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Maslov-Poisson measure and Feynman formulas for the solution of the Dirac equation. (English) Zbl 1151.81354
J. Math. Sci., New York 151, No. 1, 2767-2780 (2008); translation from Fundam. Prikl. Mat. 12, No. 6, 193-211 (2006).
Summary: As the main step, the method used by V. P. Maslov for representing a solution of the initial-value problem for the classical Schrödinger equation and admitting an application to the Dirac equation includes the construction of a cylindrical countably additive measure (that is an analog of the Poisson distribution) on a certain space of functions (= trajectories in the impulse space) whose Fourier transform coincides with the factor in the formula for representation of the solution of the Schrödinger equation by the integral in the so-called cylindrical Feynman (pseudo) measure (in the trajectory space of the configurational space for the classical system). On the other hand, in the Maslov formula for the solution of the Schrödinger equation, the exponential factor is (with accuracy up to a shift) the Fourier transform of the Feynman pseudomeasure. In the case of the Dirac equation, historically, for the first time, there arose formulas for the impulse representation that use countably additive functional distributions of the Poisson-Maslov measure type but with noncommuting (matrix) values. The paper finds generalized measures whose Fourier transforms coincide with an analog of the exponential factor under the integral sign in the Maslov-type formula for the Dirac equation and integrals with respect to which yield solutions of the Cauchy problem for this equation in the configurational space.

MSC:
81S40 Path integrals in quantum mechanics
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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