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Probability in the quantum world

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Reflections on Quanta, Symmetries, and Supersymmetries
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Abstract

Probability has been a part of quantum theory from its very beginnings. The original probabilistic interpretation of quantum mechanics was put forward by Max Born. However the idea that the understanding of Nature had to be statistical was unacceptable to a lot of people including Einstein, and there was substantial criticism of this aspect of quantum theory. Eventually, the ideas of von Neumann, Bell, Feynman, Mackey, Glea- son, and many others on the probabilistic aspects of quantum theory clarified the situation and answered the criticisms. These contributions have made the role of probability in the quantum world both far-reaching and profound.

This essay and the previous one are based on lectures given at Howard University, Washington D.C., sponsored by my friend D. Sundararaman, in the 1990s.

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References

  1. A. M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885–893.

    Google Scholar 

  2. V. S. Varadarajan, Geometry of Quantum Theory, Second Edition. Springer-Verlag, New York, 1985.

    Google Scholar 

  3. W. Heisenberg, The Physical Principles of Quantum Theory, Dover, 1930.

    Google Scholar 

  4. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955.Translated by R. T. Bayer from Von Neumann’s 1932 Springer monograph Mathematische Grundlagen der Quantenmechanik.

    Google Scholar 

  5. G. W. Mackey, Quantum mechanics and Hilbert space, Amer. Math. Monthly 64(1957) no. 8, part II, 45–57. See also

    Google Scholar 

  6. G. W. Mackey, The Mathematical Foundations of Quantum Mechanics: W., A. Benjamin, 1963. For a detailed retrospective review of Mackey’s work, see

    Google Scholar 

  7. V. S. Varadarajan, George Mackey and his work on representation theory and foundations of physics, in Group representations, ergodic theory, and mathematical physics: a tribute to George W. Mackey, 417–446, Contemp. Math., 449, (Eds.) R. S. Dornan, C. C. Moore, and R. J. Zimmer. Amer. Math. Soc., 2008.

    Google Scholar 

  8. E. G. Beltrametti, and G. Cassinelli, The Logic of Quantum Mechanics, Encyclopedia of Mathematics and its Applications, Vol 15, Addison-Wesley, 1981.

    Google Scholar 

  9. J. S. Bell,On the problem of hidden variables in quantum mechanics, Rev. Modern Phys., 38(1966), 447–452. For a collection of papers on quantum foundations see

    Article  MathSciNet  MATH  Google Scholar 

  10. J. A. Wheeler, and W. H. Zurek,Quantum Theory and Measurement, Princeton University Press, Princeton, 1983.

    Google Scholar 

  11. E. G. Beltrametti, and J. Lévy-Leblond (Eds.), Advances in Quantum Phenomena, NATO ASI Series B: Physics, Plenum, 1995.

    Google Scholar 

  12. A. Aspect, and P. Grangier, Experimental tests of Bell’s inequalities, 201–214, ibid.

    Google Scholar 

  13. A. Einstein, B. Podolsky, and N. Rosen, Can quantum mechanical description of physical reality be considered complete? Phys. Rev. 47(1935), 777–780. See p. 142 in the Wheeler-Zurek volume [7]. Bohr’s reply can be found in the same volume pp. 143–151 and was published in

    Article  MATH  Google Scholar 

  14. N. Bohr, Can quantum mechanical description of physical reality be considered complete? Phys. Rev. 48(1935), 696–702. For further insights into measurement theory and EPR experiments see

    Article  MATH  Google Scholar 

  15. P. J. Lahti, and P. Mittelstaedt, (Eds.), Symposium on the Foundations of Modern Physics: 50 years of the Einstein-Podolsky-Rosen Gedankenexperiment, World Scientific, 1985 (look at the poems and songs!!).

    Google Scholar 

  16. S. Rozental (Ed.) Niels Bohr, Interscience, 1964, pp. 128–131. See also Bohr’s article Discussions with Einstein on epistemological problems in atomic physics, in Albert Einstein: Philosopher-Scientist, P. A. Schilpp (Ed.), pp. 200–241, The library of living philosophers, Evanston, 1940. This is reproduced in

    Google Scholar 

  17. J. A. Wheeler, and W. H. Zurek,Quantum Theory and Measurement, Princeton University Press, Princeton, 1983, pp. 9–49.

    Google Scholar 

  18. R. Arens, and V. S. Varadarajan,On the concept of Einstein-Podolsky-Rosen states and their structure, J. Math. Phys., 41(2000), no. 2, 638–651.

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Bohm, Quantum Theory, Prentice-Hall, 1951.

    Google Scholar 

  20. P. A. M. Dirac,The quantum theory of emission and absorption of radiation, Proc. Royal Soc. London, Series A, 114(1927), 243–265.

    Article  Google Scholar 

  21. J. Schwinger (Ed.), Quantum Electrodynamics, Dover 1958.

    Google Scholar 

  22. H. Weyl, Theory of Groups and Quantum Mechanics, Dover 1931.

    Google Scholar 

  23. P. A. M. Dirac,The Lagrangian in quantum mechanics, Phys. Zeit. der Sowjetunion, 3(1933), 64–72.

    MATH  Google Scholar 

  24. R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Phys., 20(1948), 367–387.

    Article  MathSciNet  Google Scholar 

  25. R. P. Feynman, The concept of probability in quantum mechanics, Proceedings of the Second Berkeley Symposium, J. Neyman (Ed.), University of California Press, Berkeley, 1951.

    Google Scholar 

  26. B. Simon, Functional Integration and Quantum Physics, Academic Press, 1979.

    Google Scholar 

  27. J. Glimm, and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer, 1987.

    Google Scholar 

  28. D. Babbitt, Wiener integral representations for certain semigroups which have infinitesimal generators with matrix coefficients, Indiana Univ. Math. J. 19(1970), 1051–1067.

    Article  MathSciNet  MATH  Google Scholar 

  29. V. S. Varadarajan, and D. Weisbart,Convergence of quantum systems on grids, J. Math. Ana; Appl. 336(2007), 608–624.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Kac,Enigmas of Chance: An Autobiography, University of California Press, Berkeley and Los Angeles, 1987. Originally published by Harper and Row, 1985.

    Google Scholar 

  31. M. Kac, On distributions of certain Wiener functionals, in Probability, Number Theory, and Statistical Physics. Selected Papers, 268–280, 1979.

    Google Scholar 

  32. E. Nelson,Feynman integrals and the Schrödinger equation, Jour. Math. Phys. 5 (1964), 332–343.

    Article  MATH  Google Scholar 

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Varadarajan, V.S. (2011). Probability in the quantum world. In: Reflections on Quanta, Symmetries, and Supersymmetries. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0667-0_3

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