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Journal of the Mathematical Society of Japan

Convergence of the Feynman path integral in the weighted Sobolev spaces and the representation of correlation functions

Wataru Ichinose

Published Oct 1, 2003• 14 citations
Publisher

Abstract

There are many ways to give a rigorous meaning to the Feynman path integral. In the present paper especially the method of the time-slicing approximation determined through broken line paths is studied. It was proved that these time-slicing approximate integrals of the Feynman path integral in configuration space and also in phase space converge in L2 space as the discretization parameter tends to zero. In the present paper it is shown that these time-slicing approximate integrals converge in some weighted Sobolev spaces as well. Next as an application of this convergence result in the weighted Sobolev spaces, the path integral representation of correlation functions is studied of the position and the momentum operators. We note that their path integral representation is given in phase space. It is shown that the approximate integrals of correlation functions converge or diverge as the discretization parameter tends to zero. We note that the divergence of the approximate integrals reflects the uncertainty principle in quantum mechanics.

DOI
10.2969/jmsj/1191418759

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There are many ways to give a rigorous meaning to the Feynman path integral. In the present paper especially the method of the time-slicing approximation determined through broken line paths is studied. It was proved that these time-slicing approximate integrals of the Feynman path integral in configuration space and also in phase space converge in L2 space as the discretization parameter tends to zero. In the present paper it is shown that these time-slicing approximate integrals converge in some weighted Sobolev spaces as well. Next as an application of this convergence result in the weighted Sobolev spaces, the path integral representation of correlation functions is studied of the position and the momentum operators. We note that their path integral representation is given in phase space. It is shown that the approximate integrals of correlation functions converge or diverge as the discretization parameter tends to zero. We note that the divergence of the approximate integrals reflects the uncertainty principle in quantum mechanics.
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.
Convergence of a time-slicing approximation is studied in a general way of the Feynman path integral in phase space with an electromagnetic potential. In the present paper a new approximation of the Feynman path integral in phase space, different from the familiar one in physics, is proposed so that its convergence can be shown independently of the choice of electromagnetic potentials and that gauge invariance of the Feynman path integral defined by its limit can be shown directly. It is also shown that the Feynman path integral in phase space defined above can be expressed in the form of the Feynman path integral in configuration space defined in the familiar way in physics, which Feynman himself stated heuristically.
Discretizations of the Feynman-Kac path integral representation of the quantum mechanical density matrix are investigated. Each infinite-dimensional path integral is approximated by a Riemann integral over a finite-dimensional function space, by restricting the integration to a subspace of all admissible paths. Using this process, a wide class of methods can be derived, with each method corresponding to a different choice for the approximating subspace. The traditional ``short-time'' approximation and ``Fourier discretization'' can be recovered from this approach, using linear and spectral basis functions respectively. As an illustration, a novel method is formulated using cubic elements and is shown to have improved convergence properties when applied to a simple model problem.
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.
Path integration methods are of crucial importance to quantum mechanics and quantum field theory. There are multiple ways the path integral can be constructed, one method uses the link between the Fokker-Planck and the Langevin equations, as is covered in this thesis. This is related to the standard derivation of the path integral in physics, in which the time is discretised. We study the mathematical problems related to the Feynman path integral, in particular the impossibility of a Lebesgue-type measure on the space of paths. It is discussed how oscillatory integrals can be used to have a well-defined `integral' on an infinite dimensional space. This formalism is subsequently applied to both the harmonic and the anharmonic oscillator. We prove an infinite dimensional oscillatory integral exists and obtain a convergent expression for both these cases, under conditions. Examples of these conditions are the initial wavefunctions being Schwartz functions as well as conditions on the endtime, angular frequency and coupling constant. Finally, a possible approach to establish an oscillatory integral for the hydrogen atom is discussed. It is proven that the result is no longer independent of the sequence of projection operators, which is a key step towards a rigorous path integral for this system.
The main aim of the present paper is using a Chernoff theorem (i.e., the Chernoff formula) to formulate and to prove some rigorous results on representations for solutions of Schrödinger equations by the Hamiltonian Feynman path integrals (=Feynman integrals over trajectories in the phase space). The corresponding theorem is related to the original (Feynman) approach to Feynman path integrals over trajectories in the phase space in much the same way as the famous theorem of Nelson is related to the Feynman approach to the Feynman path integral over trajectories in the configuration space. We also give a representation for solutions of some Schrödinger equations by a series which represents an integral with respect to the complex Poisson measure on trajectories in the phase space.