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Feynman`s path integrals and Henstock`s non-absolute integration

Muldowney P.

Journal of Applied Analysis

2000

Vol. 6, nr 1

1-24

R. Feynman formulated quantum mechanics in terms of integrals over spaces of paths (Feynman path integrals). But the absolute value of Feynman's integrand is not integrable. And his integrand does not generate a measure. So Lebesgue integration theory could not be used by Feynman. To establish the equivalence of his theory with the traditional formulation of quantum mechanics, Feynman gave an argument that his path integral satisfies Schrodinger's equation. This paper gives a proof of this part of Feynman's theory. To justify Feynman's and other investigators' use of the language and concepts of integration and probability theory, and to justify taking limits under the integral sign in Feynman's integral, we use R. Henstock's approach to non-absolute integration, which does not require the measure concept, and for which the absolute value of the integrand need not be integrable.