Generalizations of Quantum Mechanics Induced by Classical Statistical Field Theory

No HeadingWe show that the Dirac-von Neumann formalism for quantum mechanics can be obtained as an approximation of classical statistical field theory. This approximation is based on the Taylor expansion (up to terms of the second order) of classical physical variables – maps f : Ω → R, where Ω is the infinite-dimensional Hilbert space. The space of classical statistical states consists of Gaussian measures ρ on Ω having zero mean value and dispersion σ2(ρ) ≈ h. This viewpoint to the conventional quantum formalism gives the possibility to create generalized quantum formalisms based on expansions of classical physical variables in the Taylor series up to terms of nth order and considering statistical states ρ having dispersion σ2(ρ) = hn (for n = 2 we obtain the conventional quantum formalism).

[1]  Gerard 't Hooft.  Determinism Beneath Quantum Mechanics , 2002 .

[2]  Andrei Khrennikov,et al.  A pre-quantum classical statistical model with infinite-dimensional phase space , 2005, quant-ph/0505228.

[3]  Louis Marchildon.  Quantum Mechanics: From Basic Principles to Numerical Methods and Applications , 2002 .

[4]  Albert Einstein,et al.  The collected papers of Albert Einstein , 1987 .

[5]  Karl Hess,et al.  A possible loophole in the theorem of Bell , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Mysteries Without Mysticism and Correlations Without Correlata: On Quantum Knowledge and Knowledge in General , 2003 .

[7]  Andrei Khrennikov,et al.  Frequency Analysis of the EPR-Bell Argumentation , 2002 .

[8]  R. Smirnov-Rueda.  On Essential Incompleteness of Hertz’s Experiments on Propagation of Electromagnetic Interactions , 2005, physics/0510015.

[9]  J. Neumann.  Mathematical Foundations of Quantum Mechanics , 1955 .

[10]  Karl Hess,et al.  Bell’s theorem: Critique of proofs with and without inequalities , 2005 .

[11]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[12]  Albert Einstein,et al.  The evolution of physics : from the early concepts to relativity and quanta , 1967 .

[13]  Andrei Khrennikov.  Interference of probabilities and number field structure of quantum models , 2001 .

[14]  A. V. Skorohod,et al.  Integration in Hilbert Space , 1974 .

[15]  P. Dirac.  Principles of Quantum Mechanics , 1982 .

[16]  Louis Marchildon,et al.  Why Should We Interpret Quantum Mechanics? , 2004 .

[17]  Willem M. de Muynck.  Interpretations of quantum mechanics, and interpretations of violation of Bell's inequality , 2001 .

[18]  Timothy H. Boyer,et al.  A Brief Survey of Stochastic Electrodynamics , 1980 .

[19]  Sergey V. Polyakov,et al.  Quantum Theory: Reconsideration of Foundations-4 , 2007 .

[20]  Andrei Khrennikov,et al.  Foundations of Probability and Physics , 2002 .

[21]  Gerard 't Hooft.  Quantum Mechanics and Determinism , 2001 .

[22]  Luis de la Peña,et al.  The quantum dice : an introduction to stochastic electrodynamics , 1996 .

[23]  A. M. Cetto,et al.  Does quantum mechanics accept a stochastic support? , 1982 .

[24]  A. Plotnitsky.  The Knowable and the Unknowable: Modern Science, Nonclassical Thought, and the "Two Cultures" , 2002 .

[25]  Harald Atmanspacher,et al.  Epistemic and Ontic Quantum Realities , 2003 .

[26]  Andrei Khrennikov.  Prequantum classical statistical model with infinite dimensional phase-space-2: complex representation of symplectic phase-space model , 2005 .

[27]  Andrei Khrennikov,et al.  Interpretations of Probability , 1999 .