FKWC-bases and geometrical identities for classical and quantum field theories in curved spacetime

TL;DR

This paper refines FKWC bases for Riemann polynomials and provides geometrical identities that facilitate calculations in classical and quantum field theories in curved spacetime, aiding researchers in gravitational physics.

Contribution

It offers a modified version of FKWC-bases and a set of dimension-independent geometrical relations to simplify complex tensor calculations.

Findings

Provides a revised FKWC-bases for scalar and tensor Riemann polynomials.

Lists geometrical identities useful for calculations in gravitational physics.

Facilitates hand calculations in quantum and classical gravity theories.

Abstract

Fulling, King, Wybourne and Cummings (FKWC) have proposed to expand systematically the Riemann polynomials encountered in the context of field theories in curved spacetime on standard bases constructed from group theoretical considerations. They have also displayed such bases for scalar Riemann polynomials of order eight or less in the derivatives of the metric tensor and for tensorial Riemann polynomials of order six or less. Here we provide a slightly modified version of the FKWC-bases as well as an important list of geometrical relations we have used in recent works. These relations are independent of the dimension of spacetime. In our opinion, they are helpful to achieve quickly and easily, by hand, very tedious calculations as well as, of course, to provide irreducible expressions for all the results obtained. They could be very helpful to people working in gravitational physics and more particularly (i) to treat some aspects of the classical theory of gravitational waves such as the radiation reaction problem and (ii) to deal with regularization and renormalization of quantum field theories, of stochastic semiclassical gravity and of higher-order theories of gravity.