Frame covariance in quantum gravity

Abstract

We develop a quantum effective action for scalar-tensor theories of gravity which is both spacetime diffeomorphism invariant and field reparameterisation (frame) invariant beyond the classical approximation. We achieve this by extending the Vilkovisky-DeWitt formalism, treating both the scalar fields and the components of the gravitational tensor field as coordinates describing a manifold. By using tensors covariant under diffeomorphisms of this manifold, we show that scalar-tensor theories can be written in a form that is manifestly frame invariant at both classical and quantum levels. In~the same context, we show that in order to maintain manifest frame invariance, we must modify the Feynman rules of theories with a non-trivial field space. We show that one such theory is General Relativity by demonstrating explicitly that it has a non-zero field-space Riemann tensor. Thus, when constructing theories of quantum gravity, we must deal not only with curved spacetime, but also with a curved field space. Finally, we address the cosmological frame problem by tracing its origin to the existence of a new model function that appears in the path integral measure. Once this function is fixed, we find that frame transformations have no effect on the quantisation of the theory. The uniqueness of our improved quantum effective action is discussed.