Path Integrals, Hyperbolic Spaces and Selberg Trace Formulae

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

Readership: 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