Spontaneously broken Lorentz symmetry for Hamiltonian gravity

Understand

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.