Topics in Mode Conversion Theory and the Group Theoretical Foundations of Path Integrals

TL;DR

This dissertation explores the phase space approach to wave mode conversion, introduces new higher order corrections, and develops a group theoretical foundation for path integrals related to operator symbols.

Contribution

It presents a novel group theoretical formulation of symbols and connects path integrals with the calculation of operator symbols in phase space.

Findings

Higher order corrections improve asymptotic matching in mode conversion

Group theoretical framework unifies symbol calculus and path integrals

Potential applications in multicomponent wave systems

Abstract

This dissertation reports research about the phase space perspective for solving wave problems, with particular emphasis on the phenomenon of mode conversion in multicomponent wave systems, and the mathematics which underlie the phase space perspective. Part I of this dissertation gives a review of the phase space theory of resonant mode conversion. We describe how the WKB approximation is related to geometrical structures in phase space, and how in particular ray-tracing algorithms can be used to construct the WKB solution. We also present new higher order corrections to the local solution for the mode conversion problem which allow better asymptotic matching to the WKB solutions. The phase space tools used in Part I rely on the Weyl symbol calculus, which gives a way to relate operators to functions on phase space. In Part II of this dissertation, we explore the mathematical foundations of the theory of symbols. We go on to present the group theoretical formulation of symbols, as developed recently by Zobin. This defines the symbol of an operator in terms of a double Fourier transform on a non-commutative group. We then show how the path integral arises when calculating the symbol of a function of an operator. We conclude with a survey of ideas for future research and describe several potential applications of this group theoretical perspective to problems in mode conversion.