Quantum Equivalence of $f(R)$ Gravity and Scalar-tensor Theories in the Jordan and Einstein Frames

TL;DR

This paper investigates whether the classical equivalence between $f(R)$ gravity and scalar-tensor theories persists at the quantum level, demonstrating their equivalence at one-loop order on shell in arbitrary dimensions.

Contribution

It proves the quantum equivalence of $f(R)$ gravity and scalar-tensor theories in Jordan and Einstein frames at one-loop level on shell, and computes the one-loop divergence in $f(R)$ gravity.

Findings

The three formulations yield the same effective action on shell.

Quantum equivalence holds at one-loop level in arbitrary dimensions.

The one-loop divergence in $f(R)$ gravity is explicitly calculated.

Abstract

The $f(R)$ gravity and scalar-tensor theory are known to be equivalent at the classical level. We study if this equivalence is valid at the quantum level. There are two descriptions of the scalar-tensor theory in the Jordan and Einstein frames. It is shown that these three formulations of the theories give the same determinant or effective action on shell, and thus they are equivalent at the quantum one-loop level on shell in arbitrary dimensions. We also compute the one-loop divergence in $f(R)$ gravity.