One-Loop Amplitudes in Euclidean Quantum Gravity

TL;DR

This paper evaluates one-loop gravitational amplitudes in Euclidean quantum gravity with boundaries, confirming covariant results and emphasizing the necessity of including all perturbative modes for gauge invariance.

Contribution

It demonstrates the equivalence of mode-by-mode and covariant calculations of amplitudes and clarifies the importance of including all modes for gauge-invariant results.

Findings

$ ext{zeta}(0)$ matches covariant calculations

Transverse-traceless restriction is insufficient

All perturbative and ghost modes are needed for gauge invariance

Abstract

This paper studies the linearized gravitational field in the presence of boundaries. For this purpose, $\zeta$-function regularization is used to perform the mode-by-mode evaluation of BRST-invariant Faddeev-Popov amplitudes in the case of flat Euclidean four-space bounded by a three-sphere. On choosing the de Donder gauge-averaging term, the resulting $\zeta(0)$ value is found to agree with the space-time covariant calculation of the same amplitudes, which relies on the recently corrected geometric formulas for the asymptotic heat kernel in the case of mixed boundary conditions. Two sets of mixed boundary conditions for Euclidean quantum gravity are then compared in detail. The analysis proves that one cannot restrict the path-integral measure to transverse-traceless perturbations. By contrast, gauge-invariant amplitudes are only obtained on considering from the beginning all perturbative modes of the gravitational field, jointly with ghost modes.