Hamiltonian formulation of gravity as a spontaneously-broken gauge theory of the Lorentz group

Abstract

A number of approaches to gravitation have much in common with the gauge theories of the standard model of particle physics. In this paper, we develop the Hamiltonian formulation of a class of gravitational theories that may be regarded as spontaneously-broken gauge theories of the complexified Lorentz group $SO(1,3)_C$ with the gravitational field described entirely by a gauge field valued in the Lie algebra of $SO(1,3)_C$ and a `Higgs field' valued in the group's fundamental representation. The theories have one free parameter $\beta$ which appears in a similar role to the inverse of the Barbero-Immirzi parameter of Einstein-Cartan theory. However, contrary to that parameter, it is shown that the number of degrees of freedom crucially depends on the value of $\beta$. For non-zero values of $\beta$, it is shown that three complex degrees of freedom propagate on general backgrounds, and for the specific values $\beta=\pm i$ an extension to General Relativity is recovered in a symmetry-broken regime. For the value $\beta=0$, the theory propagates no local degrees of freedom. A non-zero value of $\beta$ corresponds to the self-dual and anti-self-dual gauge fields appearing asymmetrically in the action, therefore in these models, the existence of gravitational degrees of freedom is tied to chiral asymmetry in the gravitational sector.