Einstein held that the formalism of quantum mechanics involves “spooky actions at a distance.” In the 1960s, Bell amplified this by showing that the predictions of quantum mechanics disagree with the results of any locally causal description. It should be appreciated that accepting nonlocal descriptions while retaining causality leads to a clash with relativity. Furthermore, the causal arrow of time by definition contradicts time-reversal symmetry. For these reasons, Wheeler and Feynman, Costa de Beauregard, Cramer, Price, and others have advocated abandoning microscopic causality. In this paper, a simplistic but concrete example of this line of thought is presented, in the form of a retro-causal toy model that is stochastic and provides an appealing description of the quantum correlations discussed by Bell. It is concluded that Einstein’s “spooky actions” may occur “in the past” rather than “at a distance,” resolving the tension between quantum mechanics and relativity and opening unexplored possibilities for future reformulations of quantum mechanics.
REFERENCES
1.
2.
W.
Heisenberg
, “Quantum-theoretical re-interpretation of kinematic and mechanical relations
,” Z. Phys.
33
, 879
–893
(1925
)[original in German; English translation available in
B. L.
van der Waerden
, Sources of Quantum Mechanics
(North-Holland
, Amsterdam
, 1967
)].3.
E.
Schrödinger
, “Quantization as a problem of proper values (part I)
,” Ann. Phys.
79
, 361
–76
(1926
)[original in German; English translation available in
E.
Schrödinger
, Collected Papers on Wave Mechanics
(Chelsea
, New York
, 1978
)].4.
R. P.
Feynman
, “Space-time approach to non-relativistic quantum mechanics
,” Rev. Mod. Phys.
20
, 367
–387
(1948
).5.
D.
Bohm
, “A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. I & II
,” Phys. Rev.
85
, 166
–179
and 180–193 (1952
).6.
E.
Nelson
, “Derivation of the Schrödinger equation from Newtonian mechanics
,” Phys. Rev.
150
, 1079
–1085
(1966
).7.
D. F.
Styer
et al., “Nine formulations of quantum mechanics
,” Am. J. Phys.
70
, 288
–297
(2002
).8.
See, for example,
L. D.
Landau
and E. M.
Lifshitz
, Quantum Mechanics: Non-Relativistic Theory
, 3rd ed. (Pergamon
, New York
, 1997
).9.
H.
Everett
III, “‘Relative state’ formulation of quantum mechanics
,” Rev. Mod. Phys.
29
, 454
–462
(1957
).10.
J. G.
Cramer
, “The transactional interpretation of quantum mechanics
,” Rev. Mod. Phys.
58
, 647
–687
(1986
).11.
For a review, see
F.
Laloë
, “Do we really understand quantum mechanics? Strange correlations, paradoxes and theorems
,” Am. J. Phys.
69
, 655
–701
(2001
).12.
Bell’s relevant publications are collected in
John S. Bell on the Foundations of Quantum Mechanics
, edited by M.
Bell
, K.
Gottfried
, and M.
Veltman
(World Scientific
, Singapore
, 2001
) and in Ref. 13.13.
J. S.
Bell
, Speakable and Unspeakable in Quantum Mechanics
, revised edition (Cambridge U. P.
, Cambridge
, 2004
).14.
J. S.
Bell
, “On the Einstein–Podolsky–Rosen paradox
,” Physics
1
, 195
–200
(1964
)(Ref. 13, Chap. 2).
15.
(Ref. 13, Chap. 23).
16.
For a recent example, see
G.
Blaylock
, “The EPR paradox, Bell’s inequality, and the question of locality
,” Am. J. Phys.
78
, 111
–120
(2010
);T.
Maudlin
, “What Bell proved: A reply to Blaylock
,” Am. J. Phys.
78
, 121
–125
(2010
).17.
J. S.
Bell
, “On the problem of hidden variables in quantum mechanics
,” Rev. Mod. Phys.
38
, 447
–452
(1966
)(Ref. 13, Chap. 1).
18.
D.
Bohm
and Y.
Aharonov
, “Discussion of experimental proof for the paradox of Einstein, Rosen and Podolsky
,” Phys. Rev.
108
, 1070
–1076
(1957
).19.
A.
Einstein
, B.
Podolsky
, and N.
Rosen
, “Can quantum-mechanical description of physical reality be considered complete?
,” Phys. Rev.
47
, 777
–780
(1935
).20.
See
J. S.
Bell
, “Bertlmann’s socks and the nature of reality
,” J. Phys. (Paris)
42
, C2
–41
(1981
) (Ref. 13, Chap. 16), especially note 10.21.
T.
Norsen
, “Against ‘Realism’
,” Found. Phys.
37
, 311
–340
(2007
).22.
The word “causality” appears in early publications, including Ref. 14, with the meaning of “determinism,” often in phrases such as “complete causality” or “ideal causality.” It was used by Bell in the sense of an arrow of time in later publications.
23.
For example,
J. S.
Bell
, “Free variables and local causality
,” Epistemological Letters
, 15
(1977
) (Ref. 13, Chap. 12) says, “‘It has been assumed that the settings of instruments are in some sense free variables….’ For me this means that the values of such variables have implications only in their future light cones.” The flaw is in adopting this interpretation of “free variables,” an ostensibly time-reversal symmetric concept, without mentioning that the causal arrow of time is involved as a separate assumption.24.
For example,
J. S.
Bell
, “La nouvelle cuisine
,” in Between Science and Technology
, edited by A.
Sarlemijn
and P.
Kroes
(Elsevier/North-Holland
, New York/Amsterdam
, 1990
), pp. 97
–115
(Ref. 13, Chap. 24).
See also
J. S.
Bell
, “How to teach special relativity
,” Progress in Scientific Culture
1
(2
), 1
–13
(1976
)(Ref. 13, Chap. 9).
He retreated from this position in
J. S.
Bell
, “Are there quantum jumps?
,” in Schrödinger. Centenary of a Polymath
(Cambridge U. P.
, Cambridge
, 1987
)(Ref. 13, Chap. 22).
26.
See, for example,
D. Z.
Albert
and R.
Galchen
, “A quantum threat to special relativity
,” Sci. Am.
300
(3
), 32
–39
(2009
).27.
H.
Price
, “Toy models of retrocausality
,” Stud. Hist. Philos. Mod. Phys.
39
, 752
–761
(2008
).28.
For example,
B.
d’Espagnat
, “Nonseperability and the tentative descriptions of reality
,” Phys. Rep.
110
, 201
–264
(1984
), refers to the work of Costa de Beauregard and states that “a ‘backwards causality’…was never explicitly defined” (p. 246).29.
Early versions of the present work are available at ⟨pirsa.org/06110017⟩ and ⟨arxiv.org/abs/0807.2041v1⟩.
30.
J. A.
Wheeler
and R. P.
Feynman
, “Interaction with the absorber as the mechanism of radiation
,” Rev. Mod. Phys.
17
, 157
–181
(1945
);J. A.
Wheeler
and R. P.
Feynman
, Rev. Mod. Phys.
“Classical electrodynamics in terms of direct interparticle action,” ibid. 21
, 425
–433
(1949
).31.
See
Y.
Aharonov
and L.
Vaidman
, “The two-state vector formalism of quantum mechanics: An updated review
,” in Time and Quantum Mechanics
, Lect. Notes Phys.
, Vol. 734
, edited by J. G.
Muga
, R.
Sala Mayato
, and I. L.
Egusquiza
(Springer
, Berlin
, 2008
), pp. 399
–447
.32.
O.
Costa de Beauregard
, “Une réponse à l’argument dirigé par Einstein, Podolsky, et Rosen contre l’interprétation bohrienne des phénomènes quantiques
,” C. R. Hebd. Seances Acad. Sci.
236
, 1632
–1634
(1953
);O.
Costa de Beauregard
, “Time symmetry and the Einstein paradox
,” Nuovo Cimento B
42
, 41
–64
(1977
);O.
Costa de Beauregard
, Nuovo Cimento B
“Time symmetry and the Einstein paradox–II,” ibid. 51
, 267
–279
(1979
).The authors of Refs. 10 and 32 have, years later, taken their arguments much further, resulting in publications, which are, at best, speculative (including discussions of signaling backward in time and of paranormal phenomena, respectively). The transition made by these authors may constitute a remarkable history-of-science phenomenon but does not detract from the validity of the original arguments.
33.
R. P.
Feynman
, “Simulating physics with computers
,” Int. J. Theor. Phys.
21
, 467
–488
(1982
).34.
R. I.
Sutherland
, “Bell’s theorem and backwards-in-time causality
,” Int. J. Theor. Phys.
22
, 377
–384
(1983
), and references therein.35.
It is interesting to note that
N.
Bohr
, “Can quantum-mechanical description of physical reality be considered complete?
,” Phys. Rev.
48
, 696
–702
(1935
), already discusses a measurement apparatus set up a la EPR, which allows switching between position or momentum detection after the detected particle has continued its flight and is at a remote location. The fact that even after the particle has left, we are “still left with a free choice whether we wish to know the momentum of the particle or its initial position” (italics in original) did not deter Bohr, as he did not associate “physical reality” with the position and momentum coordinates, nor did he explicitly refer to the value one of these variables would have at a time before the choice was made. Instead, he pointed out that the choice would merely affect “the possible types of predictions regarding the future behavior of the system.”36.
H.
Price
, “A neglected route to realism about quantum mechanics
,” Mind
103
, 303
–336
(1994
);Time’s Arrow and Archimedes’ Point
(Oxford U. P.
, Oxford
, 1996
).37.
R. I.
Sutherland
, “A corollary to Bell’s theorem
,” Nuovo Cimento B
88
, 114
–118
(1985
);38.
Wheeler and Feynman give an example of a retro-causal physical theory avoiding causal loops in Fig. 2 of the second article in Ref. 30.
39.
For example,
A.
Aspect
, J.
Dalibard
, and G.
Roger
, “Experimental test of Bell’s inequalities using time-varying analyzers
,” Phys. Rev. Lett.
49
, 1804
–1807
(1982
).40.
C. A.
Kocher
and E. D.
Commins
, “Polarization correlation of photons emitted in an atomic cascade
,” Phys. Rev. Lett.
18
, 575
–577
(1967
);C. N.
Yang
, “Selection rules for the dematerialization of a particle into two photons
,” Phys. Rev.
77
, 242
–245
(1950
).41.
That the many-worlds view is nonlocal in the sense relevant to Bell’s theorem has been disputed and should be contrasted with, for example, the conclusions in Ref. 16.
42.
A.
Shimony
, M. A.
Horne
, and J. F.
Clauser
, “Comment on ‘The theory of local beables’
,” Epistemological Lett.
, 13
(1976
) [reproduced inA.
Shimony
, M. A.
Horne
, and J. F.
Clauser
, “Dialectica
39
, 97
–102
(1985
)] suggested denying the free-variable status of the apparatus settings and by introducing a conspiratorial setup in which the experimenters fail to exercise free will in studying Bell-type correlations.In this scenario, common causes in the past determine and (and ). If this were the case, use of a distribution of the form could be justified without invoking retro-causation, that is, causes in the future (Ref. 36). This possibility has been called superdeterministic by Bell (Ref. 24). Rather than delving into a philosophical discussion of “free will,” it suffices to point out that and do play the role of free variables in standard quantum mechanics, and that the purpose here is to characterize the class of models which are capable of reproducing quantum mechanics in this sense (see also Ref. 23).
43.
See, for example,
N. D.
Mermin
, “Is the moon there when nobody looks? Reality and the quantum theory
,” Phys. Today
38
(4
), 38
–47
(1985
).44.
R. W.
Spekkens
, “Evidence for the epistemic view of quantum states: A toy theory
,” Phys. Rev. A
75
, 032110
(2007
), and references therein.45.
D.
Bouwmeester
, J. -W.
Pan
, K.
Mattle
, M.
Eibl
, H.
Weinfurter
, and A.
Zeilinger
, “Experimental quantum teleportation
,” Nature (London)
390
, 575
–579
(1997
).46.
The citation count for Ref. 14 has roughly doubled each decade and is now greater than 200/year.
47.
J. 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
).48.
J. F.
Clauser
and M. A.
Horne
, “Experimental consequences of objective local theories
,” Phys. Rev. D
10
, 526
–535
(1974
);see especially note 15 therein and note 10 in
J. S.
Bell
, “Introduction to the hidden-variable question
,” in Proceedings of the International School of Physics “Enrico Fermi,” Course IL: Foundations of Quantum Mechanics
, edited by B.
d'Espagnat
(Academic
, New York
, 1971
), pp. 171
–181
(Ref. 13, Chap. 4).
49.
G.
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
);M. A.
Rowe
, D.
Kielpinski
, V.
Meyer
, C. A.
Sackett
, W. M.
Itano
, C.
Monroe
, and D. J.
Wineland
, “Experimental violation of a Bell’s inequality with efficient detection
,” Nature (London)
409
, 791
–794
(2001
).50.
D. M.
Greenberger
, M. A.
Horne
, and A.
Zeilinger
, “Multiparticle interferometry and the superposition principle
,” Phys. Today
46
(8
), 22
–29
(1993
).51.
D.
Bouwmeester
, J. -W.
Pan
, M.
Daniell
, H.
Weinfurter
, and A.
Zeilinger
, “Observation of three-photon Greenberger–Horne–Zeilinger entanglement
,” Phys. Rev. Lett.
82
, 1345
–1349
(1999
).52.
See, for example,
A. K.
Ekert
, “Quantum cryptography based on Bell’s theorem
,” Phys. Rev. Lett.
67
, 661
–663
(1991
).53.
It has become more-or-less standard to interpret the failure of Einstein’s local realism as in the following example:
H. A.
Wiseman
, “From Einstein’s theorem to Bell’s theorem: a history of quantum non-locality
,” Contemp. Phys.
47
, 79
–88
(2006
), uses quotes of Einstein to list the possible falsehoods identified by Bell’s theorem as locality (the postulates of relativity); and the independent reality of distant events. However, focusing on causal mathematical descriptions (as the assumption of causality is obviously being made), it seems that one is merely distinguishing here between two types of nonlocality: Signals which propagate instantaneously, or nonlocal variables, which are not simply associated with particular positions in space. Both of these violate relativistic causality.See also Ref. 21.
54.
See, for example,
J. S.
Bell
, “de Broglie–Bohm, delayed-choice, double-slit experiment, and density matrix
,” Int. J. Quantum Chem., Quantum Chem. Symp.
14
, 155
–159
(1980
)(Ref. 13, Chap. 14).
55.
G.
Bacciagaluppi
and A.
Valentini
, Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference
(Cambridge U. P.
, Cambridge
, 2009
).56.
The notation allows for spin and spinors; else would have been used.
57.
For example, Fig. 1 of Ref. 10 implicitly hints at this.
See also
J. G.
Cramer
, “Generalized absorber theory and the Einstein–Podolsky–Rosen paradox
,” Phys. Rev. D
22
, 362
–376
(1980
).58.
This law was discovered well before single photons were considered and originally related to intensities of light. See, e.g.,
B.
Kahr
and K.
Claborn
, “The lives of Malus and his bicentennial law
,” ChemPhysChem
9
, 43
–58
(2008
).59.
It is interesting to note that the causal loops that can be generated with the symmetric version of the toy model involve loss of predictive power, rather than the contradictions considered in Ref. 37. The agreement with quantum mechanics guarantees that contradictions cannot occur.
See
J.
Berkovitz
, “On predictions in retro-causal interpretations of quantum mechanics
,” Stud. Hist. Philos. Mod. Phys.
39
, 709
–735
(2008
).60.
Translating the standard quantum mechanics of a single particle into a retro-causal model in this manner gives essentially the transactional interpretation of Ref. 10.
Doing the same for the two-photon system gives the description of
N.
Rosen
, “Bell’s theorem and quantum mechanics,”
Am. J. Phys.
62
, 109
–110
(1994
)(This type of description could be the one the authors of Ref. 34 had in mind.) Note that Rosen associated with the time of measurement, yielding a nonlocal description (as with the abovementioned ), and did not mention causality and the arrow of time.
61.
The table refers to the standard, nonrelativistic version of Bohmian mechanics—it is irrelevant to the present discussion that adapting Bohmian mechanics to photons may require a relativistic extension.
62.
D.
Bohm
, Causality and Chance in Modern Physics
(Routledge
, London
, 1957
);see also Ref. 6.
63.
S.
Goldstein
and R.
Tumulka
, “Opposite arrows of time can reconcile relativity and nonlocality
,” Class. Quantum Grav.
20
, 557
–564
(2003
).64.
R.
Sutherland
, “Causally symmetric Bohm model
,” Stud. Hist. Philos. Mod. Phys.
39
, 782
–805
(2008
).65.
For a recent generalization of this rule, see
M.
Pawłowski
, T.
Paterek
, D.
Kaszlikowski
, V.
Scarani
, A.
Winter
, and M.
Zukowski
, “Information causality as a physical principle
,” Nature (London)
461
, 1101
–1104
(2009
).67.
Note that although classically simulating a quantum computer is widely conjectured to require exponentially many steps, it is known not to require a large memory—see, for example,
E.
Bernstein
and U.
Vazirani
, “Quantum complexity theory
,” in Proceedings of the 25th Annual ACM Symposium on Theory of Computing
(ACM
, New York
, 1993
), pp. 11
–20
. The fact that the memory requirements are modest may indicate that the state of a quantum system, when it is not restricted to a single time, should not be so complicated.68.
J. S.
Bell
, “On the impossible pilot wave
,” Found. Phys.
12
, 989
–999
(1982
)(Ref. 13, Chap. 17).
© 2010 American Association of Physics Teachers.
2010
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
