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

Understand

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.