Various Hamiltonian formulations of f(R) gravity and their canonical relationships

TL;DR

This paper compares different Hamiltonian formulations of f(R) gravity, demonstrating that various approaches, including Ostrogradsky's method and conformal transformations, are connected through canonical transformations.

Contribution

It clarifies the relationships between different Hamiltonian formulations of f(R) gravity, showing they are canonically equivalent despite different variable choices.

Findings

Different Hamiltonian formulations are related by canonical transformations.

Ostrogradsky and conformal approaches are equivalent at the canonical level.

The equivalence clarifies the structure of f(R) gravity theories.

Abstract

Various Hamiltonian formulations of f(R) gravity can be found in the literature. Some authors follow the Ostrogradsky treatment of higher derivative theories and introduce as extra variables first order time derivatives of the metric (typically the extrinsic curvature). Some others take advantage of the conformal equivalence of f(R) theory with Einstein's gravity coupled to a scalar field and introduce as an extra variable the scalar curvature R itself, which includes second time derivatives of the metric. We show that, contrarily to some claims, these formulations are related by canonical transformations.