The method of covariant symbols in curved space-time

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.

arXiv comment28 pages, no figures. References added. To appear in European Physical Journal C

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• Renormalization of vector fields with mass-like coupling in curved spacetime
• Direct construction of the effective action of chiral gauge fermions in the anomalous sector
• The invariant factor of the chiral determinant
• Leptonic CP violating effective action for Dirac and Majorana neutrinos
• Non-minimal non-Abelian quantum vector fields in curved spacetime