The Concept of Probability pp 71-90 | Cite as

# Bell’s Theorem: A Counterexample that Agrees with the Quantum Formalism

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## Abstract

A family {Aμ} of models is here constructed whose members satisfy all the postulates of locality due to Clauser and Home (CH), and whose members *converge uniformly* to a unique *limit function* identical with the function of the quantum formalism (QF) model for the Einstein-Podolsky-Rosen-Bohm (EPRB) *ideal* experiment. This renders *invalid* Bell’s theorem. My construction establishes my proposed *local explanatory* theory of the EPRB experiment from more basic postulates of a structural character as Einstein had in mind. The theory explains, in purely local terms, *the* characteristic trait of the EPRB experiment where the directions of polarization of the single photon pairs are *chosen at random* by the process of annihilation from the singlet state, and the directions of the polarizer settings are chosen at random (or nearly so) by the switches as in the Aspect experiment. Moreover, a bona fide *specified form* of the generalized CH inequality, known as CH(4), is here constructed which is *satisfied* by the QF model itself. This *directly* demonstrates the *consistency* of CH(4) with the quantum formalism.

## Keywords

Limit Function Uniform Convergence Singlet State Universal Quantifier Photon Pair## Preview

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## References and Notes

- 1.M. Jammer
*The Philosophy of Quantum Mechanics*John Wiley, New York 1974 116Google Scholar - 2.A. Einstein
*Dialectica 2*320 1948CrossRefzbMATHGoogle Scholar - 3.J. S. Bell
*Physics I*195 (1964)Google Scholar - 4.A. Einstein, B. Podolsky, and N. Rosen,
*Phys. Rev. 47*777 1935CrossRefADSGoogle Scholar - 4.AD. J. Bohm,
*Quantum Theory*Prentice-Hall Englewood Cliffs N.J. 1951 611-623Google Scholar - 5.Th. D. Angelidis
*Phys. Rev. Lett. 51*1819 1983CrossRefADSMathSciNetGoogle Scholar - 5.ATh.D.Angelidis
*Open Questions in Quantum Physics*A. van der Merwe Reidel Dordrecht, Holland 1985 51-62CrossRefGoogle Scholar - 6.A. Aspect, J. Dalibard, G. Roger
*Phys. Rev. Lett. 49*1804 1982CrossRefADSMathSciNetGoogle Scholar - 7.B. d′Espagnat,
*In Search of Reality*Springer-Verlag London 1983Google Scholar - 8.J. F. Clauser A. Shimony
*Rep. Prog. Phys. 41*1881 1978CrossRefADSGoogle Scholar - 9.J. F. Clauser M. A. Home
*Phys. Rev. D*10 526 1974CrossRefADSGoogle Scholar - 10.P R Halmos
*Naive Set Theory*Van Nostrand Reinhold, New York, 1960 34zbMATHGoogle Scholar - 11.A. M. Gleason
*Fundamentals of Abstract Analysis*Addison-Wesley, Reading, Mass. 77, 245-248.zbMATHGoogle Scholar - 12.E. J. McShane T. A. Botts,
*Real Analysis*Van Nostrand, Princeton, London 1959 32-33, 68-69, 81-82zbMATHGoogle Scholar - 13.Th. D. Angelidis,
*Microphysical Reality and Quantum Formalism*A. van der Merwe Reidel, Dordrecht, Holland, 1988 457-478CrossRefGoogle Scholar - 14.Th. D. Angelidis
*Phys. Rev. Lett. 53*1022 1984CrossRefADSGoogle Scholar - 14.AThD Angelidis
*Phys. Rev. Lett. 53*1297 1984.CrossRefADSGoogle Scholar - 15.Th. D. Angelidis, in Causality and Locality: 50 Years of the Einstein- Podolsky-Rosen Paradox (Hellenic Physical Society and Interdiscipli-nary Research Group, University of Athens, 1987).Google Scholar
- 16.J. S. Bell, CERN preprint TH-2926 (1980).Google Scholar
- 17.R.P. Feynman, Intl. J. Theor. Phys. 21, 467 (1982). Feynman wrote: ″The only difference between a probabilistic classical world and the... quantum world is that somehow or other it appears as if the probabilities would have to go negative... that′s the fundamental problem.″Note 18 of Ref. 15 explains how I tackled this ″fundamental problem.″Google Scholar
- 18.P. A. M. Dirac,
*The Principles of Quantum Mechanics*Oxford University Press, Oxford 1958 59zbMATHGoogle Scholar - 19.J. A. Wheeler
*Ann. N.Y. Acad. Sci. 48*219 1946.CrossRefADSGoogle Scholar - 20.See Ref. 18; and I.N. Sneddon, The Use of Integral Transforms (Tata McGraw-Hill, New York, Delhi,1974), p. 499. More formally, if T(x) is a distribution defined over the set R of real numbers, and a e R, the shifted distribution T(x-a) can be defined by the equantion >T(x-a) ,(|)(x>= T(x) ,cj)(x+a)> for all test functions c|)(x) defined on the space Cl(Rn).Google Scholar
- 21.A. Church
*Introduction to Mathematical Logic*Princeton University Press Princeton New Jersey 1956 9-10, 28, 37, 41-44zbMATHGoogle Scholar - 22.My local explanatory theory, based on my proof of formula (13), shows that a defender of local realism does not have to rely on the absurd loophole of the backward light cones. This loophole, used by those who believe in non-locality to dismiss local realism, conjures up an utterly improbable conspiracy, which would leave even a Cartesian demon gaping, that physical events in the overlap of the backward light cones determine both the random directions of polarization of the photon pairs and the random directions of the settings of the polarizers. Although this loophole is an instance of what one could call “Bohr′ s principle of the total experimental conditions,″” and of the related notion of “Bohm′s undivided wholeness,” and although it it admitted by Bell, Shimony et al. [see Ref. 8] to be a loophole, my local explanatory theory shows that there is no such conspiracy. On the other hand, those who believe in non-locality do not seem to have noticed that, should the “spooky action at a distance” they uphold be true, everything in the physical world would be conspiratorial, not only the events in the backward light cones. It would be a world whose very structure would prevent us from learning more about these spooky effects at any distance, and from relying on the significance of any experiment including that of Aspect et al.Google Scholar
- 23.C.A. Kocher and E.D. Commins, Phys. Rev. Lett. 18, 575 (1967). In this experiment, one polarizer ″Pi″ is fixed and the other ″P2″ is movable. These authors state: ″We have made runs with different orientations ″settings″ of the fixed polarizer ″Pi″* obtaining in each case a correlation which depends only on the relative angle ″$″.″; A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 47, 460 (1981). These authors wrote: “We never observed any deviation from rotational ″axial″ invariance.”Google Scholar