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General Relativity and Quantum Cosmology, gr-qc,High Energy Physics - Theory, hep-th
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 or second class constraints, and either option
presents a significant obstacle to quantization. After reviewing the sources of
difficulty, we present a Lorentz covariant, real formulation free of second
class constraints. 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.
