FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 6, PAGES 193-211

The Maslov--Poisson measure and Feynman formulas for the solution of the Dirac equation

N. N. Shamarov

Abstract

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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 (which 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 representing the solution of the Schrödinger equation by the integral in the so-called cylindrical Feynman (pseudo)measure (in the trajectory space in 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 arise the 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 the integrals with respect to which yield solutions of the Cauchy problem for this equation in the configurational space.

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Last modified: February 26, 2007