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Path Integration on Hermitian Hyperbolic Space

Grosche, C. (Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik), Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)

Published Jan 1, 2004• 5 citations
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Abstract

In this paper the path integral technique is applied to the quantum motion on the Hermitian hyperbolic space HH(2). The Schroedinger equation on this space separates in 12 coordinate systems which are closely related to the coordinate systems on the two-dimensional hyperboloid. For six coordinate systems out of the twelve it is possible to find a path integral solution. (orig.)

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Path Integration on Darboux Spaces

Christian Grosche, Hamburg (Germany). Projekttraeger ''Erforschung der Kondensierten Materie: Synchrotronstrahlung'' Deutsches Elektronen-Synchrotron (DESY)

In this paper the Feynman path integral technique is applied to two-dimensional spaces of non-constant curvature: these spaces are called Darboux spaces $\\DI$--$\\DIV$. We start each consideration in terms of the metric and then analyze the quantum theory in the separable coordinate systems. The path integral in each case is formulated and then solved in the majority of cases, the exceptions being quartic oscillators where no closed solution is known. The required ingredients are the path integral solutions of the linear potential, the harmonic oscillator, the radial harmonic oscillator, the modified P\\"oschl--Teller potential, and for spheroidal wave-functions, respectively. The basic path integral solutions, which appear here in a complicated way, have been developed in recent work and are known. The final solutions are represented in terms of the corresponding Green's functions and the expansions into the wave-functions, respectively. We also sketch some limiting cases of the Darboux spaces, where spaces of constant negative and zero curvature emerge.

•Jan 1, 2004•7 citations
In this paper the path integral technique is applied to the quantum motion on the Hermitian hyperbolic space HH(2). The Schroedinger equation on this space separates in 12 coordinate systems which are closely related to the coordinate systems on the two-dimensional hyperboloid. For six coordinate systems out of the twelve it is possible to find a path integral solution. (orig.)
Quantum mechanics on the hyperbolic spaces of rank one is discussed using a path interaction technique. Hyperbolic spaces are multi-dimensional generalizations of the hyperbolic plane, i.e. the Poincare upper half-plane endowed with a hyperbolic geometry. The author evaluates the path integral on S 1 equivalent to SO(n, 1)/SO (n) and S 2 equivalent to SU(n,1)/S(U(1)*U(n)) in a particular coordinate system, yielding explicitly the wavefunctions and the energy spectrum. Furthermore, one can exploit a general property of all these spaces, namely that they can be parameterized by a pseudopolar coordinate system.
In this contribution, I show that it is possible to construct three-dimensional spaces of nonconstant curvature, i.e., three-dimensional Darboux spaces. Two-dimensional Darboux spaces have been introduced by Kalnins et al., with a path integral approach by the present author. In comparison to two dimensions, in three dimensions it is necessary to add a curvature term in the Lagrangian in order that the quantum motion can be properly defined. Once this is done, it turns out that, in the two three-dimensional Darboux spaces which are discussed in this paper, the quantum motion is similar to the two-dimensional case. In D 3d-I, we find seven coordinate systems which separate the Schrödinger equation. For the second space, D 3d-II, all coordinate systems of flat three-dimensional Euclidean space which separate the Schrödinger equation also separate the Schrödinger equation in D 3d-II. I solve the path integral on D 3d-I in the (u, v, w) system and on D 3d-II in the (u, v, w) system and in spherical coordinates.
In this paper, the Feynman path integral technique is applied to two-dimensional spaces of nonconstant curvature: these spaces are called Darboux spaces D I-D IV. We start each consideration in terms of the metric and then analyze the quantum theory in the separable coordinate systems. The path integral in each case is formulated and then solved in the majority of cases; the exceptions being the quartic oscillators where no closed solution is known. The required ingredients are the path integral solutions of the linear potential, the harmonic oscillator, the radial harmonic oscillator, the modified Pöschl-Teller potential, and the spheroidal wave functions. The basic path integral solutions, which appear here in a complicated way, have been developed in recent work and are known. The final solutions are represented in terms of the corresponding Green’s functions and the expansions into the wave functions. We also sketch some limiting cases of the Darboux spaces, where spaces of constant negative and zero curvature emerge.
In this paper the Feynman path integral technique is applied to two-dimensional spaces of non-constant curvature: these spaces are called Darboux spaces DI\\DIDI--DIV\\DIVDIV. We start each consideration in terms of the metric and then analyze the quantum theory in the separable coordinate systems. The path integral in each case is formulated and then solved in the majority of cases, the exceptions being quartic oscillators where no closed solution is known. The required ingredients are the path integral solutions of the linear potential, the harmonic oscillator, the radial harmonic oscillator, the modified P\\"oschl--Teller potential, and for spheroidal wave-functions, respectively. The basic path integral solutions, which appear here in a complicated way, have been developed in recent work and are known. The final solutions are represented in terms of the corresponding Green's functions and the expansions into the wave-functions, respectively. We also sketch some limiting cases of the Darboux spaces, where spaces of constant negative and zero curvature emerge.