Gauge Field Theories and Propagators in Curved Space-Time

TL;DR

This paper presents DeWitt's covariant formalism for quantizing gauge field theories in curved space-time, emphasizing the geometric classification of theories, the role of ghost fields, and deriving Green functions in curved backgrounds.

Contribution

It introduces a covariant framework for gauge theories in curved space-time, classifies theories by gauge algebra, and derives Green functions considering curved geometry effects.

Findings

DeWitt's formalism enables covariant quantization in curved space-time.

Classification of gauge theories based on gauge algebra geometry.

Explicit expansion of Feynman propagator in curved space-time.

Abstract

In this paper DeWitt's formalism for field theories is presented; it provides a framework in which the quantization of fields possessing infinite dimensional invariance groups may be carried out in a manifestly covariant (non-Hamiltonian) fashion, even in curved space-time. Another important virtue of DeWitt's approach is that it emphasizes the common features of apparently very different theories such as Yang-Mills theories and General Relativity; moreover, it makes it possible to classify all gauge theories in three categories characterized in a purely geometrical way, i.e., by the algebra which the generators of the gauge group obey; the geometry of such theories is the fundamental reason underlying the emergence of ghost fields in the corresponding quantum theories, too. These "tricky extra particles", as Feynman called them in 1964, contribute to a physical observable such as the stress-energy tensor, which can be expressed in terms of Feynman's Green function itself. Therefore, an entire section is devoted to the study of the Green functions of the neutron scalar meson: in flat space-time, the choice of a particular Green's function is the choice of an integration contour in the "momentum" space; in curved space-time the momentum space is no longer available, and the definition of the different Green functions requires a careful discussion itself. After the necessary introduction of bitensors, world function and parallel displacement tensor, an expansion for the Feynman propagator in curved space-time is obtained. Most calculations are explicitly shown.