  We perform the BFV quantization of the 2+1 projectable and the 3+1
nonprojectable versions of the Horava theory. This is a Hamiltonian formalism,
and noncanonical gauges can be used with it. In the projectable case, we show
that the integration on canonical momenta reproduces the quantum Lagrangian
known from the proof of renormalization of Barvinsky et al. This quantum
Lagrangian is nonlocal, its nonlocality originally arose as a consequence of
getting regular propagators. The matching of the BFV quantization with the
quantum Lagrangian reinforces the program of quantization of the Horava theory.
We introduce a local gauge-fixing condition, hence a local Hamiltonian, that
leads to the nonlocality of the Lagrangian after the integration. For the case
of the nonprojectable theory, this procedure allows us to obtain the complete
(nonlocal) quantum Lagrangian that takes into account the second-class
contraints. We compare with the integration in general relativity, making clear
the relationship between the underlying anisotropic symmetry of the Horava
theory and the nonlocality of its quantum Lagrangian.
