Article,
Path integrals and the indefiniteness of the gravitational action
Nuclear Physics B, 138 (1): 141--150 (Jun 5, 1978)
DOI: 10.1016/0550-3213(78)90161-x
Abstract
The Euclidean action for gravity is not positive definite unlike those of scalar and Yang-Mills fields. Indefiniteness arises because conformal transformations can make the action arbitrarily negative. In order to make the path integral converge one has to take the contour of integration for the conformal factor to be parallel to the imaginary axis. The path integral will then converge at least in the one-loop approximation if a certain positive action conjecture holds. We perform a zeta function regularization of the one-loop term for gravity and obtain a non-trivial scaling behaviour in cases in which the background metric has non-zero curvature tensor, and hence non-trivial topologies.
