Geometry and physics of pseudodifferential operators on manifolds

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: existence 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.