Cauchy Problem and Green's Functions for First Order Differential Operators and Algebraic Quantization

TL;DR

This paper proves existence and uniqueness of Green's functions and solutions to the Cauchy problem for first order differential operators on curved spacetimes, aiding quantum field theory constructions.

Contribution

It establishes fundamental solutions and well-posedness results for first order operators on globally hyperbolic Lorentzian manifolds, crucial for quantum field theory.

Findings

Existence of advanced and retarded Green's functions

Uniqueness of solutions to the Cauchy problem

Applicability to higher spin field equations

Abstract

Existence and uniqueness of advanced and retarded fundamental solutions (Green's functions) and of global solutions to the Cauchy problem is proved for a general class of first order linear differential operators on vector bundles over globally hyperbolic Lorentzian manifolds. This is a core ingredient to CAR-/CCR-algebraic constructions of quantum field theories on curved spacetimes, particularly for higher spin field equations.