Spontaneously broken Lorentz symmetry for Hamiltonian gravity

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.

10 pages, revtex, APS style; v2: extended discussion of relation to approaches using second class constraints in abstract/introduction, made some additional small corrections; v3: fixed abstract; v4: minimal changes to match published version

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• Lifting General Relativity to Observer Space
• Hamiltonian formulation of gravity as a spontaneously-broken gauge theory of the Lorentz group
• Linking Covariant and Canonical General Relativity via Local Observers
• Degenerate Plebanski Sector and Spin Foam Quantization
• Contracted Lorentz invariance for gravity with a preferred foliation
• Geometrodynamics and Lorentz symmetry
• Spontaneous breaking of Lorentz symmetry for canonical gravity
• The Space of Connections as the Arena for (Quantum) Gravity