Some comments on the Hamiltonian for Unimodular Gravity

TL;DR

This paper explores various Hamiltonian formulations of unimodular gravity, highlighting differences between naive and educated approaches and their classical equivalences and physical implications.

Contribution

It introduces and compares alternative first order Hamiltonian formulations of unimodular gravity, clarifying their classical equivalences and differences from the second order approach.

Findings

The educated Hamiltonian formulation is classically equivalent to the second order formulation.

Naive approach differs by treating some momenta as coordinates.

Physical differences between formulations are discussed.

Abstract

Several alternative formulations of the first order approach to unimodular gravity are presented. There is always a particular one such that it is {\em classically} equivalent to the second order formulation; this we call {\em educated}. It is often at variance with the {\em naive} approach, in which the lagrangian is taken as given exactly by the same expression as in the second order formulation; only the number and character of the independent variables changes. Namely, typically some of the momenta are now considered as coordinates. The ensuing Hamiltonians are thereby discussed and their physical differences pointed out.