18 research outputs found
Renormalization of Entanglement Entropy and the Gravitational Effective Action
The entanglement entropy associated with a spatial boundary in quantum field
theory is UV divergent, with the leading term proportional to the area of the
boundary. For a class of quantum states defined by a path integral, the
Callan-Wilczek formula gives a geometrical definition of the entanglement
entropy. We show that, for this class of quantum states, the entanglement
entropy is rendered UV-finite by precisely the counterterms required to cancel
the UV divergences in the gravitational effective action. In particular, the
leading contribution to the entanglement entropy is given by the renormalized
Bekenstein-Hawking formula, in accordance with a proposal of Susskind and
Uglum. We show that the subleading UV-divergent terms in the entanglement
entropy depend nontrivially on the quantum state. We compute new subleading
terms in the entanglement entropy and find agreement with the Wald entropy
formula for black hole spacetimes with bifurcate Killing horizons. We speculate
that the entanglement entropy of an arbitrary spatial boundary may be a
well-defined observable in quantum gravity.Comment: 26 pages, 2 figures. v2: minor corrections and clarification
Scale-dependent homogeneity measures for causal dynamical triangulations
I propose two scale-dependent measures of the homogeneity of the quantum
geometry determined by an ensemble of causal triangulations. The first measure
is volumetric, probing the growth of volume with graph geodesic distance. The
second measure is spectral, probing the return probability of a random walk
with diffusion time. Both of these measures, particularly the first, are
closely related to those used to assess the homogeneity of our own universe on
the basis of galaxy redshift surveys. I employ these measures to quantify the
quantum spacetime homogeneity as well as the temporal evolution of quantum
spatial homogeneity of ensembles of causal triangulations in the well-known
physical phase. According to these measures, the quantum spacetime geometry
exhibits some degree of inhomogeneity on sufficiently small scales and a high
degree of homogeneity on sufficiently large scales. This inhomogeneity appears
unrelated to the phenomenon of dynamical dimensional reduction. I also uncover
evidence for power-law scaling of both the typical scale on which inhomogeneity
occurs and the magnitude of inhomogeneity on this scale with the ensemble
average spatial volume of the quantum spatial geometries.Comment: 25 pages, 19 figure
Renormalization of lattice-regularized quantum gravity models I. General considerations
Lattice regularization is a standard technique for the nonperturbative
definition of a quantum theory of fields. Several approaches to the
construction of a quantum theory of gravity adopt this technique either
explicitly or implicitly. A crucial complement to lattice regularization is the
process of renormalization through which a continuous description of the
quantum theory arises. I provide a comprehensive conceptual discussion of the
renormalization of lattice-regularized quantum gravity models. I begin with a
presentation of the renormalization group from the Wilsonian perspective. I
then consider the application of the renormalization group in four contexts:
quantum field theory on a continuous nondynamical spacetime, quantum field
theory on a lattice-regularized nondynamical spacetime, quantum field theory of
continuous dynamical spacetime, and quantum field theory of lattice-regularized
dynamical spacetime. The first three contexts serve to identify successively
the particular issues that arise in the fourth context. These issues originate
in the inescability of establishing all scales solely on the basis of the
dynamics. While most of this discussion rehearses established knowledge, the
attention that I pay to these issues, especially the previously
underappreciated role of standard units of measure, is largely novel. I
conclude by briefly reviewing past studies of renormalization of
lattice-regularized quantum gravity models. In the second paper of this
two-part series, I illustrate the ideas presented here by proposing a
renormalization group scheme for causal dynamical triangulations.Comment: 27 pages; revised and updated review of past literature. This is the
much delayed first paper in the two-part serie
Quantizing Horava-Lifshitz Gravity via Causal Dynamical Triangulations
We extend the discrete Regge action of causal dynamical triangulations to
include discrete versions of the curvature squared terms appearing in the
continuum action of (2+1)-dimensional projectable Horava-Lifshitz gravity.
Focusing on an ensemble of spacetimes whose spacelike hypersurfaces are
2-spheres, we employ Markov chain Monte Carlo simulations to study the path
integral defined by this extended discrete action. We demonstrate the existence
of known and novel macroscopic phases of spacetime geometry, and we present
preliminary evidence for the consistency of these phases with solutions to the
equations of motion of classical Horava-Lifshitz gravity. Apparently, the phase
diagram contains a phase transition between a time-dependent de Sitter-like
phase and a time-independent phase. We speculate that this phase transition may
be understood in terms of deconfinement of the global gravitational Hamiltonian
integrated over a spatial 2-sphere.Comment: 24 pages; 10 figure