Extrinsic curvature in 2-dimensional Causal Dynamical Triangulation

TL;DR

This paper explores the connection between 2D Causal Dynamical Triangulations and Hořava-Lifshitz gravity by applying CDT quantisation to the latter, deriving a matching continuum Hamiltonian, and showing the lattice does not affect continuum physics.

Contribution

It demonstrates that applying CDT to Hořava-Lifshitz gravity in 2D yields a continuum Hamiltonian identical to the canonical quantisation result, with no lattice imprint.

Findings

Continuum Hamiltonian matches canonical quantisation

Lattice quantisation does not restrict configuration space

No imprint of lattice on continuum physics

Abstract

Causal Dynamical Triangulations (CDT) is a non-perturbative quantisation of general relativity. Ho\v{r}ava-Lifshitz gravity on the other hand modifies general relativity to allow for perturbative quan- tisation. Past work has given rise to the speculation that Ho\v{r}ava-Lifshitz gravity might correspond to the continuum limit of CDT. In this paper we add another piece to this puzzle by applying the CDT quantisation prescription directly to Ho\v{r}ava-Lifshitz gravity in 2 dimensions. We derive the continuum Hamiltonian and we show that it matches exactly the Hamiltonian one derives from canonically quantising the Ho\v{r}ava-Lifshitz action. Unlike the standard CDT case, here the intro- duction of a foliated lattice does not impose further restriction on the configuration space and, as a result, lattice quantisation does not leave any imprint on continuum physics as expected.