During the fall semester of 1984 I visited the State University of New York at Stony Brook and taught an advanced graduate course on wave equations and quantum fields in curved space-times. This book is based on the notes for that course. I am very grateful to the Mathematics Department of SUNY, particularly Professor Michael Taylor, for arranging my temporary faculty appointment there.

The audience for the course consisted of graduate students and faculty members, mostly in mathematics but some in physics. They were assumed to have some knowledge of differential geometry and general relativity, and therefore not much time was spent on expounding those subjects. (The major exception is a chapter on connections on vector bundles and the Synge–DeWitt formalism needed for curved-space renormalization.) More time was spent on establishing a background in quantum theory and certain aspects of analysis, notably eigenfunction expansions.

Addressing such a mixed audience forces two difficult decisions – the relatively superficial one of what language to adopt, and the deeper one of what background knowledge to assume. I have an easy way out of the first problem: because of my own mixed background, the terminology which comes most naturally to me is a roughly equal mixture of the standard vocabularies of mathematicians and physicists. I have tried to say things in several different ways, to keep as many readers comfortable as possible.