470 research outputs found
Gauge Group and Topology Change
The purpose of this study is to examine the effect of topology change in the
initial universe. In this study, the concept of -cobordism is introduced to
argue about the topology change of the manifold on which a transformation group
acts. This -manifold has a fiber bundle structure if the group action is
free and is related to the spacetime in Kaluza-Klein theory or
Einstein-Yang-Mills system. Our results revealed that fundamental processes of
compactification in -manifolds. In these processes, the initial high
symmetry and multidimensional universe changes to present universe by the
mechanism which lowers the dimensions and symmetries.Comment: 8 page
Sums over geometries and improvements on the mean field approximation
The saddle points of a Lagrangian due to Efetov are analyzed. This Lagrangian
was originally proposed as a tool for calculating systematic corrections to the
Bethe approximation, a mean-field approximation which is important in
statistical mechanics, glasses, coding theory, and combinatorial optimization.
Detailed analysis shows that the trivial saddle point generates a sum over
geometries reminiscent of dynamically triangulated quantum gravity, which
suggests new possibilities to design sums over geometries for the specific
purpose of obtaining improved mean field approximations to -dimensional
theories. In the case of the Efetov theory, the dominant geometries are locally
tree-like, and the sum over geometries diverges in a way that is similar to
quantum gravity's divergence when all topologies are included. Expertise from
the field of dynamically triangulated quantum gravity about sums over
geometries may be able to remedy these defects and fulfill the Efetov theory's
original promise. The other saddle points of the Efetov Lagrangian are also
analyzed; the Hessian at these points is nonnormal and pseudo-Hermitian, which
is unusual for bosonic theories. The standard formula for Gaussian integrals is
generalized to nonnormal kernels.Comment: Accepted for publication in Physical Review D, probably in November
2007. At the reviewer's request, material was added which made the article
more assertive, confident, and clear. No changes in substanc
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
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
Representations of the -algebra and the loop representation in -dimensions
We consider the phase-space of Yang-Mills on a cylindrical space-time () and the associated algebra of gauge-invariant functions, the
-variables. We solve the Mandelstam identities both classically and
quantum-mechanically by considering the -variables as functions of the
eigenvalues of the holonomy and their associated momenta. It is shown that
there are two inequivalent representations of the quantum -algebra. Then we
compare this reduced phase space approach to Dirac quantization and find it to
give essentially equivalent results. We proceed to define a loop representation
in each of these two cases. One of these loop representations (for ) is
more or less equivalent to the usual loop representation.Comment: 15 pages, LaTeX, 1 postscript figure included, uses epsf.sty,
G\"oteborg ITP 93-3
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
A statistical formalism of Causal Dynamical Triangulations
We rewrite the 1+1 Causal Dynamical Triangulations model as a spin system and
thus provide a new method of solution of the model.Comment: 21 pages, 19 pictures, 1 graph, Published in section: Field Theory
And Statistical System
The moduli space of isometry classes of globally hyperbolic spacetimes
This is the last article in a series of three initiated by the second author.
We elaborate on the concepts and theorems constructed in the previous articles.
In particular, we prove that the GH and the GGH uniformities previously
introduced on the moduli space of isometry classes of globally hyperbolic
spacetimes are different, but the Cauchy sequences which give rise to
well-defined limit spaces coincide. We then examine properties of the strong
metric introduced earlier on each spacetime, and answer some questions
concerning causality of limit spaces. Progress is made towards a general
definition of causality, and it is proven that the GGH limit of a Cauchy
sequence of , path metric Lorentz spaces is again a
, path metric Lorentz space. Finally, we give a
necessary and sufficient condition, similar to the one of Gromov for the
Riemannian case, for a class of Lorentz spaces to be precompact.Comment: 29 pages, 9 figures, submitted to Class. Quant. Gra
Coupling a Point-Like Mass to Quantum Gravity with Causal Dynamical Triangulations
We present a possibility of coupling a point-like, non-singular, mass
distribution to four-dimensional quantum gravity in the nonperturbative setting
of causal dynamical triangulations (CDT). In order to provide a point of
comparison for the classical limit of the matter-coupled CDT model, we derive
the spatial volume profile of the Euclidean Schwarzschild-de Sitter space glued
to an interior matter solution. The volume profile is calculated with respect
to a specific proper-time foliation matching the global time slicing present in
CDT. It deviates in a characteristic manner from that of the pure-gravity
model. The appearance of coordinate caustics and the compactness of the mass
distribution in lattice units put an upper bound on the total mass for which
these calculations are expected to be valid. We also discuss some of the
implementation details for numerically measuring the expectation value of the
volume profiles in the framework of CDT when coupled appropriately to the
matter source.Comment: 26 pages, 9 figures, updated published versio
Crossing the c=1 barrier in 2d Lorentzian quantum gravity
In an extension of earlier work we investigate the behaviour of
two-dimensional Lorentzian quantum gravity under coupling to a conformal field
theory with c>1. This is done by analyzing numerically a system of eight Ising
models (corresponding to c=4) coupled to dynamically triangulated Lorentzian
geometries. It is known that a single Ising model couples weakly to Lorentzian
quantum gravity, in the sense that the Hausdorff dimension of the ensemble of
two-geometries is two (as in pure Lorentzian quantum gravity) and the matter
behaviour is governed by the Onsager exponents. By increasing the amount of
matter to 8 Ising models, we find that the geometry of the combined system has
undergone a phase transition. The new phase is characterized by an anomalous
scaling of spatial length relative to proper time at large distances, and as a
consequence the Hausdorff dimension is now three. In spite of this qualitative
change in the geometric sector, and a very strong interaction between matter
and geometry, the critical exponents of the Ising model retain their Onsager
values. This provides evidence for the conjecture that the KPZ values of the
critical exponents in 2d Euclidean quantum gravity are entirely due to the
presence of baby universes. Lastly, we summarize the lessons learned so far
from 2d Lorentzian quantum gravity.Comment: 21 pages, 18 figures (postscript), uses JHEP.cls, see
http://www.nbi.dk/~ambjorn/lqg2 for related animated simulation
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