The equivalence theorem in the generalized gravity of f(R)-type and canonical quantization II

TL;DR

This paper reviews the equivalence between f(R)-type gravity and Einstein gravity with a scalar field, explores the extent of this equivalence in classical and quantum contexts, and discusses related foundational issues.

Contribution

It provides a self-contained review of the equivalence theorem and analyzes the quantum and surface term issues within a canonical formalism, extending prior work.

Findings

Clarified the necessity of the Gibbons-Hawking surface term in f(R) gravity.

Reviewed the quantum equivalence problem in a canonical framework.

Commented on the physical metric identification and non-commutative spacetime implications.

Abstract

We first review the equivalence theorem of the f(R)-type gravity to Einstein gravity with a scalar field by deriving it in a self-contained and pedagogical way. Then we describe the problem of to what extent the equivalence holds. Main problems are (i) Is the surface term given by Gibbons and Hawking which is necessary in Einstein gravity also necessary in the f(R)-type gravity? (ii) Does the equivalence hold also in quantum theory? (iii) Which metric is physical, i.e., which metric should be identified with the observed one? In this work, we clarify the problem (i) and review the problem (ii) in a canonical formalism which is the generalization of the Ostrogradski one. We briefly comment on the problem (iii). Some discussions are given on one of the results of (ii) concerning the general relativity in non-commutative spacetime.