625 research outputs found
Lattice knot theory and quantum gravity in the loop representation
We present an implementation of the loop representation of quantum gravity on
a square lattice. Instead of starting from a classical lattice theory,
quantizing and introducing loops, we proceed backwards, setting up constraints
in the lattice loop representation and showing that they have appropriate
(singular) continuum limits and algebras. The diffeomorphism constraint
reproduces the classical algebra in the continuum and has as solutions lattice
analogues of usual knot invariants. We discuss some of the invariants stemming
from Chern--Simons theory in the lattice context, including the issue of
framing. We also present a regularization of the Hamiltonian constraint. We
show that two knot invariants from Chern--Simons theory are annihilated by the
Hamiltonian constraint through the use of their skein relations, including
intersections. We also discuss the issue of intersections with kinks. This
paper is the first step towards setting up the loop representation in a
rigorous, computable setting.Comment: 23 pages, RevTeX, 14 figures included with psfi
Semi-classical limit and minimum decoherence in the Conditional Probability Interpretation of Quantum Mechanics
The Conditional Probability Interpretation of Quantum Mechanics replaces the
abstract notion of time used in standard Quantum Mechanics by the time that can
be read off from a physical clock. The use of physical clocks leads to apparent
non-unitary and decoherence. Here we show that a close approximation to
standard Quantum Mechanics can be recovered from conditional Quantum Mechanics
for semi-classical clocks, and we use these clocks to compute the minimum
decoherence predicted by the Conditional Probability Interpretation.Comment: 8 pages, references adde
Classical and quantum general relativity: a new paradigm
We argue that recent developments in discretizations of classical and quantum
gravity imply a new paradigm for doing research in these areas. The paradigm
consists in discretizing the theory in such a way that the resulting discrete
theory has no constraints. This solves many of the hard conceptual problems of
quantum gravity. It also appears as a useful tool in some numerical simulations
of interest in classical relativity. We outline some of the salient aspects and
results of this new framework.Comment: 8 pages, one figure, fifth prize of the Gravity Research Foundation
2005 essay competitio
Canonical quantum gravity in the Vassiliev invariants arena: II. Constraints, habitats and consistency of the constraint algebra
In a companion paper we introduced a kinematical arena for the discussion of
the constraints of canonical quantum gravity in the spin network representation
based on Vassiliev invariants. In this paper we introduce the Hamiltonian
constraint, extend the space of states to non-diffeomorphism invariant
``habitats'' and check that the off-shell quantum constraint commutator algebra
reproduces the classical Poisson algebra of constraints of general relativity
without anomalies. One can therefore consider the resulting set of constraints
and space of states as a consistent theory of canonical quantum gravity.Comment: 20 Pages, RevTex, many figures included with psfi
Unified model of loop quantum gravity and matter
We reconsider the unified model of gravitation and Yang--Mills interactions
proposed by Chakraborty and Peld\'an, in the light of recent formal
developments in loop quantum gravity. In particular, we show that one can
promote the Hamiltonian constraint of the unified model to a well defined
anomaly-free quantum operator using the techniques introduced by Thiemann, at
least for the Euclidean theory. The Lorentzian version of the model can be
consistently constructed, but at the moment appears to yield a correct weak
field theory only under restrictive assumptions, and its quantization appears
problematic.Comment: 4 pages, dedicated to Michael P. Ryan on the occasion of his sixtieth
birthda
Planck-Length Phenomenology
This author's recent proposal of interferometric tests of
Planck-scale-related properties of space-time is here revisited from a strictly
phenomenological viewpoint. The results announced previously are rederived
using elementary dimensional considerations. The dimensional analysis is then
extended to the other two classes of experiments (observations of neutral kaons
at particle accelerators and observations of the gamma rays we detect from
distant astrophysical sources) which have been recently considered as
opportunities to explore "foamy" properties of space-time. The emerging picture
suggests that there is an objective and intuitive way to connect the
sensitivities of these three experiments with the Planck length. While in
previous studies the emphasis was always on some quantum-gravity scenario and
the analysis was always primarily aimed at showing that the chosen scenario
would leave a trace in a certain class of doable experiments, the analysis here
reported takes as starting point the experiments and, by relating in a direct
quantitative way the sensitivities to the Planck length, provides a
model-independent description of the status of Planck-length phenomenology.Comment: Paper awarded an ``honorable mention'' in the Annual Competition of
the Gravity Research Foundation for the year 2000 (LaTex, 7 pages, no
figures
Finite, diffeomorphism invariant observables in quantum gravity
Two sets of spatially diffeomorphism invariant operators are constructed in
the loop representation formulation of quantum gravity. This is done by
coupling general relativity to an anti- symmetric tensor gauge field and using
that field to pick out sets of surfaces, with boundaries, in the spatial three
manifold. The two sets of observables then measure the areas of these surfaces
and the Wilson loops for the self-dual connection around their boundaries. The
operators that represent these observables are finite and background
independent when constructed through a proper regularization procedure.
Furthermore, the spectra of the area operators are discrete so that the
possible values that one can obtain by a measurement of the area of a physical
surface in quantum gravity are valued in a discrete set that includes integral
multiples of half the Planck area. These results make possible the construction
of a correspondence between any three geometry whose curvature is small in
Planck units and a diffeomorphism invariant state of the gravitational and
matter fields. This correspondence relies on the approximation of the classical
geometry by a piecewise flat Regge manifold, which is then put in
correspondence with a diffeomorphism invariant state of the gravity-matter
system in which the matter fields specify the faces of the triangulation and
the gravitational field is in an eigenstate of the operators that measure their
areas.Comment: Latex, no figures, 30 pages, SU-GP-93/1-
Classical Loop Actions of Gauge Theories
Since the first attempts to quantize Gauge Theories and Gravity in the loop
representation, the problem of the determination of the corresponding classical
actions has been raised. Here we propose a general procedure to determine these
actions and we explicitly apply it in the case of electromagnetism. Going to
the lattice we show that the electromagnetic action in terms of loops is
equivalent to the Wilson action, allowing to do Montecarlo calculations in a
gauge invariant way. In the continuum these actions need to be regularized and
they are the natural candidates to describe the theory in a ``confining
phase''.Comment: LaTeX 14 page
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