John Stewart Bell’s famous theorem is widely regarded as one of the most important developments in the foundations of physics. Yet even as we approach the 50th anniversary of Bell’s discovery, its meaning and implications remain controversial. Many workers assert that Bell’s theorem refutes the possibility suggested by Einstein, Podolsky, and Rosen (EPR) of supplementing ordinary quantum theory with “hidden” variables that might restore determinism and/or some notion of an observer-independent reality. But Bell himself interpreted the theorem very differently—as establishing an “essential conflict” between the well-tested empirical predictions of quantum theory and relativistic local causality. Our goal is to make Bell’s own views more widely known and to explain Bell’s little-known formulation of the concept of relativistic local causality on which his theorem rests. We also show precisely how Bell’s formulation of local causality can be used to derive an empirically testable Bell-type inequality and to recapitulate the EPR argument.

1.
Max
Born
,
The Born-Einstein Letters
, translated by
Irene
Born
(
Walker and Company
,
New York
,
1971
).
2.
Mary B.
Hesse
,
Forces and Fields: The Concept of Action at a Distance in the History of Physics
(
Dover
,
Mineola, NY
,
2005
).
3.
Ernan
McMullin
, “
The Explanation of Distant Action: Historical Notes
,” in
Philosophical Consequences of Quantum Theory
, edited by
James T.
Cushing
and
E.
McMullin
(
University of Notre Dame Press
,
Notre Dame
,
1989
).
4.
Isaac
Newton
, February 25, 1693 letter to Richard Bentley. See, for example, Andrew Janiak, “
Newton’s philosophy
,” in
The Stanford Encyclopedia of Philosophy
, edited by
Edward N.
Zalta
, <plato.stanford.edu/archives/win2009/entries/newton-philosophy/>.
5.
I.
Bernard Cohen
, “
A guide to Newton’s Principia
,” in
Isaac Newton, The Principia
, translated by
I. B.
Cohen
and
Anne
Whitman
(
University of California Press
,
Berkeley
,
1999
). See especially pp.
60
64
and
277
.
6.
Albert
Einstein
,
Relativity: The Special and the General Theory
(
Penguin Classics
,
New York
,
2006
), p.
47
.
7.
John S.
Bell
, “
La nouvelle cuisine
,” in
Between Science and Technology
, edited by
A.
Sarlemijn
and
P.
Kroes
(
Elsevier Science
,
Amsterdam
,
1990
); reprinted in Ref. 8, pp.
232
248
.
8.
John S.
Bell
,
Speakable and Unspeakable in Quantum Mechanics
, 2nd ed. (
Cambridge U.P.
,
Cambridge
,
2004
).
9.
Albert
Einstein
,
Boris
Podolsky
, and
Nathan
Rosen
, “
Can quantum-mechanical description of reality be considered complete?
,”
Phys. Rev.
47
,
777
780
(
1935
).
10.
Travis
Norsen
, “
Einstein’s boxes
,”
Am. J. Phys.
73
(
2
),
164
176
(
2005
).
11.
The terminology of “hidden variables” is unfortunate because, at least in the one existing example of a serious hidden variables theory (the de Broglie-Bohm “pilot wave” theory, which adds to the standard quantum mechanical wave function definite particle positions obeying a deterministic evolution law), the “hidden variables” are not hidden. In Ref. 18, Bell remarked that “it would be appropriate to refer to the ×s as ‘exposed variables’ and to ψ as a ‘hidden variable.’ It is ironic that the traditional terminology is the reverse of this.” Similarly in Ref. 12, he writes “Although [in Bohmian mechanics] Ψ is a real field it does not show up immediately in the result of a single ‘measurement,’ but only in the statistics of many such results. It is the de Broglie-Bohm variable X that shows up immediately each time. That X rather than Ψ is historically called a ‘hidden’ variable is a piece of historical silliness.”12 It is also relevant that the wave function ψ is “hidden” in the sense of being not accessible via experiment, even in orthodox quantum theory, which is the primary example of a non-hidden-variable theory. See
Roderich
Tumulka
, “
Understanding Bohmian mechanics: A dialogue
,”
Am. J. Phys.
72
(
9
),
1220
1226
(
2004
).
12.
John S.
Bell
, “
On the impossible pilot wave
,”
Found. Phys.
12
,
989
99
(
1982
); reprinted in Ref. 8, pp. 159–168.
13.
John S.
Bell
, “
Bertlmann’s socks and the nature of reality
,”
J. Physique, Colloque C2, suppl. au numero 3, Tome
42
,
C2
41
61
(
1981
); reprinted in Ref. 8, pp. 139–158.
14.
John S.
Bell
, “
On the Einstein-Podolsky-Rosen paradox
,”
Physics
1
,
195
200
(
1964
); reprinted in Ref. 8, pp. 14–21.
15.
John S.
Bell
, “
Speakable and unspeakable in quantum mechanics
,” introductory remarks at Naples-Amalfi meeting, May 7,
1984
; reprinted in Ref. 8, pp.
169
172
.
16.
Alain
Aspect
,
J.
Dalibard
, and
G.
Roger
, “
Experimental test of Bell’s inequalities using time-varying analyzers
,”
Phys. Rev. Lett.
49
,
1804
1807
(
1982
).
17.
Gregor
Weihs
,
T.
Jennewein
,
C.
Simon
,
H.
Weinfurter
, and
A.
Zeilinger
, “
Violation of Bell’s inequality under strict Einstein locality conditions
,”
Phys. Rev. Lett.
81
,
5039
5043
(
1998
).
18.
John S.
Bell
, “
Quantum mechanics for cosmologists
,” in
Quantum Gravity 2
, edited by
C.
Isham
,
R.
Penrose
, and
D.
Sciama
(
Clarendon Press
,
Oxford
,
1981
); reprinted in Ref. 8, pp.
117
138
.
19.
The Ghost in the Atom
, interview with J. S. Bell, edited by
P. C. W.
Davies
and
J. R.
Brown
(
Cambridge U.P.
,
Cambridge
,
1986
), Chap. 3.
20.
See
J. S.
Bell
, “
How to teach special relativity
,”
Progress in Scientific Culture
1
(
1976
); reprinted in Ref. 8, pp.
67
80
.
21.
Huw
Price
,
Time’s Arrow and Archimedes’ Point
(
Oxford U.P.
,
Oxford
,
1997
).
22.
John S.
Bell
, “
Are there quantum jumps?
,” in
Schrödinger: Centenary celebration of a Polymath
, edited by
C. W.
Kilmister
(
Cambridge U.P.
,
Cambridge
,
1987
); reprinted in Ref. 8, pp.
201
212
.
23.
Roderich
Tumulka
, “
A relativistic version of the Ghirardi-Rimini-Weber model
,”
J. Stat. Phys.
125
,
821
840
(
2006
). See also Tim Maudlin, “Non-local correlations in quantum theory: how the trick might be done,” in Einstein, Relativity and Absolute Simultaneity, edited by W. L. Craig and Q. Smith (Routledge, New York, 2008).
24.
Abner
Shimony
, “
Bell’s theorem
,” in
The Stanford Encyclopedia of Philosophy
, edited by
Edward N.
Zalta
, <plato.stanford.edu/archives/sum2009/entries/bell-theorem/>.
25.
Eugene P.
Wigner
, “
Unterpretation of quantum mechanics
,” in
Quantum Theory and Measurement
, edited by
J. A.
Wheeler
and
W. H.
Zurek
(
Princeton U.P.
,
Princeton
,
1983
).
26.
N.
David Mermin
, “
Hidden variables and the two theorems of John Bell
,”
Rev. Mod. Phys.
65
(
3
),
803
815
(
1993
).
27.
Jon
Jarrett
, “
Bell’s theorem: A guide to the implications
,” in
Philosophical Consequences of Quantum Theory
, edited by
J. T.
Cushing
and
E.
McMullin
(
University of Notre Dame Press
,
Notre Dame
,
1989
).
28.
Anton
Zeilinger
, “
The message of the quantum
,”
Nature
438
,
743
(
2005
).
29.
George
Greenstein
and
A.
Zajonc
,
The Quantum Challenge
, 2nd ed. (
Jones and Bartlett
,
Sudbury, MA
,
2005
).
30.
David J.
Griffiths
,
Introduction to Quantum Mechanics
(
Prentice Hall
,
Upper Saddle River, NJ
,
1995
).
31.
John S.
Townsend
,
A Modern Approach to Quantum Mechanics
(
McGraw-Hill
,
New York
,
1992
).
32.
J. J.
Sakurai
,
Modern Quantum Mechanics
(
Addison-Wesley
,
Boston
,
1994
).
33.
For Bell, there was no important distinction between “locality” and “local causality.” For example, Bell first used the phrase “local causality” in print in Ref. 36. In the same paper, he refers to the inequality (which he has shown how to derive from “local causality”) as “the locality inequality” and remarks that the detailed discussion of “local causality” in Sec. 2 was “an attempt to be rather explicit and general about the notion of locality, along lines only hinted at in previous publications,” see Ref. 36. This usage is consistent with his later publications. See, for example, Ref. 8, pp. xi–xii and Ref. 7.
34.
Charles
Mann
and
Robert
Crease
, “
John Bell, particle physicist”
(interview),
Omni
10
(
8
),
84
92
and 121 (
1988
).
35.
John S.
Bell
, “
Beables for quantum field theory
,” CERN-TH 4035/84, August 2, 1984; reprinted in Ref. 8, pp.
173
180
.
36.
John S.
Bell
, “
The theory of local beables
,”
Epistemological Lett.
9
,
11
24
(
1976
); reprinted in Ref. 8, pp. 52–62.
37.
John S.
Bell
, “
Against ‘measurement’
,” in
62 Years of Uncertainty
, edited by
A. I.
Miller
(
Plenum Publishers
,
New York
,
1989
); reprinted in Ref. 8, pp.
213
231
.
38.
Travis
Norsen
, “
Against ‘Realism’
,”
Found. Phys.
37
(
3
),
311
340
(
2007
).
39.
John F.
Clauser
,
M. A.
Horne
,
A.
Shimony
, and
R. A.
Holt
, “
Proposed experiment to test local hidden-variable theories
,”
Phys. Rev. Lett.
23
,
880
884
(
1969
).
40.
John S.
Bell
, “
Free variables and local causality
,”
Epistemological Lett.
15
(
1977
); reprinted in Dialectica 39 103–106 (1985) and in Ref. 8, pp.
100
104
.
41.
Jeremy
Butterfield
, “
Bell’s theorem: What it takes
,”
British J. Philos. Sci.
42
,
41
83
(
1992
)
42.
Harvey R.
Brown
,
Physical Relativity
(
Oxford University Press
,
Oxford
,
2005
).
43.
Tim
Maudlin
,
Quantum Non-Locality and Relativity
, 2nd ed. (
Blackwell
,
Malden, MA
,
2002
).
44.
What we call “local signaling” is also sometimes called “no signaling” or “signal locality.”
45.
See, for example, Refs. 33 and 34 and
Leslie E.
Ballentine
and
Jon P.
Jarrett
, “
Bell’s theorem: Does quantum mechanics contradict relativity?
Am. J. Phys.
55
(
8
),
696
701
(
1987
).
46.
For a more detailed version of this argument see
Travis
Norsen
, “
Bell Locality and the nonlocal character of nature
,”
Found. Phys. Lett.
19
,
633
655
(
2006
).
47.
N.
David Mermin
, “
Bringing home the atomic world: Quantum mysteries for anybody
,”
Am. J. Phys.
49
,
940
943
(
1981
).
48.
Abner
Shimony
,
M. A.
Horne
, and
J. F.
Clauser
, “
Comment on ‘The theory of local beables’
,”
Epistemological Lett.
9
(
1976
); reprinted as “An exchange on local beables,” Dialectica39,
97
101
(1985).
49.
About the de Broglie-Bohm theory, Bell wrote: “No one can understand this theory until he is willing to think of ψ as a real objective field rather than just a ‘probability amplitude.’ Even though it propagates not in 3-space but in 3N-space” in Ref. 18. Some proponents of the de Broglie-Bohm pilot wave theory prefer to interpret the wave function in that theory not as a beable, but rather as a law. See, for example, S. Goldstein and N. Zanghi, 1618 “Reality and the role of the wavefunction in quantum theory,” forthcoming in D. Albert and A. Ney, eds. The Wave Function: Essays in the Metaphysics of Quantum Mechanics (Cambridge, UP., 2011).
50.
Travis
Norsen
, “
The theory of (exclusively) local beables
,”
Found. Phys.
40
,
1858
1884
(
2010
).
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