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On the problem of a local extension of the quantum formalism
1.J. von Neumann, Mathematische Grundlagen der Quantenmechanic (Springer, Berlin, 1932);
1.English translation (Princeton University, Princeton, NJ, 1955), pp. 325, 327–328.
2.L. de Broglie, in Rapport au V’ieme Congres de Physique Solvay (Gauthier-Villars, Paris, 1930);
2.and Tentative d’Interpretation Causale et Non-Lineaire de la Mechanique Ondulatoire (Gauthier-Villars, Paris, 1956).
3.A. Einstein, in Rapports et Discussions du Cinquieme Congres de Physique Solvay (Gauthier-Villars, Paris, 1928). Here, Einstein spoke of the difficulty with the description given by the quantum state |ψ〉, which he said had to be extended to a more detailed specification of the localization of a particle during its propagation. Furthermore, Einstein thought de Broglie was right in searching in this direction.
4.N. Bohr, Atomic Theory and the Description of Nature (Cambridge University, Cambridge, 1934);
4.W. Heisenberg, The Physical Principles of Quantum Theory (University of Chicago, Chicago, 1930).
5.G. Hermann, Abhandlungen der Fries’schen Schule 6, 75 (1935);
5.cf. also abstract in G. Hermann, Die Naturwissenschaften 42, 718 (1935).
6.M. Jammer, The Philosophy of Quantum Mechanics (Wiley, New York, 1974), pp. 207–210, 272–275.
7.Concerning the question of what von Neumann actually demonstrated, we suggest that the von Neumann proof can be regarded as a consistency proof of the von Neumann set of postulates (including the additivity postulate) in the sense that the class of all quantum states constitutes a model of this set of postulates. Cf. also Ref. 6, footnote 45, pp. 273–274 for an affiliated assessment, even though our assessment differs on a number of essential points.
8.A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
9.A. Einstein, Dialectica 2, 320 (1948).
10.D. J. Bohm, Phys. Rev. 85, 166, 180 (1952).
11.J. S. Bell, Rev. Mod. Phys. 38, 447 (1966).
12.D. J. Bohm, Causality and Chance in Modern Physics (Routledge and Kegan Paul, London, 1957), p. 110.
13.D. J. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, NJ, 1951), pp. 611–623.
14.P. R. Holland, Phys. Rept. 169, 316 (1988). This author of the de Broglie-Bohm school admits that Bohm’s theory has no consistent relativistic extension. We further suggest that, even if one were to concede provisionally that could be consistently constructed, would have no licence to induce a causal influence acting instantaneously at any distance in Minkowski space-time.
15.D. J. Bohm, in Microphysical Reality and Quantum Formalism, Vol. 2, edited by A. van der Merwe et al. (Kluwer Academic, Dordrecht, 1988), p. 11.
16.We suggest that Bohm’s “causal interpretation” cannot be reconciled with the causal structure of Minkowski space-time. Whence, the name “causal” seems rather inappropriate.
17.A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, 1804 (1982).
18.J. S. Bell, Physics (N.Y.) 1, 195 (1964).
19.A. Shimony, Found. Phys. 19, 1426 (1989).
20.Th. D. Angelidis, Phys. Rev. Lett. 51, 1819 (1983);
20.and in Open Questions in Quantum Physics, edited by A. van der Merwe et al. (Reidel, Dordrecht, 1985), pp. 51–62.
21.J. S. Bell, in Foundations of Quantum Mechanics, edited by B. d’Espagnat (Academic, New York, 1971), p. 178.
21.Here, Bell wrote: “…no local…hidden-variable theory can reproduce all the…predictions of quantum mechanics.
22.J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526 (1974).
23.Popper recognized and emphasized the significance of UC as crucial. Cf. K. R. Popper, in Open Questions in Quantum Physics, edited by A. van der Merwe et al. (Reidel, Dordrecht, 1985), pp. 22–23;
23.and in Determinism in Physics, edited by E. Bitsakis et al. (Gutenberg, Athens, 1985), pp. 17, 27–28;
23.and in Microphysical Reality and Quantum Formalism, Vol. 1, edited by A. van der Merwe et al. (Kluwer Academic, Dordrecht, 1988), p. 414.
24.E. T. Jaynes, in Maximum Entropy and Bayesian Methods, edited by J. Skilling (Kluwer Academic, Dordrecht, 1989), pp. 8–15. Jaynes also recognized that “the class of Bell theories does not include all local causal theories.” This is essentially what our disproof of UC established (cf. Ref. 20). Furthermore, although we agree with Jaynes’ assessment that a local causal theory need not reproduce the QF probabilities in the shape of the syntactical form (2), we should point out that the local (and causal) theory proposed here is a “Bell theory” and reproduces the QF probability function as the syntactical form (2) precisely requires, where a “Bell theory” is one whose postulates satisfy (cf. Sec. III). Thus, belongs to the intersection of the two classes.
25.The subsequent debate on UC has been clarified a little in Ref. 38. There, we also reinstated some unpublished material which had been omitted, inadvertently or otherwise, from that debate impairing its integrity. (We do thank Shimony et al. for having apparently corrected their initial belief in the purported validity of the stronger UC of Bell et al.).
26.Th. D. Angelidis, in The Concept of Probability, edited by E. Bitsakis and C. Nicolaides (Kluwer Academic, Dordrecht, 1989), pp. 71–90.
27.Th. D. Angelidis, Proceedings of the Athens Academy 66, 292 (1991).
28.J. L. Bell and M. Machover, A Course In Mathematical Logic (North-Holland, Amsterdam, 1977), pp. 34–35, 51, 109, 165, 170.
29.A. Church, Introduction to Mathematical Logic (Princeton University, Princeton, NJ, 1956), pp. 9–10, 28, 30–31.
30.R. P. Feynman, Int. 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. 38 explains how we tackled this “fundamental problem.
31.J. S. Bell, CERN preprint TH-2926 (1980), p. 13;
31.published in J. dePhysique 42, C2, 41–61 (1981).
32.R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. 3. (Addison-Wesley, Reading, MA, 1965), Chap. 11, pp. 11–12.
33.Y. C. Bruhat, C. W. Morette, and M. D. Bleick, Analysis, Manifolds and Physics (North-Holland, Amsterdam, New York, 1977), pp. 356, 358.
34.P. R. Halmos, Naive Set Theory (Van Nostrand Reinhold, New York, 1960), p. 34.
35.In fact, rotating the polarizers about their common axis is used to test the axial invariance of the distribution of the photon pairs emitted by the source. Cf. C. A. Kocher and E. D. Commins, Phys. Rev. Lett. 18, 575 (1967).
35.In this experiment, one polarizer is held fixed and the other is movable. These authors wrote: “We have made runs with different orientations [settings] of the fixed polarizer obtaining in each case a correlation which depends only on the relative angle [θ].” Cf. also 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.
36.E. J. McShane and T. A. Botts, Real Analysis (Van Nostrand, Princeton, 1959), pp. 32–33, 68–69, 81–82.
37.A. M. Gleason, Fundamentals of Abstract Analysis (Addison-Wesley, Reading, MA, 1966), pp. 77, 245–248.
38.Th. D. Angelidis, in Causality and Locality in Microphysics, edited by E. Bitsakis (Hellenic Physical Society, Athens, 1988), pp. 131–171.
39.T. H. Hassan, A. J. Duncan, W. Perrie, H. Kleinpoppen, and M. Merzbacher, Phys. Rev. Lett. 62, 237 (1989).
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