A functional integral with respect to a countably additive measure representing a solution of the Dirac equation

Author:
N. N. Shamarov

Translated by:
J. Wiegold

Original publication:
Trudy Moskovskogo Matematicheskogo Obshchestva, tom **66** (2005).

Journal:
Trans. Moscow Math. Soc. **2005**, 243-255

MSC (2000):
Primary 28B99; Secondary 46G10, 81Q05

DOI:
https://doi.org/10.1090/S0077-1554-05-00150-0

Published electronically:
November 16, 2005

MathSciNet review:
2193434

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Abstract | References | Similar Articles | Additional Information

Abstract: A family of cylindrical measures is constructed on the space of functions (trajectories) such that for every the formula

(1) |

represents a solution of the Cauchy problem

(2) |

(with respect to the required function , ), for the general evolution equation from some class containing the classical Dirac equation and the Schrödinger equation in its impulse representation, with ``model'' potentials that are independent of , and are Fourier transforms of countably additive (and in general matrix-valued) measures in the space variables.

The images of the measures obtained by restricting trajectories to finite intervals have bounded variation and are countably additive.

The integral kernels (``Green's functions'') of the corresponding solution operators, which can be approximated (using Trotter's formula) by integrals of finite multiplicity of the expressions explicitly defined by the ingredients of the original equation, are (matrix-valued) transition measures that give cylindrical measures similarly to the way Markov transition probabilities give the distribution of a Markov process.

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Additional Information

**N. N. Shamarov**

Affiliation:
Mechanics and Mathematics Department, Moscow State University, Vorob’evy Gory, Moscow 119234, Russia

DOI:
https://doi.org/10.1090/S0077-1554-05-00150-0

Published electronically:
November 16, 2005

Additional Notes:
This work was supported by the Russian Foundation for Fundamental Research, Grant No. 02-01-01074.

Article copyright:
© Copyright 2005
American Mathematical Society