Spontaneously broken Lorentz symmetry for Hamiltonian gravity

TL;DR

This paper introduces a Lorentz covariant, real Hamiltonian formulation of gravity that avoids second class constraints by using an observer field to break symmetry, enabling a Cartan geometric perspective.

Contribution

It presents a novel real, Lorentz covariant Hamiltonian formulation of gravity that employs an observer field to break symmetry without second class constraints.

Findings

A real, Lorentz covariant formulation avoids complex fields.

Symmetry breaking via an observer field leads to Cartan geometrodynamics.

The approach maintains Lorentz covariance despite symmetry breaking.

Abstract

In Ashtekar's Hamiltonian formulation of general relativity, and in loop quantum gravity, Lorentz covariance is a subtle issue that has been strongly debated. Maintaining manifest Lorentz covariance seems to require introducing either complex-valued fields, presenting a significant obstacle to quantization, or additional (usually second class) constraints whose solution renders the resulting phase space variables harder to interpret in a spacetime picture. After reviewing the sources of difficulty, we present a Lorentz covariant, real formulation in which second class constraints never arise. Rather than a foliation of spacetime, we use a gauge field y, interpreted as a field of observers, to break the SO(3,1) symmetry down to a subgroup SO(3)_y. This symmetry breaking plays a role analogous to that in MacDowell-Mansouri gravity, which is based on Cartan geometry, leading us to a picture of gravity as 'Cartan geometrodynamics.' We study both Lorentz gauge transformations and transformations of the observer field to show that the apparent breaking of SO(3,1) to SO(3) is not in conflict with Lorentz covariance.