Canonical loop quantization of the lowest-order projectable Horava gravity

TL;DR

This paper develops a loop quantum gravity framework for the lowest-order projectable Horava gravity, deriving a well-defined Hamiltonian operator and physical states using a dust field as a clock.

Contribution

It extends loop quantum gravity techniques to $ ext{lambda}$-$R$ gravity, providing a canonical and quantum formulation with a physical Hamiltonian and states.

Findings

The constraint algebra of $ ext{lambda}$-$R$ gravity forms a Lie algebra.

A connection-dynamical formalism with $su(2)$-connections is established.

A physical Hamiltonian operator and states are constructed using a dust clock.

Abstract

The Hamiltonian formulation of the lowest-order projectable Horava gravity, namely the so-called $\lambda$-$R$ gravity, is studied. Since a preferred foliation has been chosen in projectable Horava gravity, there is no local Hamiltonian constraint in the theory. In contrast to general relativity, the constraint algebra of $\lambda$-$R$ gravity forms a Lie algebra. By canonical transformations, we further obtain the connection-dynamical formalism of the $\lambda$-R gravity theories with real $su(2)$-connections as configuration variables. This formalism enables us to extend the scheme of non-perturbative loop quantum gravity to the $\lambda$-$R$ gravity. While the quantum kinematical framework is the same as that for general relativity, the Hamiltonian constraint operator of loop quantum $\lambda$-$R$ gravity can be well defined in the diffeomorphism-invariant Hilbert space. Moreover, by introducing a global dust degree of freedom to represent a dynamical time, a physical Hamiltonian operator with respect to the dust can be defined and the physical states satisfying all the constraints are obtained.