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Journal of Applied Probability

Feynman integral and phase space probability

James G. Gilson

Published Aug 1, 1968• 5 citations
Publisher

Abstract

The relation between “forward” and “backward” definitions of quantum velocity is discussed. A connection between the Feynman Integral and Wigner's phase space distributions which was noted in an earlier paper is explored further and some new and useful formulae are derived.
DOI
10.1017/s002190020011006x

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A concise derivation of all uncertainty relations is given entirely within the context of phase-space quantization, without recourse to operator methods, to the direct use of Weyl’s correspondence, or to marginal distributions of x and p. Phase-space quantization is the third autonomous and logically complete formulation of quantum mechanics beyond the conventional ones based on operators in Hilbert space or path integrals [1, 2, 3]. It is free of operators and wavefunctions: observables and matrix elements are computed through phasespace integrals of c-number functions (“classical kernels”) weighted by a Wigner function (WF) [4, 3]. This is a phase-space distribution function which is not positive semi-definite, and constitutes the Weyl correspondent [5] of the density matrix in the conventional formulation, fmn(x,p) = 1 2π dye −iyp ψ ∗ m(x − � 2 y)ψn(x + � 2 y) = f ∗ nm(x,p). (1)