Wave equations on Lorentzian manifolds and quantization

by Christian Bär, Nicolas Ginoux and Frank Pfäffle

ESI Lectures in Mathematics and Physics, EMS Publishing House (2007)

Available here and there

Summary: This book provides a detailed introduction to linear wave equations on Lorentzian manifolds (for vector-bundle valued fields). After a collection of preliminary material in the first chapter one finds in the second chapter the construction of local fundamental solutions together with their Hadamard expansion. The third chapter establishes the existence and uniqueness of global fundamental solutions on globally hyperbolic spacetimes and discusses Green's operators and well-posedness of the Cauchy problem. The last chapter is devoted to field quantization in the sense of algebraic quantum field theory. The necessary basics on C*-algebras and CCR-representations are developed in full detail.

Nicolas Ginoux, 10.03.2011