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Journal of Physics A Mathematical and General

Path integration on hyperbolic spaces

Ch. Grosche

Published Aug 7, 1992• 10 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/25/15/028

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Journal of Physics A Mathematical and General•Nov 7, 1990•19 citations
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 paper rigorous path integral treatments are presented of free motion on the Poincaré disc, the Poincaré upper half-plane and on the hyperbolic strip, three spaces which are analytically equivalent to each other. Whereas the path integral treatments on the disc and on the strip are new, two further path integral treatments are discussed for the Poincaré upper half-plane to the existing one. All the calculations are mainly based on Fourier-expansions of the Feynman kernels which can be easily performed. The remaining path integrals on D, U and S can be reduced to the path integral problems on the pseudosphere Δ2, Liouville quantum mechanics and the modified Pöschl-Teller potential problem, respectively, where the results by means of path integrals are known. The corresponding normalized wave-functions and the energy-spectrum are derived. The energy-spectra are the same in all three spaces and read E = (1/2m) (p2 + 1/4) (p -momentum). The “zero-point” energy E0 = 1/8m is discussed which can be interpreted in terms of the Heisenberg uncertainty relation. The equivalences between the Feynman kernels on Δ2, D, U and S are also discussed.
DeWitt’s covariant formulation of path integration [B. De Witt, “Dynamical theory in curved spaces. I. A review of the classical and quantum action principles,” Rev. Mod. Phys. 29, 377–397 (1957)] has two practical advantages over the traditional methods of “lattice approximations;” there is no ordering problem, and classical symmetries are manifestly preserved at the quantum level. Applying the spectral theorem for unbounded self-adjoint operators, we provide a rigorous proof of the convergence of certain path integrals on Riemann surfaces of constant curvature −1. The Pauli–DeWitt curvature correction term arises, as in DeWitt’s work. Introducing a Fuchsian group Γ of the first kind, and a continuous, bounded, Γ-automorphic potential V, we obtain a Feynman–Kac formula for the automorphic Schrödinger equation on the Riemann surface Γ\ℍ. We analyze the Wick rotation and prove the strong convergence of the so-called Feynman maps [K. D. Elworthy, Path Integration on Manifolds, Mathematical Aspects of Superspace, edited by Seifert, Clarke, and Rosenblum (Reidel, Boston, 1983), pp. 47–90] on a dense set of states. Finally, we give a new proof of some results in C. Grosche and F. Steiner, “The path integral on the Poincare upper half plane and for Liouville quantum mechanics,” Phys. Lett. A 123, 319–328 (1987).
Representations of propagators by means of path integrals over velocities are discussed both in nonrelativistic and relativistic quantum mechanics. It is shown that all the propagators can only be expressed through bosonic path integrals over velocities of space-time coordinates. In the representations the integration over velocities is not restricted by any boundary conditions; matrices, which have to be inverted in course of doing Gaussian integrals, do not contain any derivatives in time, and spinor and isospinor structures of the propagators are given explicitly. One can define universal Gaussian and quasi-Gaussian integrals over velocities and rules of handling them. Such a technique allows one effectively calculate propagators in external fields. Thus, Klein-Gordon propagator is found in a constant homogeneous electromagnetic field and its combination with a plane wave field.
The author discusses how to separate variables in path integrals. It is assumed that a one-dimensional problem with potential V(x) has an exact solution with energy levels E lambda and wavefunctions Psi lambda . In order to perform the separation of variables, a time transformation is performed back and forth in the path integral which allows one to insert the path integral solution corresponding to the potential V(x). The author illustrates the method by discussing some specific potential problems on the Poincare upper half-plane.