CDT meets Hořava–Lifshitz gravity

Abstract

The theory of causal dynamical triangulations (CDT) attempts to define a nonperturbative theory of quantum gravity as a sum over space-time geometries. One of the ingredients of the CDT framework is a global time foliation, which also plays a central role in the quantum gravity theory recently formulated by Hořava. We show that the phase diagram of CDT bears a striking resemblance with the generic Lifshitz phase diagram appealed to by Hořava. We argue that CDT might provide a unifying nonperturbative framework for anisotropic as well as isotropic theories of quantum gravity.

The transfer matrix in four-dimensional CDT

“…For these choices of coupling constants the shape of the universe is consistent with an interpretation as an (Euclidean) de Sitter space, at least as long as one looks at the scale factor [49,50]. This is the region of coupling constants which will have our interest (for other choices of the coupling constants one obtains more degenerate configurations [51][52][53]). …”

“…The phase diagram now depends on κ 0 and ∆ and we refer to [3,[51][52][53] for a detailed discussion. Here we will be working in the interesting, so-called de Sitter phase where we, for a given (large) N 4 and sufficient large t tot , observe a (Euclidean) de Sitter universe, i.e.…”

A functional renormalization group equation for foliated spacetimes

“…Following the ideas [57,58,59,56] one could, e.g., use the foliated flow equation to study diffusion processes on spatial slices and examine if these can be matched to the CDT data recently reported in [60]. This may shed new light on the question whether the continuum limit of CDT corresponds to an isotropic or anisotropic gravity theory as suggested in [61,62]. The trace of the heat kernel has the well-known asymptotic expansion for s → 0…”

On a renormalization group scheme for causal dynamical triangulations

“…The value of a dimensionful physical observable at some scale is also a real number, namely the ratio of the physical observable to a standard unit of measure. 2 This reasoning also applies to the scale at which a physical observable is measured. To measure a dimensionful physical observable, one thus requires an appropriate standard unit of measure.…”

“…The A-C phase transition is of first order, the B-C phase transition is of second order, and the order of the A-B phase transition has not been ascertained [8,9]. The three phases are hypothesized to meet at a triple point [2]. My proposed renormalization group scheme applies only to phase C. For this phase there are several observables whose extensive study has linked their phenomenology to a theoretical model, allowing for the formulation of a renormalization group scheme.…”