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Publications of the Research Institute for Mathematical Sciences

Remarks on the Feynman Representation

Brian Jefferies

Published Dec 31, 1985• 3 citations
Publisher

Abstract

It is shown that there exists a complete space 3L 1 (Mf, Mf) of integrable functions such that for any potential V with zero // 0 ~bound relative to the free Hamiltonian operator H 0 C* of a finite non-relativistic quantum system, the function exp[-i \ V°X S ds] belongs to Jo dL l (Mf, Mf), and the Feynman representation e ~ic * 0+|r)l =^ exp[-i{* V°X,ds] dMf is valid.

DOI
10.2977/prims/1195178518

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It is shown that there exists a complete space 3L 1 (Mf, Mf) of integrable functions such that for any potential V with zero // 0 ~bound relative to the free Hamiltonian operator H 0 C* of a finite non-relativistic quantum system, the function exp[-i \ V°X S ds] belongs to Jo dL l (Mf, Mf), and the Feynman representation e ~ic * 0+|r)l =^ exp[-i{* V°X,ds] dMf is valid.
In the setting of Cameron and Storvick's recent theory we show that the solution of an integral equation formally equivalent to the Schroedinger equation is expressible as the analytic Feynman integral of a function on ^-dimensional Wiener space of the form F(X) = exp{/ o '0(ίs, X(s) + I) ds}ψ(X(t\+ ?).Here Xis an Revalued continuous function on [0, /] such that Z(0) = 0, ?G R\ and ψ and θ(s, ) are Fourier-Stieltjes transforms.REMARK 3.2.Corollary 3.2 and Lemma 3.2 will yield the measurability properties that we will need as we continue.For example, let θ be in % and for λ > 0 and fin W let G: [0, ί 0 ] X C[0, t 0 ] -> C be defined by The Pacific
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.
It was an interested and important problem to give the description of quantization, i.e. of passing from classical physical systems to the corresponding quantum ones, from the moment that quantum mechanics came into existence. In the end we succeeded in giving it as follows: Let L be a Lagrangian function. Then the Hamiltonian function H is defined through the Legendre transformation of L. The Hamiltonian operator ( ) at time in quantum mechanics is defined from H. It should be noted that ( ) has ordering ambiguities (cf. [14]). Let be a probability amplitude at time . Then its temporal evolution can be given by the solution of the Schrodinger equation
The canonical operator exp [−β(H − μN)] associated with the Bardeen-Cooper-Schrieffer (BCS) model Hamiltonian of superconductivity is represented as a functional integral by the use of Feynman's ordering parameter. General properties of the partition function in this representation are discussed. Taking the inverse volume of the system as an expansion parameter, it is possible to calculate the thermodynamic potential including terms independent of the volume. This yields a new proof that the BCS variational value is asymptotically exact. The behavior of the canonical operator for large volume is described and related to the state of free quasiparticles. A study of the terms of the thermodynamic potential which are of smaller order in the volume in the low-temperature limit, shows that the ground state energy is nondegenerate and belongs to a number eigenstate.