Wave Equations on Lorentzian Manifolds and Quantization

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.

199 pages, 43 figures, order information for a hardcopy: http://www.ems-ph.org/order.php

References in corpus (1)

• The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes

Cited by in corpus (22)

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• On the initial conditions and solutions of the semiclassical Einstein equations in a cosmological scenario
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• Global Existence of Solutions of the Semiclassical Einstein Equation for Cosmological Spacetimes
• Operads for algebraic quantum field theory
• Linear Yang-Mills theory as a homotopy AQFT
• Electromagnetic Potential in Pre-Metric Electrodynamics: Causal Structure, Propagators and Quantization
• Modular Theory, Non-Commutative Geometry and Quantum Gravity
• Dynamical locality of the free Maxwell field
• The Fermionic Signature Operator in De Sitter Spacetime
• The Regularized Hadamard Expansion
• Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes
• A unified field theory I: The quantization of gravity
• Supergeometry in locally covariant quantum field theory
• A unified quantum theory II: gravity interacting with Yang-Mills and spinor fields
• The Cosmological Semiclassical Einstein Equation as an Infinite-Dimensional Dynamical System
• On the Scalar Spectrum of the Y^{p,q} Manifolds
• Infrared problem vs gauge choice: scattering of classical Dirac field
• On the global "two-sided" characteristic Cauchy problem for linear wave equations on manifolds
• Closed conformal Killing-Yano initial data
• On causal properties of symmetric Lorentzian spaces