Wave Equations on Lorentzian Manifolds and Quantization
This book is devoted to global solutions of the wave equation on Lorentzian manifolds and to a quantization of fields in general relativity. Fields with values in Hermitean or Riemannian finite rank vector bundles are allowed. Whereas the theory of local solutions to the wave equation is quite well developed, a detailed global approach is almost impossible to find in contemporary literature. The book gives a detailed introduction to the subject summarizing and unifying particular results. After recalling basic notions on distributions on manifolds and Lorentzian geometry, local solutions are constructed using the Riesz distribution and specific methods for obtaining convergence of the so-called formal solution are derived. Using local results, global existence and uniqueness are stated and proved in the case of globally hyperbolic manifolds. The notion of a globally hyperbolic manifold turned out to be the right replacement of the notion of a complete manifold, which is quite unsuitable in the Lorentzian geometry. Related geometric notions (e.g. causal future, causal past, past compact set, causally compatible set and Cauchy hypersurfaces) are defined and illustrated in several nice figures.
At the beginning of the last chapter (devoted to quantization) the theory of C*-algebras and CCR-representations are developed in detail. Analytic results on global solutions to the wave equation are then used to obtain field quantization functors. In the appendix, necessary notions from categories, functional analysis, differential geometry and further needed topics are collected. The book is written very carefully and its contents is self-contained. It is recommended not only to mathematicians working in the area of Lorentzian geometry or global analysis but also to physicists working in general relativity or more generally to those who want to get familiar with (hyperbolic) partial differential equations on manifolds.