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CRC Press eBooks

Feynman and Wiener Path Integrals Representations of the Liouville Evolution

Ioannis E. Antoniou, Oleg Georgievich Smolyanov

Published Apr 24, 2019• 0 citations
Publisher

Abstract

We show that the Liouville equation for any Hamiltonian system can be identified with the Schrödinger equation obtained by the Schrödinger-Dirac quantization of some “extended” Hamiltonian system whose phase space is the product of two copies of the initial phase space and hence whose configuration space coincides with the initial phase space. This identification allows us to represent some solutions of the Cauchy problem for the Liouville equation by Feynman path integrals [ 1 ] over trajectories in the phase space of the extended Hamiltonian systems.

DOI
10.1201/9780429187810-24

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We show that the Liouville equation for any Hamiltonian system can be identified with the Schrödinger equation obtained by the Schrödinger-Dirac quantization of some “extended” Hamiltonian system whose phase space is the product of two copies of the initial phase space and hence whose configuration space coincides with the initial phase space. This identification allows us to represent some solutions of the Cauchy problem for the Liouville equation by Feynman path integrals [ 1 ] over trajectories in the phase space of the extended Hamiltonian systems.
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 Feynman formula is a representation of the solution of an evolutionary equation by a limit of some multiple integrals over Cartesian products of the classical configuration space, or of the classical phase space, when the multiplicity of integrals tends to infinity. From the Feynman formula one can deduce the Feynman–Kac formula, i.e. a representation of the solution by an integral over trajectories. In this paper we consider representations, by the Feynman formula, of solutions both of some Schrödinger type equations and of the corresponding diffusion equations. Introduction The first of Feynman's papers [7], on what one now calls Feynman path integrals over trajectories in the configuration space, contains three main observations. Firstly, it is shown that the solution of the Cauchy problem for a Schrödinger equation can be represented by a limit of a sequence of integrals over Cartesian products of the classical configuration space when the multiplicity of integrals tends to infinity. Secondly, the limit is interpreted as an integral over trajectories in the configuration space. And finally, it is noticed that the integrand contains the exponent of the classical action. Feynman's definition of the Feynman path integrals over trajectories in the phase space, which is formulated in his second paper on the subject, has a similar structure but, in contrast to the preceding definition, the Lagrangian in the classical action is expressed through the Hamiltonian function.
A generalized Feynman-Kac formula based on the Wiener measure is presented. Within the setting of a quantum particle in an electromagnetic field it yields the standard Feynman-Kac formula for the corresponding Schrödinger semigroup. In this case rigorous criteria for its validity are compiled. Finally, phase-space path-integral representations for more general quantum Hamiltonians are derived. These representations rely on a generalized Lie-Trotter formula which takes care of the operator-ordering multiplicity, but in general is not related to a path measure.
A class of functions ℋnF was chosen, which are boundary values of holomorphic functions. It was shown that if the potential V∈ℋnF and the initial wave function ψ∈ℋnF then the solution of the Schrödinger equation is ψt∈ℋnF. The unitary evolution ψ → ψt can be expressed by an integral with respect to the Wiener measure. To each ψt∈ℋnF is associated a complex Markov process qt fulfilling stochastic Hamilton equations. The time evolution of ψ as well as multitime correlation functions of arbitrary observables in quantum mechanics take the form of expectation values with respect to qt. Some statistical and ergodic aspects of quantum mechanics resulting from the stochastic interpretation of Feynman’s sum over trajectories are discussed.