In Ashtekar's Hamiltonian formulation of general relativity, and in loopquantum gravity, Lorentz covariance is a subtle issue that has been stronglydebated. Maintaining manifest Lorentz covariance seems to require introducingeither complex-valued fields, presenting a significant obstacle toquantization, or additional (usually second class) constraints whose solutionrenders the resulting phase space variables harder to interpret in a spacetimepicture. After reviewing the sources of difficulty, we present a Lorentzcovariant, real formulation in which second class constraints never arise.Rather than a foliation of spacetime, we use a gauge field y, interpreted as afield 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-Mansourigravity, which is based on Cartan geometry, leading us to a picture of gravityas 'Cartan geometrodynamics.' We study both Lorentz gauge transformations andtransformations of the observer field to show that the apparent breaking ofSO(3,1) to SO(3) is not in conflict with Lorentz covariance.
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