Quantization of the Horava theory at the kinetic-conformal point

The Horava theory depends on several coupling constants. The kinetic term of its Lagrangian depends on one dimensionless coupling constant lambda. For the particular value lambda = 1/3 the kinetic term becomes conformal invariant, although the full Lagrangian does not have this symmetry. For any value of lambda the nonprojectable version of the theory has second-class constraints that play a central role in the process of quantization. Here we study the complete nonprojectable theory, including the Blas-Pujolas-Sibiryakov interacting terms, at the kinetic-conformal point lambda = 1/3. The generic counting of degrees of freedom indicates that this theory propagates the same physical degrees of freedom of general relativity. We analyze this point rigorously, taking into account all the z = 1, 2, 3 terms that contribute to the action describing quadratic perturbations around the Minkowski spacetime. We show that the constraints of the theory and equations determining the Lagrange multipliers are strongly elliptic partial differential equations, an essential condition for a constrained phase-space structure in field theory. We show how their solutions lead to the two independent tensorial physical modes propagated by the theory. We also obtain the reduced Hamiltonian. These arguments strengthen the consistency of the theory. We find the restrictions on the space of coupling constants to ensure the positiveness of the reduced Hamiltonian. We obtain the propagator of the physical modes, showing that there are not ghosts and that the propagator effectively acquires the z = 3 scaling for all physical degrees of freedom at the high-energy regime. By evaluating the superficial degree of divergence, taking into account the second-class constraints, we show that the theory is power-counting renormalizable. We analyze, in the path-integral formulation of the theory, the measure associated to the second-class constraints both in the canonical and the Lagrangian (foliation-preserving diffeomorphisms group-covariant) formalisms.