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!

Regards,

Saul Youssef

February, 2018.

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

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

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

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

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

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

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

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

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

- 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. - Richard Feynman,
*Negative Probabilities*, in*Quantum Implications*, eds B.J. Hiley and F.David Peat (Routledge and Kegan Paul, 1987).

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

- F.H.Frohner,
*The Riesz-Fejer Theorem: Missing Link between Probability Theory and Quantum Mechanics*, Forschungszentrum Karlsruhe, FZKA 5888, May, 1997. - 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).

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

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

- W.Garczynski, Photon's Propagator from Maxwell Electrodynamics and the Quantum Cauchy Stochastic Process, December 2010 [revised, January 2012].
- W.Garczynski, Photon's Propagator from Maxwell Electrodynamics Part 2, August 2013.
- 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.

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

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

- 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).

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

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

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

- Yehud Izhakian and Zur Izhakian,
*Phantom Probability*, arXiv:0901.0902, (2009).

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

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

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

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

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

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

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

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

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

- 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).

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

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

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

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

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

- John Skilling and Kevin H. Knuth, The symmetrical foundation of Measure, Probability and Quantum theories, December, 2017 (arXiv:1712.09725, 2017)

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

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

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

- 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). - S.K. Srinivasan,
*Quantum Mechanics via Extended Measures*, J.Phys.A (23) 8297, (1997). - S.K. Srinivasan,
*Complex Measure, Coherent State and Squeezed State Representation*, J.Phys.A(?) - S.K. Srinivasan,
*Complex Measureable Processes and Path Integrals*, preprint. - S.K. Srinivasan,
*Theory of quantum phenomena via extended measures: geometric features*, J.Phys. A 35(3755) 2002. - S.K. Srinivasan, Assignment of Probability to the History of Paths in Quantum Phenomena, preprint, 2006.
- S.K. Srinivasan, Quantum phenomena via complex measure: Holomorphic extension, Fortschr. Phys., 1-22,(2006).

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

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

- Saul Youssef,
*A Reformulation of Quantum Mechanics*, Mod.Phys.Lett. A6, 225-236 (1991). - Saul Youssef,
*Quantum Mechanics as Complex Probability Theory*, Mod.Phys.Lett A9, 2571 (1994). - 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.
- 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. - Saul Youssef,
*Is Complex Probability Theory Consistent with Bell's Theorem?*, Phys.Lett. A204, 181(1995). - Saul Youssef,
*Physics with exotic probability theory*, hep-th/0110253, (2001).

- 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).

- 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.