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Journal of Mathematical Physics

A dominated-type convergence theorem for the Feynman integral

G. W. Johnson, Jeong-Gyoo Kim

Published May 1, 2000• 1 citation
Publisher

Abstract

A dominated-type convergence theorem is proved for the operator-valued Feynman integral defined via the Trotter product formula. This definition is intimately related to the one of Feynman and the physical setting is the same as in his original paper. The convergence result given here provides the final piece of a unifying and rather satisfactory picture of three approaches to the Feynman integral.

DOI
10.1063/1.533294

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A dominated-type convergence theorem is proved for the operator-valued Feynman integral defined via the Trotter product formula. This definition is intimately related to the one of Feynman and the physical setting is the same as in his original paper. The convergence result given here provides the final piece of a unifying and rather satisfactory picture of three approaches to the Feynman integral.
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The Feynman path integral is defined over the space RT\mathbb{R}^TRT of all possible paths; it has been a powerful tool to develop Quantum Mechanics. The absolute value of Feynman's integrand is not integrable, then Lebesgue integration theory could not be used by Feynman. However, it exists formally as a Henstock integral (which does not require the measure concept) and is a suitable alternative to the ordinary integrals that normally appear in path integrals. Feynman proved the equivalence of his theory with the traditional formulation of Quantum Mechanics, since his path integral satisfies Schrödinger's equation. On the other hand, Feynman's path integral is related to the diagrams of Feynman. For the application of this integral in Feynman's diagrams it is necessary to exchange the integral ∫RT\int_{\mathbb{R}^T}∫RT​ and the series. We discuss the impossibility to exchange the integral and the sum, considering integral of Henstock and the version of Dominated Convergence Theorem. Even it has not been proved through the several mathematical formalisms that have been used.
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