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New curvilinear integrals along paths of Feynman path integral

Naoto Kumano‐go

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

We define new curvilinear integrals along paths of Feynman path integral. In order to define curvilinear integrals along paths on a path space, Ito integrals use initial points of line segments of broken line paths, and Stratonovich integrals use middle points of them. However, our new curvilinear integrals use classical curvilinear integrals along broken line paths. Therefore, the classical fundamental theorem of calculus holds in Feynman path integral.

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We define new curvilinear integrals along paths of Feynman path integral. In order to define curvilinear integrals along paths on a path space, Ito integrals use initial points of line segments of broken line paths, and Stratonovich integrals use middle points of them. However, our new curvilinear integrals use classical curvilinear integrals along broken line paths. Therefore, the classical fundamental theorem of calculus holds in Feynman path integral.
The definitions of the Feynman path integral for the Pauli equation and more general equations in configuration space and in phase space are proposed, probably for the first time. Then it is proved rigorously that the Feynman path integrals are well-defined and are the solutions to the corresponding equations. These Feynman path integrals are defined by the time-slicing method through broken line paths, which is familiar in physics. Our definitions of these Feynman path integrals and our results give the extension of ones for the Schrödinger equation.
We study the convergence of the Feynman path integral formulated through broken line paths in nonrelativistic quantum mechanics. The rigorous proof of its convergence had been given little except for special cases for a long time. In the preceding paper the author showed for a class of potentials that this path integral converges and gives the probability amplitude, i.e. the solution of the Schrödinger equation. In the present paper we generalize this result, showing the boundedness theorem on the weighted Sobolev spaces for some integral operators and joining this boundedness theorem to the method in the preceding paper. We note that the result obtained is gauge invariant.
A new axiomatic formulation of path integrals is used to construct a path integral solution of the Schrödinger equation in curvilinear coordinates. An important feature of the formalism is that a coordinate transformation in the variables of the wavefunction does not imply a change of variable of integration in the path integral. Consequently, a transformation from Euclidean to curvilinear coordinates is simple to handle; there is no need to introduce ‘‘quantum corrections’’ into the action functional. Furthermore, the paths are differentiable: hence, issues related to stochastic paths do not arise. The procedure for constructing the path integral solution of the Schrödinger equation is straightforward. The case of the Schrödinger equation in spherical coordinates for a free particle is presented in detail.