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Transactions of the Moscow Mathematical Society

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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
Published electronically: November 16, 2005
MathSciNet review: 2193434
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Abstract | References | Similar Articles | Additional Information

Abstract: A family $ \{M_{p}\}_{p\in\mathbb{R}^d}$ of cylindrical measures is constructed on the space of functions ($ =$trajectories) $ f: [0,+\infty)\to\mathbb{R}^d$ such that for every $ t\geq0$ the formula

$\displaystyle \psi(t,p)= \int M_{p}(df)\,\psi_0(f( t))$ (1)

represents a solution of the Cauchy problem

$\displaystyle \frac\partial{\partial t} \psi = i \widehat H \psi \quad (t\geq0),\quad \psi(0,\cdot)=\psi_0$ (2)

(with respect to the required function $ \psi: [0,\infty)\times\mathbb{R}^d\to\mathbb{C}^s$, $ {d,s\in\mathbb{N}}$), 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 $ t$, and are Fourier transforms of countably additive (and in general matrix-valued) measures in the space variables.

The images of the measures $ M_p$ obtained by restricting trajectories to finite intervals $ [0,T]$ 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 $ M_p$ 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