The method of covariant symbols in curved space-time

TL;DR

This paper extends the covariant symbol method to curved space-time with gauge and coordinate connections, enabling covariant calculations of operators for effective Lagrangians and currents.

Contribution

It develops a covariant symbol extension for curved space-time and computes symbols of key operators to fourth order in derivatives, improving manifest covariance in calculations.

Findings

Computed covariant symbols for external fields, derivatives, and Laplacian to fourth order.

Derived the covariant symbol of a general operator to fourth order.

Illustrated the method by calculating a diagonal matrix element to second order.

Abstract

Diagonal matrix elements of pseudodifferential operators are needed in order to compute effective Lagrangians and currents. For this purpose the method of symbols is often used, which however lacks manifest covariance. In this work the method of covariant symbols, introduced by Pletnev and Banin, is extended to curved space-time with arbitrary gauge and coordinate connections. For the Riemannian connection we compute the covariant symbols corresponding to external fields, the covariant derivative and the Laplacian, to fourth order in a covariant derivative expansion. This allows to obtain the covariant symbol of general operators to the same order. The procedure is illustrated by computing the diagonal matrix element of a nontrivial operator to second order. Applications of the method are discussed.