1,082 research outputs found
Consistent discretization and loop quantum geometry
We apply the ``consistent discretization'' approach to general relativity
leaving the spatial slices continuous. The resulting theory is free of the
diffeomorphism and Hamiltonian constraints, but one can impose the
diffeomorphism constraint to reduce its space of solutions and the constraint
is preserved exactly under the discrete evolution. One ends up with a theory
that has as physical space what is usually considered the kinematical space of
loop quantum geometry, given by diffeomorphism invariant spin networks endowed
with appropriate rigorously defined diffeomorphism invariant measures and inner
products. The dynamics can be implemented as a unitary transformation and the
problem of time explicitly solved or at least reduced to as a numerical
problem. We exhibit the technique explicitly in 2+1 dimensional gravity.Comment: 4 pages, Revtex, no figure
Dirac-like approach for consistent discretizations of classical constrained theories
We analyze the canonical treatment of classical constrained mechanical
systems formulated with a discrete time. We prove that under very general
conditions, it is possible to introduce nonsingular canonical transformations
that preserve the constraint surface and the Poisson or Dirac bracket
structure. The conditions for the preservation of the constraints are more
stringent than in the continuous case and as a consequence some of the
continuum constraints become second class upon discretization and need to be
solved by fixing their associated Lagrange multipliers. The gauge invariance of
the discrete theory is encoded in a set of arbitrary functions that appear in
the generating function of the evolution equations. The resulting scheme is
general enough to accommodate the treatment of field theories on the lattice.
This paper attempts to clarify and put on sounder footing a discretization
technique that has already been used to treat a variety of systems, including
Yang--Mills theories, BF-theory and general relativity on the lattice.Comment: 11 pages, RevTe
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
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-
Realistic clocks, universal decoherence and the black hole information paradox
Ordinary quantum mechanics is formulated on the basis of the existence of an
ideal classical clock external to the system under study. This is clearly an
idealization. As emphasized originally by Salecker and Wigner and more recently
by other authors, there exist limits in nature to how ``classical'' even the
best possible clock can be. When one introduces realistic clocks, quantum
mechanics ceases to be unitary and a fundamental mechanism of decoherence of
quantum states arises. We estimate the rate of universal loss of unitarity
using optimal realistic clocks. In particular we observe that the rate is rapid
enough to eliminate the black hole information puzzle: all information is lost
through the fundamental decoherence before the black hole can evaporate. This
improves on a previous calculation we presented with a sub-optimal clock in
which only part of the information was lost by the time of evaporation.Comment: 3 Pages, RevTex, no figure
A Geometric Representation for the Proca Model
The Proca model is quantized in an open-path dependent representation that
generalizes the Loop Representation of gauge theories. The starting point is a
gauge invariant Lagrangian that reduces to the Proca Lagrangian when certain
gauge is selected.Comment: 10 pages, Late
The Lagrangian Loop Representation of Lattice U(1) Gauge Theory
It is showed how the Hamiltonian lattice can be cast
straightforwardly in the Lagrangian formalism. The procedure is general and
here we present the simplest case: pure compact QED. This connection has been
shaded by the non canonical character of the algebra of the fundamental loop
operators. The loops represent tubes of electric flux and can be considered the
dual objects to the Nielsen-Olesen strings supported by the Higgs broken phase.
The lattice loop classical action corresponding to the Villain form is
proportional to the quadratic area of the loop world sheets and thus it is
similar to the Nambu string action. This loop action is used in a Monte Carlo
simulation and its appealing features are discussed.Comment: 13 pp, UAB-FT-341/9
Interacting Particles and Strings in Path and Surface Representations
Non-relativistic charged particles and strings coupled with abelian gauge
fields are quantized in a geometric representation that generalizes the Loop
Representation. We consider three models: the string in self-interaction
through a Kalb-Ramond field in four dimensions, the topological interaction of
two particles due to a BF term in 2+1 dimensions, and the string-particle
interaction mediated by a BF term in 3+1 dimensions. In the first case one
finds that a consistent "surface-representation" can be built provided that the
coupling constant is quantized. The geometrical setting that arises corresponds
to a generalized version of the Faraday's lines picture: quantum states are
labeled by the shape of the string, from which emanate "Faraday`s surfaces". In
the other models, the topological interaction can also be described by
geometrical means. It is shown that the open-path (or open-surface) dependence
carried by the wave functional in these models can be eliminated through an
unitary transformation, except by a remaining dependence on the boundary of the
path (or surface). These feature is closely related to the presence of
anomalous statistics in the 2+1 model, and to a generalized "anyonic behavior"
of the string in the other case.Comment: RevTeX 4, 28 page
Matter Fields in the Lagrangian Loop Representation: Scalar QED
We present the extension of the Lagrangian loop gauge invariant
representation in such a way to include matter fields. The partition function
of lattice compact U(1)-Higgs model is expressed as a sum over closed as much
as open surfaces. We have simulated numerically the loop action equivalent to
the Villain form of the action and mapped out the beta-gamma phase diagram of
this model.Comment: 10 pages, LaTe
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