ONE-LOOP AMPLITUDES IN EUCLIDEAN QUANTUM-GRAVITY

This paper studies the linearized gravitational held in the presence o f boundaries. For this purpose, zeta-function regularization is used t o perform the mode-by-mode evaluation of BRST-invariant Faddeev-Popov amplitudes in the case of hat 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 calculat ion of the same amplitudes, which relies on the recently corrected geo metric formulas for the asymptotic heat kernel in the case of mixed bo undary 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 obtai ned on considering from the beginning all perturbative modes of the gr avitational field, jointly with ghost modes.