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.

Location: http://mech.math.msu.su/~fpm/eng/k06/k066/k06612h.htm

Last modified: February 26, 2007