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The Hamiltonian formulation of the lowest-order projectable Horava gravity,namely the so-called $\lambda$-$R$ gravity, is studied. Since a preferredfoliation has been chosen in projectable Horava gravity, there is no localHamiltonian constraint in the theory. In contrast to general relativity, theconstraint algebra of $\lambda$-$R$ gravity forms a Lie algebra. By canonicaltransformations, we further obtain the connection-dynamical formalism of the$\lambda$-R gravity theories with real $su(2)$-connections as configurationvariables. This formalism enables us to extend the scheme of non-perturbativeloop quantum gravity to the $\lambda$-$R$ gravity. While the quantumkinematical framework is the same as that for general relativity, theHamiltonian constraint operator of loop quantum $\lambda$-$R$ gravity can bewell defined in the diffeomorphism-invariant Hilbert space. Moreover, byintroducing a global dust degree of freedom to represent a dynamical time, aphysical Hamiltonian operator with respect to the dust can be defined and thephysical states satisfying all the constraints are obtained.

Canonical loop quantization of the lowest-order projectable Horava gravity