A note on the Hamiltonian constraint in canonical GR

TL;DR

This paper demonstrates that the Hamiltonian constraint in canonical general relativity with matter fields can be expressed as a Poisson bracket involving a dimensionless functional and the volume, with implications for quantum gravity eigenstates.

Contribution

It reveals a new Poisson bracket structure for the Hamiltonian constraint in coupled Einstein-Yang-Mills-Higgs systems, highlighting potential restrictions on quantum volume eigenstates.

Findings

Hamiltonian constraint expressed as a Poisson bracket.

Potential restriction on volume eigenstates in quantum gravity.

Implication for the structure of canonical quantum gravity.

Abstract

The Hamiltonian constraint of the coupled Einstein-Yang-Mills-Higgs system with a cosmological constant is shown to be a pure Poisson bracket of a dimensionless functional on the phase space and the volume of the three-space. One of its potential consequences, a restriction on the eigenstates of the volume operator in a class of canonical quantum gravity theories, is also pointed out.