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arXiv:hep-th/0205239 · WORLD SCIENTIFIC eBooks

PATH INTEGRAL FOR SEPARABLE HAMILTONIANS OF LIOUVILLE-TYPE

Kazuo Fujikawa

Published Jun 1, 2001• 0 citations
arXivHTMLPDF

Abstract

A general path integral analysis of the separable Hamiltonian of Liouville-type is reviewed. The basic dynamical principle used is the Jacobi's principle of least action for given energy which is reparametrization invariant, and thus the gauge freedom naturally appears. The choice of gauge in path integral corresponds to the separation of variables in operator formalism. The gauge independence and the operator ordering are closely related. The path integral in this formulation sums over orbits in space instead of space-time. An exact path integral of the Green's function for the hydrogen atom in parabolic coordinates is ilustrated as an example, which is also interpreted as one-dimensional quantum gravity with a quantized cosmological constant.

DOI
10.1142/9789812811240_0009

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A general path integral analysis of the separable Hamiltonian of Liouville-type is reviewed. The basic dynamical principle used is the Jacobi's principle of least action for given energy which is reparametrization invariant, and thus the gauge freedom naturally appears. The choice of gauge in path integral corresponds to the separation of variables in operator formalism. The gauge independence and the operator ordering are closely related. The path integral in this formulation sums over orbits in space instead of space-time. An exact path integral of the Green's function for the hydrogen atom in parabolic coordinates is ilustrated as an example, which is also interpreted as one-dimensional quantum gravity with a quantized cosmological constant.
The general treatment of a separable Hamiltonian of Liouville-type is well-known in operator formalism. A path integral counterpart is formulated if one starts with the Jacobi's principle of least action, and a path integral evaluation of the Green's function for the hydrogen atom by Duru and Kleinert is recognized as a special case. The Jacobi's principle of least action for given energy is reparametrization invariant, and the separation of variables in operator formalism corresponds to a choice of gauge in path integral. The Green's function is shown to be gauge independent,if the operator ordering is properly taken into account. These properties are illustrated by evaluating an exact path integral of the Green's function for the hydrogen atom in parabolic coordinates.
The path integral evaluation of the hydrogen atom by Duru and Kleinert is studied by recognizing it as a special case of the general treatment of the separable Hamiltonian of Liouville-type. The basic dynamical principle involved is identified as the Jacobi’s principle of least action for given energy which is reparametrization invariant, and thus the appearance of a gauge freedom is naturally understood. Unlike the conventional Feynman path integral,which deals with a space-time picture of particle motion, the path integral on the basis of the Jacobi’s principle sums over orbits in space. We illustrate these properties by evaluating an exact path integral of the hydrogen atom in parabolic coordinates. In 3-dimensions the parabolic coordinates do not quite lead to a Liouville-type Hamiltonian, but we can devise asimple trick which converts it to a Liouville-type Hamiltonian and thus avoiding the use of the Kustaanheimo-Stiefel transformation. In the present formulation, the Hamiltonian for Stark effect is converted to the one for anharmonic oscillators with an unstable quartic coupling. We also study the hydrogen atom path integral from a view point of one-dimensional quantum gravity coupled to matter fields representing the electron coordinates. A simple BRST analysis of the problem is presented. 1 1
Abstract In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.
A path-integral formulation of quantum mechanics is investigated which is closely related to that of Feynman. It differs from Feynman's formulation in that it involves the Hamiltonian function of the canonically conjugate coordinates and momenta. The classical limit yields the variational principle: $\ensuremath{\delta}ff(p\ifmmode\ddagger\else\textdaggerdbl\fi{}\ensuremath{-}N)dt=0$. A path-integral formula is also obtained for the energy eigenstate projection operator associated with the time-independent Schr\"odinger equation. The classical limit of the projection operator formula yields a modified form of the well-known variational principle for the phase-space orbit of given energy. Relativistically covariant Hamiltonian variational principles are analyzed and lead naturally to a relativistic scalar wave equation which involves a proper time variable which is canonically conjugate to the mass in the same manner as the ordinary time variable is conjugate to the energy in nonrelativistic quantum theory.