Path Integrals, Hyperbolic Spaces And Selberg Trace Formulae (2nd Edition)

Christian Grosche

• Publisher: WSPC
• Year: 2013
• eBook ISBN: 9789814460095
• Edition: 2
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

In this second edition, a comprehensive review is given for path integration in two- and three-dimensional (homogeneous) spaces of constant and non-constant curvature, including an enumeration of all the corresponding coordinate systems which allow separation of variables in the Hamiltonian and in the path integral. The corresponding path integral solutions are presented as a tabulation. Proposals concerning interbasis expansions for spheroidal coordinate systems are also given. In particular, the cases of non-constant curvature Darboux spaces are new in this edition.The volume also contains results on the numerical study of the properties of several integrable billiard systems in compact domains (i.e. rectangles, parallelepipeds, circles and spheres) in two- and three-dimensional flat and hyperbolic spaces. In particular, the discussions of integrable billiards in circles and spheres (flat and hyperbolic spaces) and in three dimensions are new in comparison to the first edition.In addition, an overview is presented on some recent achievements in the theory of the Selberg trace formula on Riemann surfaces, its super generalization, their use in mathematical physics and string theory, and some further results derived from the Selberg (super-) trace formula. Contents:

• Introduction
• Path Integrals in Quantum Mechanics
• Separable Coordinate Systems on Spaces of Constant Curvature
• Path Integrals in Pseudo-Euclidean Geometry
• Path Integrals in Euclidean Spaces
• Path Integrals on Spheres
• Path Integrals on Hyperboloids
• Path Integral on the Complex Sphere
• Path Integrals on Hermitian Hyperbolic Space
• Path Integrals on Darboux Spaces
• Path Integrals on Single-Sheeted Hyperboloids
• Miscellaneous Results on Path Integration
• Billiard Systems and Periodic Orbit Theory
• The Selberg Trace Formula
• The Selberg Super-Trace Formula
• Summary and Discussion

Graduate and researchers in mathematical physics. Key Features:

• The 2nd edition brings the text up to date with new developments and results in the field
• Contains enumeration of many explicit path integrals solutions

Chapter 1

Introduction

Path Integrals.

Contrary to common believe, the proper quantum potential in the path integral in quantum mechanics is in general not just a constant proportional to the curvature. There may be a formulation, where this is the case, but not necessarily. In particular, if the path integral is formulated in terms of the classical Lagrangian, thus giving rise to an effective Lagrangian, then the quantum potential is explicitly coordinate-dependent.

Our first paper [248] was followed by other instructive examples of path integrals which could be treated by this theory in a consistent way. Among them were the path integral on the Poincaré upper half-plane [247], and its related conformally equivalent formulations, the Poincaré disc and the hyperbolic strip [202], and the pseudosphere [249]. Some potential problems [198, 204] and the incorporation of magnetic fields [197, 200] could also be discussed in this context, among them the Kepler problem on the pseudosphere [201]. Here a useful lattice formulation of the path integral was extensively used, which I have called “product form” [196]. In comparison to the often used (arithmetic) mid-point formulation, this lattice prescription is basically a geometric mid-point formulation. Also the already in [248] improved space-time transformation (also Duru-Kleinert transformation) technique could be further developed in [218] by the incorporation of explicitly time-dependent transformations.

The part of this volume concerned with path integrals is designed as follows: In Chapter 2 I review the definition of path integrals on curved manifolds. This includes the explicit construction of the path integral in its lattice definition. The two most important lattice prescriptions, mid-point and product-form are presented with the emphasize on the latter. Other lattice representations are not discussed, and neither the Vielbein approach of Kleinert [352]. Furthermore, transformation techniques are outlined. This includes point canonical transformations, space-time transformations, pure time transformations, and separation of variables. Some of the path integral investigations were done in joint work with Frank Steiner [248]. It must be noted that in recent years several review articles and textbooks following the classical books of Feynman and Hibbs [164] and Schulman [460] on exactly solvable path integrals with many examples have been published, e.g., Albeverio et al. [5, 3], Dittrich and Reuter [131], Glimm and Jaffe [182], Inomata et al. [289], Khandekar and Lawande [337], Kleinert [352], Roepstorff [454]. Simon [470], and Wiegel [519].

In Chapter 3 I give a summary of the classification of coordinate systems in spaces of constant curvature. This includes some remarks about the physical significance concerning separation of variables and breaking of symmetry, a general classification scheme, and an overview of the coordinate systems in Euclidean and Minkowski spaces and on spheres and hyperboloids.

In the next nine Chapters the path integral representations in several classes of (homogeneous) spaces are discussed. It includes the two- and three-dimensional Minkowski or pseudo-Euclidean spaces (Chapter 4), the two- and three-dimensional Euclidean spaces (Chapter 5), the two- and three-dimensional spheres (Chapter 6) and hyperboloids (Chapter 7), the two- and three-dimensional complex sphere (Chapter 8), hermitian hyperbolic space (Chapter 9), and Darboux spaces (Chapter 10). Additional results for the two-dimensional single-sheeted hyperboloid are presented in Chapter 11 and for more general homogeneous and hyperbolic spaces in Chapter 12. This includes the single-sheeted two-dimensional hyperboloid, the hyperbolic space corresponding to SO(p, q) and SU(p, q), and the case of hyperbolic spaces of rank one.

In comparison to the first edition of this monograph, the Chapters 8, 9 and 10 are entirely new. They contain results, which have been achieved in the course of studying more general cases as just real spaces, respectively real spaces of constant curvature, respectively spaces whose curvature is not constant, i.e. the Darboux spaces. However, I will not discuss the so-called Koenig-spaces which emerge from, say, usual flat space by multiplying the metric with a super-integrable potential in this flat space. This can be done in two- and three-dimensional Euclidean space, respectively. In two dimensions these potentials are the isotropic harmonic oscillator, the Holt-potential, and the Coulomb potential, e.g. [241]. The quantum motion then can be analyzed in the usual way by path integration [235, 237]. However, the quantization conditions for the energy-levels turn out to be rather complicated. They require the solution of an equation of eighth order in the energy E. Surprisingly, special cases of the Koenig-spaces turn to be Darboux spaces, spaces of constant (negative) curvature, and, of course, Euclidean space.

Koenig spaces which emerge form an analogous way from, say, Minkowski space, or spaces with constant (positive and negative) curvature with their corresponding super-integrable potentials have not been discussed yet. Their construction should be straightforward, though, including a path integral evaluation.

The cases of two- and three-dimensional Darboux spaces have been partly done in collaboration with George Pogosyan [244], in particular in the context of super-integrable potentials in these spaces. This was an extension of earlier work of super-integrable potentials on spaces of constant curvature [241]–[243].

Generally, I denote by “u” coordinates with indefinite metric, and by “q” coordinates with a positive definite metric. I start with the case of the pseudo-Euclidean space, because the proposed path integral solutions are entirely new. Some of the path integral solutions in the remaining three other spaces have been already reviewed in [223], and I do not discuss all the solutions in detail once more. Only the new solutions are treated more explicitly. In particular, I concentrate on the path integral solutions which can be obtained by means of interbasis expansions. This includes the case of elliptic coordinates in two-dimensional Euclidean space, on the sphere and on the pseudosphere, the case of spheroidal coordinates in three-dimensional Euclidean space, and some cases of ellipsoidal coordinates in spaces of constant curvature in three dimensions. As we will see, all the developed path integral techniques will come into play. I have cross-checked the solutions with the ones available in the literature achieved by other means. My hope is that my presentation will serve as a table for path integral representations in homogeneous spaces. I did not intend a discussion according to Camporesi [94] with a generalization to higher dimensions, applications of my results in cosmology, zeta-function regularization, etc. In addition, some results of path integration on generalized hyperbolic spaces are given.

Periodic Orbit Theory and Selberg Trace Formulæ.

For about twenty years trace formulæ have played a major role in mathematical physics and string theory. Generally, trace formulæ relate the classical and the quantum properties of a given system to each other. This can be very easily visualized with a simple example, the drum: The periodic orbits are the classical trajectories on the drum, and the energy eigenvalues are related to its modes (hence the question “Can one hear the shape of a drum?” [296]). This is true for every system one wants to study, however, trace formulæ become particularly important and useful when the system under consideration is classically chaotic. A necessary condition that a classical system can be solved exactly is that the phase space separates into invariant tori. If this is not the case, the usual tools of a perturbative approach break down and because of the exponentially diverging distance of initially nearby trajectories described by the Lyapunov exponent, no statement about the long-time behaviour of the system can be made. In the mathematical literature explicit statements of this feature were first made by Hadamard [261] and Poincaré [445]. They considered classical motion on spaces of constant negative curvature.

Surprisingly enough it was Einstein [147] who pointed out that any attempt to quantize a generic classical system runs into trouble if there are not enough constants of motion in this system. The existence of constants of motion in conservative classical systems - the energy E being just one constant of motion among others - cause that the phase space corresponding to this system separates into invariant tori. Einstein made this observation in connection to the “old” quantum theory, and he considered the problem under which conditions the quantization rule

p · dq = nħ makes sense. It makes sense if one can find in

^D, say, a coordinate system such that for any generalized coordinate q_a one can find a generalized conjugate momentum p^a with p^a a conserved quantity. In other words, we must find in a D-dimensional space (D > 1) at least one coordinate system which separates the classical equations of motions or the Laplacian, respectively. Finding such a coordinate system is equivalent in finding a set of observables. If this is not the case a quantization procedure cannot be found in the usual way by introducing position and momentum operators and impose commutation relations among them. A minimum of two dimensions is required in order that this feature can occur. Therefore the only systems which can be quantized semiclassically are those whose classical phase space consists of D-fold separating invariant tori. Among them are many well-known standard systems as the harmonic oscillator, the hydrogen atom, anharmonic oscillators like the Morse- or the Pöschl–Teller oscillators, and all one-dimensional systems. Excluded are the motion on spaces of constant negative curvature, billiard systems with boundaries which have defocusing properties, and many others like a hydrogen atom in a uniform magnetic field or the anisotropic Kepler problem. All these systems are classically chaotic.

In the 1960’s Gutzwiller [258] was the first who developed by means of path integrals a semiclassical theory for systems, which are classically chaotic and cannot be qua...

Table of contents

• Cover
• Halftitle
• Title Page
• Copyright Page
• Contents
• List of Tables
• List of Figures
• Preface
• 1  Introduction
• 2  Path Integrals in Quantum Mechanics
• 3  Separable Coordinate Systems on Spaces of Constant Curvature
• 4  Path Integrals in Pseudo-Euclidean Geometry
• 5  Path Integrals in Euclidean Spaces
• 6  Path Integrals on Spheres
• 7  Path Integrals on Hyperboloids
• 8  Path Integral on the Complex Sphere
• 9  Path Integrals on Hermitian Hyperbolic Space
• 10  Path Integrals on Darboux Spaces
• 11    Path Integrals on Single-Sheeted Hyperboloids
• 12    Miscellaneous Results on Path Integration
• 13    Billiard Systems and Periodic Orbit Theory
• 14    The Selberg Trace Formula
• 15    The Selberg Super-Trace Formula
• 16    Summary and Discussion
• Bibliography
• Index