Quantum Lagrangian of the Horava theory and its nonlocalities

TL;DR

This paper performs BFV quantization of Horava gravity in 2+1 and 3+1 dimensions, revealing the nonlocal quantum Lagrangian structure and its relation to gauge choices and constraints, thus advancing the understanding of its quantization.

Contribution

It provides a Hamiltonian BFV quantization framework for Horava gravity, deriving the nonlocal quantum Lagrangian and analyzing gauge fixing and constraints.

Findings

Reproduces known nonlocal quantum Lagrangian from BFV quantization.

Introduces a local gauge fixing leading to nonlocality after integration.

Clarifies the relationship between anisotropic symmetry and nonlocality in quantum Lagrangian.

Abstract

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.