and the dependences thereon of A and B are unrestricted. In a complete physical theory of the
type envisaged by Einstein, the hidden variables would have dynamical significance and laws of motion;
our X can then be thought of a s initial values of these variabfes a t some suitable instant.

I II.

Illustration

T h e proof of the main result is quite simple. Before giving it, however, a number of illustrations may
s e r v e to put i t in perspective.
Firstly, there i s no difficulty in giving a hidden variable account of spin measurements on a s i n g l e

particle. Suppose we have a spin half particle in a pure spin sta? with polarization denoted by a unit
vector p'. L e t the hidden variable b e (for example) a unit vector A with uniform probability distribution
-*
over the hemisphere -p' > 0. Specify that the result of measurement of a component U * a is

x

sign

X. ;1'

,

(4)

where i' is 0 unit vector depending on 2 and p' in a way t o b e specified, and the sign function is,+1 or
-1 according to the sign of i t s argument. Actually t h i s leaves the result undetermined when A * a f 0,
but a s the probability of this i s zero we will not make special prescriptions for it. Averaging over A the
expectation value is

where 8' is the angle between
until

G1

and

;.

Suppose then that

1

2'

is obtained from

2 e'
-= case

2

by rotation towards

(6)

B

where 8 is the angle between

2

and

s. Then we have the desired result

SO in t h i s simple c a s e there i s no difficulty in the view that t h e result of every measurement is determined
by the value of a n extra variable, and that the s t a t i s t i c a l features of quantum mechanics arise because the
value of this variable is unknown in individual instances.

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9
Vol. 1, No. 3

ON THE EINSTEIN PODOLSKY ROSEN PARADOX

197

Secondly, there i s no difficulty in reproducing, in the form (2), the only features of (3) commonly used
in verbal discussions of this problem:

For example, let A now b e unit vector

X,

with uniform probability distribution over all directions, and take

This gives

where 8 is the angle between a and b, and (10) has the properties (8). For comparison, consider the result of a modified theory 161 in which the pure singlet s t a t e is replaced in the course of time by an isotropic mixture of product states; this gives the correlation function

It is probably less easy, experimentally, to distinguish (10) from (3), than (11) from (3).
Unlike (3), the function (10) i s not stationary a t the minimum value - l ( a t 6 = 0). It will be seen
that this i s characteristic of functions of type (2).
Thirdly, and finally, there is n o difficulty in reproducing the quantum mechanical correlation (3) if the
results A and B in (2) are allowed to depend on
and 2 respectively a s well as on
and is. For example, replace ;;in (9) by :', obtained from 2 by rotation towards 1: until

t

1-

-2e ' = case,
n

t.

where 8' i s the angle between 2' and
However, for given values of the hidden variables, the results
of measurements with one magnet now depend on the setting of the distant magnet, which i s just what we
would wish to avoid.

IV.

Contradiction

The main result will now be proved. Because p i s a normalized probability distribution,

h A p ( A 1 = 1,
and because of the properties (I), P in (2) cannot be less than

(12)

- 1. It can reach - 1a t

A(;, A) = - B ( s , A)

.+

a =

3

only if

(13)

except a t a set of points A of zero probability. Assuming this, (2) can be rewritten

P(s,

= -JAp(A)

A(:, A) A ( & A).

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(14)


J . S. BELL

198

It follows that

2

i s another unit vector

P ( 2 , Z) -P(;,

z)

=

-

s

Vol. I , No. 3

d A p ( A ) [ A G , A) A ( % A) -A(:, A) A(:, A)]
A(;, A) Atif, A) [ A ( & A) A(:, A) -11

=/Ap(A)

using (l),whence

IJ'(2, if) - P ( z , ;)I
T h e second term on t h e right is P(& :),

_
[I - A ( & A) A(:, A)]

whence

1 + P(Z,

32

IPG, %) - P G ,

31

(15)

a-;

z)

Unless P i s constant, the right hand s i d e is i n general of order I $-:I
for small I
1 . Thus P(Z,
cannot be stationary a t t h e minimum value (- 1 a t 7: = t ) and cannot equal the quantum mechanical
value (3).
Nor can the quantum mechanical correlation (3) be arbitrarily closely approximated by t h e form (2).
T h e formal proof of this may be set out a s follows. We would not worry about failure of the approximation
a t isolated points, so let us consider instead of (2) and (3) the functions

p(2, %) and - 2 . 2
where t h e bar denotes independent averaging of P ( g : t') and -2' %' over vectors 2' and 3' within specified small angles of 2 and 2. Suppose that for all ;and if the difference is bounded by :

li%,Z)+:.zI5c
Then it will b e shown that 6 cannot be made arbitrarily small,
Suppose that for a l l a and b

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(16)


1 1

Vol. 1, No. 3

199

Using (20) then

Then using (19) and 21)

Finally, using (18)‘

or

+

+

Take for example a * c = 0,

2- t

-9

=

*

c =

l/$F

Then

Therefore, for small finite 8, c cannot be arbitrarily small.
Thus, the quantum mechanical expectation value cannot be represented, either accurately or arbitrarily closely, in the form (2).

V. Generalizotion
The example considered above h a s the advantage that i t requires little imagination to envisage the
measurements involved actually being made. In B m o r e formal way, assuming (71 that any Herrnitian operator with a complete s e t of eigenstates is an “observable”, the tesuIt i s easily extended to other systems.
If the two systems have s t a t e spaces of dimensionality greater than 2 we can always consider two dimensional subspaces end define, in their direct product, operators d , and Ti, formally analogous to those
used above and which are zero for states outside the product subspace. Then for a t least one quantum
mechanical state, the “singlet” s t a t e in the combined subspaces, the statistical predictions of quantum
mechanics are incompatible with separable predetermination.

VI. Conclusion
In a theory in which parameters are added to quantum mechanics to determine the results of individual
measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover,
the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant.
Of course, the situation is different if the quantum mechanical predictions are of limited validity.
Conceivably they might apply only to experiments in which the settings of the instruments are made sufficiently in advance to allow them to reach some mutual rapport by exchange of signals with velocity less
than or equal to that of light. In that connection, experiments of the type proposed by Bohm and Aharonov
161, in which the settings are changed during the flight of the particles, are crucial.
1 am indebted to Drs. M. Bander and J. K. Perring fur very useful discussions of this problem. The

first draft of the paper was written during B stay a t Brandeis University; I am indebted to colleagues there
and at the University of Wisconsin for their interest and hospitality.

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200

J. S.

BELL

Vol. 1, No. 3

References
1. A. EINSTEIN, N. ROSEN and B. PODOLSKY, P h y s . Rev. 47, 777 (1935); see also N. BOHR, Ibid. 48,
696 (1935)' W. H. FURRY, ibid. 49,393 and 476 (1936), and D. R. INGLIS,Rev. Mod. P h y s . 33, 1
(196 1).
2. "But on one supposition we should, in my opinion, absolutely hold fast: the real factual situation of
the system S , is independent of what i s done with the system S, , which is spatially separated from
the former." A. EINSTEIN in Albert Einstein, Philosopher Scientist, (Edited by P. A. SCHILP) p. 85,
Library of Living Philosophers, Evanston, Illinois (1949).
3. J. VON NEUMANN, Mathematishe Grundlagen der Quanten-mechanik. Verlag Julius-Springer, Berlin
(1932). [English translation: Princeton University Press (1955)l; J . M. JAUCH and C. PIRON, Helv.
P h y s . Acta 36, 827 (1963).
4. J . S. BELL, to be published.
5. D. BOHM, Phys. Rev. 85, 166 and 180 (1952).
6. D. BOHM and Y. AHARONOV, P h y s . Rev. 108, 1070 (1957).
7. P, A. M. DIRAC, The Principles of Quantum Mechanics (3rd Ed.) p. 37. The Clarendon P r e s s , Oxford

(1947).

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13
REPRINTED F R O M :

P R E L U D E S IN
THEORETICAL PHYSICS
IN HONOR OF V. F. WEISSKOPF
edited by

A. DE-SHALlT
Deparfnieni vf Nirclcrir Physics,
The Weizmarrn ltrstirutc vf Scictrce, R ~ Ivoth,
O Israel

H. F E S H B A C H
Depcirttnetit of Physics aird Laboratory for Nuclcur Scietrce,
MIT, Curribridgc, Muss., USA

L. VAN HOVE
Theoretical Swdy Divisioir, C E RN, Geneva, Switzerlartd

1966

-

NORTH- HOLLAND PUBLISHING COMPANY AMSTERDAM

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14

THE MORAL ASPECT O F
QUANTUM MECHANICS
J. S. B E L L
CERN, Getreva

and

M. N A U E N B E R G
Slotr/ord Utrioersiry

(Received

Jittie

3, 1965)

The notion of morality appears to have been introduccd into quantum
theory by Wigner, as reported by Goldbcrger and Watson [I]. The
question at issue is the fanious “reduction of the wave packet”. Thcre
are, ultimately, no mechanical arguments for this proccss, and the
arguments that are actually used may wcll be callcd moral. This is a
popular account of the subject. Very practical people not interested in
logical questions should not read it. It is a pleasure for 11sto dcdicate
the paper t o Professor Weisskopf, for whom intense interest in the
latest developments of detail has not dulled concern with fundamentals.

Suppose that some quantity F is measured 011 a quantum mechanical system, and a result f obtained. Assume that immediate repetition
of the measurement must give the same result. Thcn, after the first
measurement, the system must be in an eigenstate of /;with eigenvaliie
f. In general, the measurement will be “incomplete”, i.e., there will
be more than one eigenstate wiih the observed eigcnvalue, so that the
latter does not suffice to specify completely the state resulting from
the measurement. Let the relevant set of eigenstates be denoted by
4fg.The extra indcx g may be rcgarded as the cigenvaiue of a second
observable G that commutes with F and so can be measured at the
same time. Given that f is observed for F, the relative probabilities
of observing various g in a simultaneous measurement of G are given
by the squares of the moduli of the inncr products

hi?
’ $1

’

where Ic/ is the initial state of the systcm. Let us now make the plausible
assumption that these relative probabilities would be the same if G
“9

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15

J. S. Bell and M . Nauenberg

280

were measured not simultaneously with F but immediately afterwards.
Then we know something more about the state resulting from the
measurement of F. One state with the desired properties is clearly

c 4 h ( h $1
@

where N is a normalization factor. It is readily shown that this is the
only state [2] for which the probability of obtaining a given value for
any quantity commuting with F is the same whether the measurement
is made at the same time or immediately after. Thus, we arrive at the
genera1 formulation for the “reduction of the wave packet” following
measurement [3]: expand the initial state in eigenstates of the observed
quantity, strike out the contributions from eigenstates which do not
have the observed eigenvalue, and renormalize the remainder. This
preserves the original phase and intensity relations between the relevant eigenstates. It therefore does the minimum damage to the original state consistent with the requirement that an immediate repetition of the measurement gives the same result. All this is very ethical,
and we will refer to the particular reduction just defined as “the moral
process”.
Now morality is not universally observed, and it is easy to think of
measuring processes for which the above account would be quite
inappropriate. Suppose for example the momentum of a neutron is
measured by observing a recoil proton. The momentum of the neutron
is altered in the process, and in a head on collision actually reduced to
zero. The subsequent state of the neutron is by no means a combination (the spin here provides the degeneracy) of states with the
observed momentum. How then is one to know whether a given measurement is moral [4] or not? Clearly, one must investigate the physics
of the process. Instead of tracing through a realistic example we will
follow voii Neumann [3] here in considering a simple model.
Suppose the system I to be observed has co-ordinates R. Suppose
that the measuring instrument, 11, has a single relevant co-ordinate

Q-a pointer position. Suppose that the measurement is effected by
switching on instantaneously an interaction between I and II

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28 1

Mornl aspect

where t is time. The simplification here, where the system of interest
acts directly on a pointer reading without intervention of circuitry, is
gross. If I is in the state # ( R ) before the measurement, and the pointer
reading is zero, the initial state of I + I t is

+

The state of I I1 immediately after t = 0 can be obtained by solving
the Schrodinger equation. i n this only the interaction term in the
Hamiltoiiian is significant, because of its impulsive character. Thc
resulting state is [ S ]

CB $dRX#fg , @>a(Q-f>

J-9

where f i s an eigenvalue of F, q5fg a corresponding eigenfunction, and
g any extra index needed to enumerate these eigenfunctions. If now

an observer reads the pointer on the instrument, and finds a particular
value f, and if this meusurement of the pointer readitig is moral, tlicn
the state reduces to

N

c +sg(R)(4vg #P(Q-.fh
7

B

The part referring to system I alone,
N

c

4!fg(wkJ

, J/)

B

is precisely the result of applying the moral process to I directly, after
the measurement of the quantity F. So we have here a dynamical modcl
of a moral measurement of F. This depends on the detailed nature of
the interaction between the system and the measuring instrument. I t
would have been equally easy to choose an interaction for which a
moral measurement of the pointer reading would imply an immoral
measurement of F.

Thus, if the morality of measurements of macroscopic pointer
readings is granted, there is no real ambiguity in practice in applying
quantum mechanics. Onc must simply understand well enough the
structure of the systems involved, including.the instruments, and work
out the consequences. This situation is not peculiar to quantum mechanics. Moreover, we are readily disposed 10 accept the nioral charactcr
of observing macroscopic pointers, for we fcel convinced from common

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