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

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 further review how to analyze the phenomena of mode conversion from the phase space perspective. By making an expansion of the dispersion matrix about the mode conversion point in phase space, a local coupled wave equation is obtained. The solution of this local problem then provides a way to asymptotically match the WKB solutions on either side of the mode conversion region. We describe this theory in the context of a pedagogical example; that of a pair of coupled harmonic oscillators undergoing resonant conversion. Lastly, we 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 first review the theory of representations of groups, and the concept of a group Fourier transform. The Fourier transform for commutative groups gives the ordinary transform, while the Fourier transform for non-commutative groups relates operators to functions on the group. 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 give examples of this new type of symbol, using the discrete Heisenberg-Weyl group to construct the symbol of a matrix. We then go on to show how the path integral arises when calculating the symbol of a function of an operator. We also show how the phase space and configuration space path integrals arise when considering reductions of the regular representation of the Heisenberg-Weyl group to the primary representations and irreducible representations, respectively. We also show how the path integral can be interpreted as a Fourier transform on the space of measures, opening up the possibility of using tools from statistical mechanics (such as maximum entropy techniques) to analyze the path integral. 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. To my parents, Andy and Diane Richardson. They gave me a love for learning, and encouraged me to pursue that love.