Exotic Probability Theories and Quantum Mechanics: References

Dear Friends,

I thought that it might be useful to collect references relevant to exotic probability theories and their relation to quantum mechanics. If you see something missing, please let me know.

I just added a couple of new references. I'm planning to come back to this subject soon!


Saul Youssef
November, 2018.

  1. A.V. Belinskii, How could you measure a negative probability?, JETP letters, 59, 301 (1994).

  1. B. Bidabad, B. Bidabad, Complex probability and Markov stochastic processes, Iranian Statistics Conference, Isfahan University of Technology, 1992.

  1. E.I. Bitsakis, Nicolaidis, Efthymios (eds.), The Concept of Probability, Proceedings of the Delphi Conference, Greece, October 1987, ISBN 978-94-009-1175-8. (Thanks to Karin Herm for this reference!).

  1. Mark Burgin, Interpretations of Negative Probabilities, August, 2010.
  2. Mark Burgin, Extended Probabilities: Mathematical Foundations, December, 2009.

  1. G.Chiribella, G.M. D'Ariano, P.Perinotti, , arXiv:1011.6451 [quant-ph], November, 2010.

  1. Thomas Curtright and Luca Mezincescu, Biorthogonal Quantum Systems, arXiv:quant-ph/0507015, July, 2005.

  1. Rob Clifton, Jeffrey Bub, Hans Halvorson, , November 2002.

  1. D.J.Miller, Realism and Time Symmetry in Quantum Mechanics, Phys. Lett. A 1996.

  1. Ariel Caticha, Consistency and Linearity in Quantum Theory, Phys.Rev. A57, 1572 (1998).
  2. Ariel Caticha, Consistency, Amplitudes and Probabilities in Quantum Theory, preprint, 1998.
  3. Ariel Caticha, Entropic Inference and the Foundations of Physics, 2012.

  1. Borivoje Dakic, Caslav Brukner, , November, 2009, arXiv:0911.0695 [quant-ph]

  1. George T. Diderrich, A Weyl Creation Algebra Approach to the Riemann Hypothesis, arXiv:1302.5033, (2013).

  1. Paul Dirac, On the Analogy Between Classical and Quantum Mechanics, Reviews of Modern Physics, 17, 195, (1945)

  1. B.H. Feintzeig and S.C. Fletcher On Noncontextual, Non-Kolmogorovian Hidden Variable Theories, arXiv:1608.03518, August, 2016.

  1. Richard Feynman, The Concept of Probability Theory in Quantum Mechanics, in the Second Berkeley Symposium on Mathematical Statistics and Probability Theory, University of California Press, Berkeley, California, 1950.
  2. Richard Feynman, Negative Probabilities, in Quantum Implications, eds B.J. Hiley and F.David Peat (Routledge and Kegan Paul, 1987).

  1. David Finkelstein, Josef M. Jauch, Samuel Schiminovich and David Speiser, Foundations of Quaternion Quantum Mechanics, J.Math.Phys. 3, 207 (1962).

  1. F.H.Frohner, The Riesz-Fejer Theorem: Missing Link between Probability Theory and Quantum Mechanics, Forschungszentrum Karlsruhe, FZKA 5888, May, 1997.
  2. F.H.Frohner, Quantum Mechanics - How Weird for Bayesians?, in Maximum Entropy and Bayesian Methods, ed. A.Mohammad-Djafari and G. Demoments, Kluwer Academic Publishers, (1993).

  1. Christopher A. Fuchs Charting the Shape of Quantum-State Space, AIP Conf.Proc.1363, pp. 305-314, 2011.
  2. Christopher A. Fuchs , arXiv:quant-ph/0105039, May, 2001.
  3. Christopher A. Fuchs and Rudiger Schack, Quantum-Bayesian Coherence, Rev. Mod. Phys. 85,1693 (2013).

  1. Ben Goertzel, Multiboundary Algebra as Pregeometry, Electronic Journal of Theoretical Physics, 4, No. 16 (2007).

  1. W.Garczynski, Photon's Propagator from Maxwell Electrodynamics and the Quantum Cauchy Stochastic Process, December 2010 [revised, January 2012].
  2. W.Garczynski, Photon's Propagator from Maxwell Electrodynamics Part 2, August 2013.
  3. W.Garczynski, Application of the Lie-Trotter-Kato (LTK) product formula to calculation of propagators and the generating functions Z(J) in the Quantum Mechanics and Quantum Field Theory, (Exotic) stochastic aspect, August 2017.

  1. Stanley Gudder, A theory of amplitudes, J.Math.Phys. 29, 9 (1988).
  2. Stanley Gudder, Realism in Quantum Mechanics, Found. Phys. 19, 949 (1989).
  3. Stanley Gudder, A new formulation of quantum mechanics, Int. Journ.Theor.Phys. 31(1992) 15.
  4. Stanley Gudder, Realistic Spin, Found. Phys. Vol 22, 1 (1992).

  1. J.J. Halliwell, Negative Probabilities, Fine's Theorem and Linear Positivity, Phys. Rev. A 87, 022114, 2013.

  1. Yeong Deok Han, Won Yong Hwang and Gyu Koh, Explicit solutions for negative-probability measures for all entangled states, Physics Letters A, 221, 283-286 (1996).

  1. Lucien Hardy, Quantum Theory from Five Reasonable Axioms, arXiv:quant-ph/0101012, September, 2001.
  2. Lucien Hardy, Reformulating and Reconstructing Quantum Theory, arXiv:1104.2066, April, 2011.

  1. J.J.Halliwell and J.M. Yearsley, Negative Probabilities, Fine's Theorem and Linear Positivity, November, 2012.

  1. James Hartle, Quantum mechanics with extended probabilities, Phys. Rev. A 78, (2008).

  • Rabha W. Ibrahim and Maslina Darus, Analytic Study of Complex Fractional Tsallis' Entropy with Applications in CNNs, Entropy, September (2018).
    1. Yehud Izhakian and Zur Izhakian, Phantom Probability, arXiv:0901.0902, (2009).

    1. Abdo Abou Jaoude, The Paradigm of Complex Probability and the Brownian Motion, Systems Science & Control Engineering, (2015).

    1. Kevin H. Knuth, Lattices and Their Consistent Quantification, Annalen der Physik, 2017.

    1. Ryszard Pawel Kostecki Quantum theory as inductive inference, AIP Conf. Proc. 1305 (2011), 24-35.

    1. Jeffrey S. Lundeen, Generalized Measurement and Post-selection in Optical Quantum Information, Ph.D. Thesis, University of Toronto, 2006.
    2. Jeffrey S. Lundeen and Charles Bamber, Weak measurement of the Dirac distribution, December, 2011.

    1. Andrew Khrennikov, p-adic probability distributions of hidden variables, SFB 237 Preprint 257, 1995.
    2. Andrew Khrennikov, p-adic probability interpretation of Bell's inequality, Phy. Lett. A 200(1995).

    1. Thomas Marlow, A Bayesian Analogue of Gleason's Theorem, quant-ph/0603065.
    2. Thomas Marlow, Bayesian Probabilities and the Histories Algebra, Int.J.Theor.Physics, 45, 2006.
    3. Thomas Marlow, A Bayesian account of quantum histories, Annals of Physics, Volum 321, issue 5, May 2006, pp. 1103-1125.
    4. Thomas Marlow, Relationalism vs Bayesianism, arXiv:gr-qc/0603015v1.
    5. Thomas Marlow, On the Probabilistic Compatibility of Special Relativity and Quantum Mechanics, quant-ph/0501131.

    1. J.E.Moyal, Quantum Mechanics as a Statistical Theory, Proceedings of the Cambridge Philosophical Society, Vol 45. 1949.
    2. Dirac-Moyal correspondence, 1944-1946.

    1. W.Muckenheim et al., Phys. Rep. 133, 339 (1983). (An interesting review.)
    2. W.Muckenheim, On quasi-realistic local spin models and extended probabilities, Phys. Lett. A 175(1993).

    1. Tilman Neumann, Bayesian Inference Featuring Entropic Priors, 2007.

    1. Itamar Pitowsky, Resolution of the Einstein-Podolsky-Rosen and Bell Paradoxes, Phys.Rev.Lett. 48, 1299 (1982).
    2. Itamar Pitowsky, Deterministic model of spin and statistics, Phys. Rev. D27, 2316 (1983).

    1. Jiravatt Rewrujirek, Arit Hutem and Sutee Boonchui, Calculation of the tunneling time using the extended probability of the quantum histories approach, Physics Letters A378, 985 (2014).

    1. Carlos C. Rodrigues, Unreal Probabilities: Partial Truth with Cliffor Numbers, arXiv:physics 9808010, 1998.

    1. Slater, P.B. Quantum Coin-Tossing in a Bayesian Jeffreys Framework, Phys. Lett. A 206 (1995).

    1. Antonio Sciarretta, A realistic interpretation of quantum mechanics. Asymmetric random walks in discrete spacetime., February, 2008.

    1. M.O.Scully, H.Walther and W.Schleich, Feynman's approach to negative probability in quantum mechanics, Phys.Rev. A49, 1562(1994).

    1. Luke Schaeffer, A Physically Universal Quantum Cellular Automaton, Lecture Notes in Computer Science, Vol 9099, pp 46-58 (2015).

    1. John Skilling and Kevin H. Knuth, The symmetrical foundation of Measure, Probability and Quantum theories, December, 2017 (arXiv:1712.09725, 2017)
    2. John Skilling and Kevin H. Knuth, The ABC of Physics, Presented at the 40th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, online, 4-9 July 2021.

    1. Angelina Ilic Stepic and Zoran Ognjanovic, Complex Valued Probability Logics, Novelle serie, tome 95(2014), 73-86.

    1. Aephraim Steinberg, How Much Time Does a Tunneling Particle Spend in the Barrier Region? Phys.Rev.Lett. 13, 2405 (1995).
    2. Aephraim Steinberg, Conditional Probabilities in Quantum Theory and the Tunneling Time Controversy Phys.Rev. A52, 32 (1996).

    1. R.F. Streater, Classical and Quantum Probability, math-ph/0002049 (2000).

    1. S.K.Srinivasan and E.C.G. Sudarshan, Complex measures and amplitudes, generalized stochastic processes and their application to quantum mechanics., J.Phys. A. Math Gen. 27 (1994).
    2. S.K. Srinivasan, Quantum Mechanics via Extended Measures, J.Phys.A (23) 8297, (1997).
    3. S.K. Srinivasan, Complex Measure, Coherent State and Squeezed State Representation, J.Phys.A(?)
    4. S.K. Srinivasan, Complex Measureable Processes and Path Integrals, preprint.
    5. S.K. Srinivasan, Theory of quantum phenomena via extended measures: geometric features, J.Phys. A 35(3755) 2002.
    6. S.K. Srinivasan, Assignment of Probability to the History of Paths in Quantum Phenomena, preprint, 2006.
    7. S.K. Srinivasan, Quantum phenomena via complex measure: Holomorphic extension, Fortschr. Phys., 1-22,(2006).

    1. Angelina Stepic and Zoran Ognjanovic, Complex Valued Probability Logics, Novelle Serie, tome 95(2014), 73-86.

    1. Y. Tikochinsky, Feynman Rules for Probability Amplitudes, Int.J.Theor.Phys., 27, 543 (1988).
    2. Y. Tikochinsky, On the generalized multiplication and addition of complex numbers, J.Math.Phys. 29 (1988).

    1. Saul Youssef, A Reformulation of Quantum Mechanics, Mod.Phys.Lett. A6, 225-236 (1991).
    2. Saul Youssef, Quantum Mechanics as Complex Probability Theory, Mod.Phys.Lett A9, 2571 (1994).
    3. Saul Youssef, Is Quantum Mechanics an Exotic Probability Theory?, in Fundamental Problems in Quantum Theory; Conference in Honor of Professor John A. Wheeler, ed: D.M. Greenberger and A. Zeilinger, Annals of the New York Academy of Sciences, Volume 755, April, 1995.
    4. Saul Youssef, Quantum Mechanics as an Exotic Probability Theory, proceedings of the Fifteenth International Workshop on Maximum Entropy and Bayesian Methods, ed. K.M.Hanson and R.N.Silver, Santa Fe, August, 1995.
    5. Saul Youssef, Is Complex Probability Theory Consistent with Bell's Theorem?, Phys.Lett. A204, 181(1995).
    6. Saul Youssef, Physics with exotic probability theory, hep-th/0110253, (2001).

    1. D. Weingarten, Complex probabilities on R(N) as real probabilities on C(N) and an application to path integrals, Phys. Rev. Lett. Dec. 2002 (24).

    1. M. Yudin, The system with discrete interactions I: Some comments about the principles of quantum theory, May 2003.

    If you know of more papers related to this subject, please let me know.