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.

Highlights

  • Horava theory [1,2] is a geometrical field theory that may be used to study quantum gravity since it is power-counting renormalizable and unitary

  • We focus on the path integral quantization, since it is more adaptable to a gravitational field theory as the Horava theory

  • We consider in the Lagrangian all the inequivalent terms that are compatible with the foliation-preserving diffeomorphisms (FDiff) symmetry, up to the minimal order in spatial derivatives required by power-counting renormalizability, which is z 1⁄4 2 in the 2 þ 1 theory [1]

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Summary

INTRODUCTION

Horava theory [1,2] is a geometrical field theory that may be used to study quantum gravity since it is power-counting renormalizable and unitary. In the case of the nonprojectable 3 þ 1 Horava gravity, the Lagrangian includes a number of the order of 102 different terms that are compatible with the FDiff gauge symmetry. An essential feature of the nonprojectable theory is the presence of second-class constraints This fact forces us to consider the quantization in rather different approaches to ones used in general relativity, projectable Horava theory, and gauge theories in general with only first-class constraints. We consider in the Lagrangian all the inequivalent terms that are compatible with the FDiff symmetry, up to the minimal order in spatial derivatives required by power-counting renormalizability, which is z 1⁄4 2 in the 2 þ 1 theory [1]. More consequences on the value of the energy and the role of the higher order terms were considered in that reference

THE NONPROJECTABLE HOŘAVA THEORY IN 2 SPATIAL DIMENSIONS
Hamiltonian and constraints
Algebra of the constraints
Linearized theory
The reduced Hamiltonian and the propagator of the physical mode
Definition of the measure
The measure of the gauge sector
The second-class-constraint measure
Perturbations in the path integral
CONCLUSIONS

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