Path Integrals in Quadratic Gravity

TL;DR

This paper reformulates quadratic gravity path integrals in FLRW spacetime using invariant variables, demonstrating their equivalence to Wiener measure and computing average scale factors perturbatively.

Contribution

It introduces a new approach to path integrals in quadratic gravity using invariant variables and establishes their equivalence to Wiener measure, with explicit perturbative calculations.

Findings

Path integrals can be expressed as Wiener measures in quadratic gravity.

The measure's equivalence to Wiener measure is proven.

Average scale factor computed perturbatively in the new framework.

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.