On the S-matrix of Liouville theory

The S-matrix for each chiral sector of Liouville theory on a cylinder is computed from the loop expansion of correlation functions of a one-dimensional field theory on a circle with a non-local kinetic energy and an exponential potential. This action is the Legendre transform of the generating function of semiclassical scattering amplitudes. It is derived from the relation between asymptotic in- and out-fields. Its relevance for the quantum scattering process is demonstrated by comparing explicit loop diagrams computed from this action with other methods of computing the S-matrix, which are also developed.

The Classical-to-Semiclassical Connection

This chapter discusses the connection between the classical and semiclassical domains of scattering. Scattering phenomena may be described via three regimes: the scattering of waves by objects with small, large, or comparable sizes with the wavelength of the incident (plane wave) radiation. All three regions can be related to three domains: the classical domain (geometrical optics, particle and particle/ray-like trajectories); the wave domain (physical optics, acoustic and electromagnetic waves, quantum mechanics); and the semiclassical domain (the vast intermediate region between the first and second domain). The chapter first provides an overview of classical and semiclassical scattering domains before beginning with an analysis of the semiclassical formulation. It also considers the radial equation, scattering by a one-dimensional potential barrier, and the radially symmetric problem. Solutions for phase shifts and the potential well are presented.

Introduction

This book deals with rays, waves, and scattering and covers many of the mathematical concepts, structures, and techniques used to study them. The subject of rays is explored in an atmosphere–sea–earth sequence, while waves are examined via the reverse sequence earth–sea–atmosphere. The book also considers the relationship between the elegance of the classical Lagrangian and Hamiltonian formulations of mechanics and optics; Kepler's laws of planetary motion, developed from gravitational scattering; surface gravity waves; diffraction; acoustics; electromagnetic scattering, including the Mie solution; and the WKB(J) approximation and its application to some simple one-dimensional potentials. The book concludes with an analysis of the salient properties of Sturm-Liouville systems with particular reference to the time-independent Schrödinger equation. This chapter provides an overview of the rainbow directory, rays, waves, classical and semiclassical scattering, and caustics and diffraction catastrophes.