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We 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).
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Khrennikov, A. Generalizations of Quantum Mechanics Induced by Classical Statistical Field Theory. Found Phys Lett 18, 637–650 (2005). https://doi.org/10.1007/s10702-005-1317-y
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DOI: https://doi.org/10.1007/s10702-005-1317-y