Quantization of the nonprojectable 2+1D Horava theory: The second-class constraints

TL;DR

This paper develops a quantization method for the 2+1D nonprojectable Horava gravity, addressing second-class constraints and their impact on the path integral measure, leading to regularized propagators.

Contribution

It introduces a Hamiltonian quantization framework for the nonprojectable Horava theory with second-class constraints, including the treatment of the path integral measure and auxiliary variables.

Findings

Propagators of lapse and metric become regular

Path integral measure is adapted for second-class constraints

Auxiliary variables have nonregular propagators

Abstract

We present the quantization of the 2+1 dimensional nonprojectable Horava theory. The central point of the approach is that this is a theory with second-class constraints, hence the quantization procedure must take account of them. We consider all the terms in the Lagrangian that are compatible with the foliation-preserving-diffeomorphisms symmetry, up to the z=2 order which is the minimal order indicated by power-counting renormalizability. The measure of the path integral must be adapted to the second-class constraints, and this has consequences in the quantum dynamics of the theory. Since this measure is defined in terms of Poisson brackets between the second-class constraints, we develop all the Hamiltonian formulation of the theory with the full Lagrangian. We found that the propagator of the lapse function (and the one of the metric) acquires a totally regular form. The quantization requires the incorporation of a Lagrange multiplier for a second-class constraint and fermionic ghosts associated to the measure of the second-class constraints. These auxiliary variables have still nonregular propagators.