Path Integrals in Quadratic Gravity

Abstract

Using the invariance of Quadratic Gravity in FLRW metric under the group of diffeomorphisms of the time coordinate, we rewrite the action $A$ of the theory in terms of the invariant dynamical variable $g(\tau)\,.$ We propose to consider the path integrals $\int\,F(g)\,\exp\left\{-A \right\}dg$ as the integrals over the functional measure $\mu(g)=\exp\left\{-A_{2} \right\}dg\,,\ $ where $A_{2}$ is the part of the action $A$ quadratic in $R\,.$ The rest part of the action stands in the exponent in the integrand as the "interaction" term. We prove the measure $\mu(g)$ to be equivalent to the Wiener measure, and, as an example, calculate the averaged scale factor in the first nontrivial perturbative order.