Geometry and physics of pseudodifferential operators on manifolds

TL;DR

This paper reviews the mathematical framework of pseudodifferential operators on Riemannian manifolds, highlighting their symbolic calculus, derivatives, and physical applications like quantum field theory and Hawking radiation.

Contribution

It provides a comprehensive overview of the calculus for pseudodifferential operators on manifolds, including existence theorems and symbolic analysis, with connections to physics.

Findings

Existence theorem for generalized phase functions

Development of symbolic calculus including torsion and curvature

Application to Green functions in quantum field theory

Abstract

A review is made of the basic tools used in mathematics to define a calculus for pseudodifferential operators on Riemannian manifolds endowed with a connection: esistence theorem for the function that generalizes the phase; analogue of Taylor's theorem; torsion and curvature terms in the symbolic calculus; the two kinds of derivative acting on smooth sections of the cotangent bundle of the Riemannian manifold; the concept of symbol as an equivalence class. Physical motivations and applications are then outlined, with emphasis on Green functions of quantum field theory and Parker's evaluation of Hawking radiation.