Abstract
The quantum dynamics of the gravitational field non-minimally coupled to an (also dynamical) scalar field is studied in the {\em broken phase}. For a particular value of the coupling the system is classically conformal, and can actually be understood as the group averaging of Einstein-Hilbert's action under conformal transformations. Conformal invariance implies a simple Ward identity asserting that the trace of the equation of motion for the graviton is the equation of motion of the scalar field. We perform an explicit one-loop computation to show that the DeWitt effective action is not UV divergent {\em on shell} and to find that the Weyl symmetry Ward identity is preserved {\em on shell} at that level. We also discuss the fate of this Ward identity at the two-loop level --under the assumption that the two-loop UV divergent part of the effective action can be retrieved from the Goroff-Sagnotti counterterm-- and show that its preservation in the renormalized theory requires the introduction of counterterms which exhibit a logarithmic dependence on the dilaton field.
Highlights
When gravitational fields are dynamical, the corresponding symmetry is Weyl invariance, local rescalings of the spacetime metric
We perform an explicit oneloop computation to show that the DeWitt effective action is not UV divergent on shell and to find that the Weyl symmetry Ward identity is preserved on shell at that level
We discuss the fate of this Ward identity at the two-loop level — under the assumption that the two-loop UV divergent part of the effective action can be retrieved from the Goroff-Sagnotti counterterm — and show that its preservation in the renormalized theory requires the introduction of counterterms which exhibit a logarithmic dependence on the dilaton field
Summary
The simplest way to proceed in order to compute the divergences of any action involving the gravitational field is to use heat kernel techniques pioneered by Bryce de Witt. It can be shown [11] that this is equivalent to the assumption that the singular part of the propagator is of Hadamard type. The gauge fixing for diffeomorphism (Diff on) invariance will be chosen with an eye put on being able to implement heat kernel techiques in the simplest possible way This indicates that we shall try to cancel any non-minimal contribution to the kinetic term.
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