374 research outputs found
Spectral dimension of quantum geometries
The spectral dimension is an indicator of geometry and topology of spacetime
and a tool to compare the description of quantum geometry in various approaches
to quantum gravity. This is possible because it can be defined not only on
smooth geometries but also on discrete (e.g., simplicial) ones. In this paper,
we consider the spectral dimension of quantum states of spatial geometry
defined on combinatorial complexes endowed with additional algebraic data: the
kinematical quantum states of loop quantum gravity (LQG). Preliminarily, the
effects of topology and discreteness of classical discrete geometries are
studied in a systematic manner. We look for states reproducing the spectral
dimension of a classical space in the appropriate regime. We also test the
hypothesis that in LQG, as in other approaches, there is a scale dependence of
the spectral dimension, which runs from the topological dimension at large
scales to a smaller one at short distances. While our results do not give any
strong support to this hypothesis, we can however pinpoint when the topological
dimension is reproduced by LQG quantum states. Overall, by exploring the
interplay of combinatorial, topological and geometrical effects, and by
considering various kinds of quantum states such as coherent states and their
superpositions, we find that the spectral dimension of discrete quantum
geometries is more sensitive to the underlying combinatorial structures than to
the details of the additional data associated with them.Comment: 39 pages, 18 multiple figures. v2: discussion improved, minor typos
correcte
Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results
We review some recent attempts to extract information about the nature of
quantum gravity, with and without matter, by quantum field theoretical methods.
More specifically, we work within a covariant lattice approach where the
individual space-time geometries are constructed from fundamental simplicial
building blocks, and the path integral over geometries is approximated by
summing over a class of piece-wise linear geometries. This method of
``dynamical triangulations'' is very powerful in 2d, where the regularized
theory can be solved explicitly, and gives us more insights into the quantum
nature of 2d space-time than continuum methods are presently able to provide.
It also allows us to establish an explicit relation between the Lorentzian- and
Euclidean-signature quantum theories. Analogous regularized gravitational
models can be set up in higher dimensions. Some analytic tools exist to study
their state sums, but, unlike in 2d, no complete analytic solutions have yet
been constructed. However, a great advantage of our approach is the fact that
it is well-suited for numerical simulations. In the second part of this review
we describe the relevant Monte Carlo techniques, as well as some of the
physical results that have been obtained from the simulations of Euclidean
gravity. We also explain why the Lorentzian version of dynamical triangulations
is a promising candidate for a non-perturbative theory of quantum gravity.Comment: 69 pages, 16 figures, references adde
Monte Carlo simulations of 4d simplicial quantum gravity
Dynamical triangulations of four-dimensional Euclidean quantum gravity give
rise to an interesting, numerically accessible model of quantum gravity. We
give a simple introduction to the model and discuss two particularly important
issues. One is that contrary to recent claims there is strong analytical and
numerical evidence for the existence of an exponential bound that makes the
partition function well-defined. The other is that there may be an ambiguity in
the choice of the measure of the discrete model which could even lead to the
existence of different universality classes.Comment: 16 pages, LaTeX, epsf, 4 uuencoded figures; contribution to the JMP
special issue on "Quantum Geometry and Diffeomorphism-Invariant Quantum Field
Theory
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