Abstract
This work has two objectives. The first is to begin a mathematical formalism appropriate to treating particles which only interact with each otherindirectly due to hypothesized memory effects in a stochastic medium. More specifically we treat a situation in which a sequence of particles consecutively passes through a region (e.g., a measuring apparatus) in such a way that one particle leaves the region before the next one enters. We want to study a situation in which a particle may interact with other particles that previously passed through the system via disturbances made in the region by these previous particles.
Second, we apply the type of stochastic process appearing in this context to the stochastic interpretation of quantum mechanics to obtain a modified version of this interpretation. This version is free of many of the criticisms made against the stochastic interpretation of quantum mechanics.
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References
A. F. Prado de Andrade, “Particulas que Interagim Indiretamente e a Interpretação Estocástica de Mecânica Quântica,” PhD Thesis, Instituto de Matemática, Universidade Estadual de Campinas, Campinas, S.P., Brazil.
L. F. Clauser and A. Shimony, “Bell's theorem: Experimental tests and implications,”Rep. Prog. Phys. 41, 1881 (1978).
F. Selleri and G. Tarozzi, “Quantum mechanics, reality and separability,”Riv. Nuovo Cimento 4, 1 (1981).
V. Buonomano, “A limitation on Bell's inequality,”Ann. Inst. Henri Poincaré 29, 379 (1978).
J. R. Kinney, “Continuity properties of sample functions of Markov processes,”Trans. Am. Math. Soc. 74, 280 (1953), Theorem VII.
L. M. Graves,The Theory of Functions of Real Variables, (McGraw-Hill, New York, 1965), Chap. VII.
G. C. Ghirardi, C. Omero, A. Rimini, and T. Weber, “The stochastic interpretation of quantum mechanics: A critical review,”Riv. Nuovo Cimento 1 1 (1978).
E. Nelson,Bull. Am. Math. Soc. 84, 121 (1978).
E. Nelson,Dynamical Theories of Brownian motion (Princeton University Press, Princeton, 1967).
H. Grabert, P. Hanggi, and P. Talkner, “Is quantum mechanics equivalent to a classical process?”,Phys. Rev. A 19, 2440 (1979).
M. Davidson, “A generalization of the Fényes-Nelson Stochastic model of quantum mechanics,”Lett. Math. Phys. 3, 271 (1979).
V. Buonomano, “Quantum mechanics as a non-ergodic classical statistical theory,”Nuovo Cimento B 57, 146 (1980).
V. Buonomano, “The non-ergodic interpretation of quantum mechanics,” inMicrophysical Reality and Quantum Formalism, A. van der Merwe, F. Selleri, and G. Tarozzi, eds. (Kluwer Academic, Dordrecht, 1988).
L. Ballentine,Rev. Mod. Phys. 42, 358 (1979).
R. J. Glauber, “Optical coherence and photon statistics,” inQuantum Optics and Electronics, C. de Witt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965), p. 63.
L. Mandel and E. Wolf, “Coherence properties of optical fields,”Rev. Mod. Phys. 37, 231 (1965).
H. Margenau, “Measurements and quantum states: Part I,”Philos. Soc. 30, 1 (1963).
V. Buonomano, “The number of counting measurements, the visibility and the intensity,”Lett. Nuovo Cimento 43, 69 (1985); Also “Testing the ergodic assumption in the low-intensity interference experiments,” with F. Bartmann,Il Nuovo Cimento 95B, 99 (1986).
J. G. Gilson, “On stochastic theories of quantum mechanics,”Proc. Camb. Philos. Soc. 64, 1061 (1968).
E. Onofre, “The stochastic interpretation of quantum mechanics: A reply to Ghirardi,”Lett. Nuovo Cimento 24, 253 (1979)_.
E. Onofre, “Schrödinger's equation,” inGroup Theoretical Methods in Physics, A. Jammer, T. Janssen, and M. Boon, eds. (Springer, Berlin, 1976), p. 582.
A. F. Kracklauer, “Comment on derivation of the Schrödinger equation from Newtonian mechanics,”Phys. Rev. D 10, 1358 (1974).
B. H. Lavenda, “On the equivalence between classical Markov processes and quantum mechanics,”Lett. Nuovo Cimento 27, 433 (1980).
S. Albeverio and R. Hoegh-Krohn, “A remark on the connection between stochastic mechanics and the heat equation,”J. Math. Phys. 15, 1745 (1974).
B. Mielnik and G. Tengstrand, “Nelson-Brown motion: Some question marks,”Int. J. Theory. Phys. 19, 239 (1980).
E. Nelson, “Derivation of the Schrödinger equation from Newtonian mechanics,”Phys. Rev. 150, 1079 (1966).
S. M. Moore, “Can stochastic physics be a complete theory of nature?”,Found. Phys. 9, 237 (1979).
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Buonomano, V., Prado de Andrade, A.F. Stochastic processes for indirectly interacting particles and stochastic quantum mechanics. Found Phys 18, 401–426 (1988). https://doi.org/10.1007/BF00732547
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DOI: https://doi.org/10.1007/BF00732547