Wave Equations on Lorentzian Manifolds and Quantization

TL;DR

This book offers a comprehensive introduction to wave equations on Lorentzian manifolds, covering fundamental solutions, well-posedness, and quantum field theory, serving as a resource for students and researchers in related fields.

Contribution

It provides a self-contained, detailed treatment of wave equations and quantization on Lorentzian manifolds, integrating mathematical and physical perspectives.

Findings

Construction of local and global fundamental solutions

Existence and uniqueness of solutions on globally hyperbolic spacetimes

Development of algebraic quantum field theory framework

Abstract

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. The text provides a self-contained introduction to these topics addressed to graduate students in mathematics and physics. At the same time it is intended as a reference for researchers in global analysis, general relativity, and quantum field theory.