530 research outputs found
THE VOLUME OPERATOR IN DISCRETIZED QUANTUM GRAVITY
We investigate the spectral properties of the volume operator in quantum
gravity in the framework of a previously introduced lattice discretization. The
presence of a well-defined scalar product in this approach permits us to make
definite statements about the hermiticity of quantum operators. We find that
the spectrum of the volume operator is discrete, but that the nature of its
eigenstates differs from that found in an earlier continuum treatment.Comment: 15 pages, TeX, 3 figures (postscript, compressed and uu-encoded), May
9
Independent Loop Invariants for 2+1 Gravity
We identify an explicit set of complete and independent Wilson loop
invariants for 2+1 gravity on a three-manifold , with
a compact oriented Riemann surface of arbitrary genus . In the
derivation we make use of a global cross section of the -principal
bundle over Teichm\"uller space given in terms of Fenchel-Nielsen coordinates.Comment: 11pp, 2 figures (postscript, compressed and uu-encoded), TeX,
Pennsylvania State University, CGPG-94/7-
Quantum widening of CDT universe
The physical phase of Causal Dynamical Triangulations (CDT) is known to be
described by an effective, one-dimensional action in which three-volumes of the
underlying foliation of the full CDT play a role of the sole degrees of
freedom. Here we map this effective description onto a statistical-physics
model of particles distributed on 1d lattice, with site occupation numbers
corresponding to the three-volumes. We identify the emergence of the quantum
de-Sitter universe observed in CDT with the condensation transition known from
similar statistical models. Our model correctly reproduces the shape of the
quantum universe and allows us to analytically determine quantum corrections to
the size of the universe. We also investigate the phase structure of the model
and show that it reproduces all three phases observed in computer simulations
of CDT. In addition, we predict that two other phases may exists, depending on
the exact form of the discretised effective action and boundary conditions. We
calculate various quantities such as the distribution of three-volumes in our
model and discuss how they can be compared with CDT.Comment: 19 pages, 13 figure
Counting a black hole in Lorentzian product triangulations
We take a step toward a nonperturbative gravitational path integral for
black-hole geometries by deriving an expression for the expansion rate of null
geodesic congruences in the approach of causal dynamical triangulations. We
propose to use the integrated expansion rate in building a quantum horizon
finder in the sum over spacetime geometries. It takes the form of a counting
formula for various types of discrete building blocks which differ in how they
focus and defocus light rays. In the course of the derivation, we introduce the
concept of a Lorentzian dynamical triangulation of product type, whose
applicability goes beyond that of describing black-hole configurations.Comment: 42 pages, 11 figure
A String Field Theory based on Causal Dynamical Triangulations
We formulate the string field theory in zero-dimensional target space
corresponding to the two-dimensional quantum gravity theory defined through
Causal Dynamical Triangulations. This third quantization of the quantum gravity
theory allows us in principle to calculate the transition amplitudes of
processes in which the topology of space changes in time, and to include
non-trivial topologies of space-time. We formulate the corresponding
Dyson-Schwinger equations and illustrate how they can be solved iteratively.Comment: 29 pages, 4 figure
Modelling gravity on a hyper-cubic lattice
We present an elegant and simple dynamical model of symmetric, non-degenerate
(n x n) matrices of fixed signature defined on a n-dimensional hyper-cubic
lattice with nearest-neighbor interactions. We show how this model is related
to General Relativity, and discuss multiple ways in which it can be useful for
studying gravity, both classical and quantum. In particular, we show that the
dynamics of the model when all matrices are close to the identity corresponds
exactly to a finite-difference discretization of weak-field gravity in harmonic
gauge. We also show that the action which defines the full dynamics of the
model corresponds to the Einstein-Hilbert action to leading order in the
lattice spacing, and use this observation to define a lattice analogue of the
Ricci scalar and Einstein tensor. Finally, we perform a mean-field analysis of
the statistical mechanics of this model.Comment: 5 page
Random walks on combs
We develop techniques to obtain rigorous bounds on the behaviour of random
walks on combs. Using these bounds we calculate exactly the spectral dimension
of random combs with infinite teeth at random positions or teeth with random
but finite length. We also calculate exactly the spectral dimension of some
fixed non-translationally invariant combs. We relate the spectral dimension to
the critical exponent of the mass of the two-point function for random walks on
random combs, and compute mean displacements as a function of walk duration. We
prove that the mean first passage time is generally infinite for combs with
anomalous spectral dimension.Comment: 42 pages, 4 figure
Emergence of a 4D World from Causal Quantum Gravity
Causal Dynamical Triangulations in four dimensions provide a
background-independent definition of the sum over geometries in nonperturbative
quantum gravity, with a positive cosmological constant. We present evidence
that a macroscopic four-dimensional world emerges from this theory dynamically.Comment: 11 pages, 3 figures; some short clarifying comments added; final
version to appear in Phys. Rev. Let
(Broken) Gauge Symmetries and Constraints in Regge Calculus
We will examine the issue of diffeomorphism symmetry in simplicial models of
(quantum) gravity, in particular for Regge calculus. We find that for a
solution with curvature there do not exist exact gauge symmetries on the
discrete level. Furthermore we derive a canonical formulation that exactly
matches the dynamics and hence symmetries of the covariant picture. In this
canonical formulation broken symmetries lead to the replacements of constraints
by so--called pseudo constraints. These considerations should be taken into
account in attempts to connect spin foam models, based on the Regge action,
with canonical loop quantum gravity, which aims at implementing proper
constraints. We will argue that the long standing problem of finding a
consistent constraint algebra for discretized gravity theories is equivalent to
the problem of finding an action with exact diffeomorphism symmetries. Finally
we will analyze different limits in which the pseudo constraints might turn
into proper constraints. This could be helpful to infer alternative
discretization schemes in which the symmetries are not broken.Comment: 32 pages, 15 figure
The 1/N expansion of colored tensor models in arbitrary dimension
In this paper we extend the 1/N expansion introduced in [1] to group field
theories in arbitrary dimension and prove that only graphs corresponding to
spheres S^D contribute to the leading order in the large N limit.Comment: 4 pages, 3 figure
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