45 research outputs found
Automorphism covariant representations of the holonomy-flux *-algebra
We continue an analysis of representations of cylindrical functions and
fluxes which are commonly used as elementary variables of Loop Quantum Gravity.
We consider an arbitrary principal bundle of a compact connected structure
group and following Sahlmann's ideas define a holonomy-flux *-algebra whose
elements correspond to the elementary variables. There exists a natural action
of automorphisms of the bundle on the algebra; the action generalizes the
action of analytic diffeomorphisms and gauge transformations on the algebra
considered in earlier works. We define the automorphism covariance of a
*-representation of the algebra on a Hilbert space and prove that the only
Hilbert space admitting such a representation is a direct sum of spaces L^2
given by a unique measure on the space of generalized connections. This result
is a generalization of our previous work (Class. Quantum. Grav. 20 (2003)
3543-3567, gr-qc/0302059) where we assumed that the principal bundle is
trivial, and its base manifold is R^d.Comment: 34 pages, 1 figure, LaTeX2e, minor clarifying remark
Polymer and Fock representations for a Scalar field
In loop quantum gravity, matter fields can have support only on the
`polymer-like' excitations of quantum geometry, and their algebras of
observables and Hilbert spaces of states can not refer to a classical,
background geometry. Therefore, to adequately handle the matter sector, one has
to address two issues already at the kinematic level. First, one has to
construct the appropriate background independent operator algebras and Hilbert
spaces. Second, to make contact with low energy physics, one has to relate this
`polymer description' of matter fields to the standard Fock description in
Minkowski space. While this task has been completed for gauge fields, important
gaps remained in the treatment of scalar fields. The purpose of this letter is
to fill these gaps.Comment: 13 pages, no figure
Background independent quantizations: the scalar field I
We are concerned with the issue of quantization of a scalar field in a
diffeomorphism invariant manner. We apply the method used in Loop Quantum
Gravity. It relies on the specific choice of scalar field variables referred to
as the polymer variables. The quantization, in our formulation, amounts to
introducing the `quantum' polymer *-star algebra and looking for positive
linear functionals, called states. The assumed in our paper homeomorphism
invariance allows to determine a complete class of the states. Except one, all
of them are new. In this letter we outline the main steps and conclusions, and
present the results: the GNS representations, characterization of those states
which lead to essentially self adjoint momentum operators (unbounded),
identification of the equivalence classes of the representations as well as of
the irreducible ones. The algebra and topology of the problem, the derivation,
all the technical details and more are contained in the paper-part II.Comment: 13 pages, minor corrections were made in the revised versio
Diffeomorphism covariant representations of the holonomy-flux star-algebra
Recently, Sahlmann proposed a new, algebraic point of view on the loop
quantization. He brought up the issue of a star-algebra underlying that
framework, studied the algebra consisting of the fluxes and holonomies and
characterized its representations. We define the diffeomorphism covariance of a
representation of the Sahlmann algebra and study the diffeomorphism covariant
representations. We prove they are all given by Sahlmann's decomposition into
the cyclic representations of the sub-algebra of the holonomies by using a
single state only. The state corresponds to the natural measure defined on the
space of the generalized connections. This result is a generalization of
Sahlmann's result concerning the U(1) case.Comment: 37 pages, no figures, LaTeX2e, to be published in Class. Quant. Grav;
typos corrected, minor clarifying remark
Hilbert space built over connections with a non-compact structure group
Quantization of general relativity in terms of SL(2,C)-connections (i.e. in
terms of the complex Ashtekar variables) is technically difficult because of
the non-compactness of SL(2,C). The difficulties concern the construction of a
diffeomorphism invariant Hilbert space structure on the space of cylindrical
functions of the connections. We present here a 'toy' model of such a Hilbert
space built over connections whose structure group is the group of real
numbers. We show that in the case of any Hilbert space built analogously over
connections with any non-compact structure group (this includes some models
presented in the literature) there exists an obstacle which does not allow to
define a *-representation of cylindrical functions on the Hilbert space by the
multiplication map which is the only known way to define a diffeomorphism
invariant representation of the functions.Comment: 45 pages, no figures, LaTeX2e, the discussion of results extende
Background independent quantizations: the scalar field II
We are concerned with the issue of quantization of a scalar field in a
diffeomorphism invariant manner. We apply the method used in Loop Quantum
Gravity. It relies on the specific choice of scalar field variables referred to
as the polymer variables. The quantization, in our formulation, amounts to
introducing the `quantum' polymer *-star algebra and looking for positive
linear functionals, called states. Assumed in our paper homeomorphism
invariance allows to derive the complete class of the states. They are
determined by the homeomorphism invariant states defined on the CW-complex
*-algebra. The corresponding GNS representations of the polymer *-algebra and
their self-adjoint extensions are derived, the equivalence classes are found
and invariant subspaces characterized. In the preceding letter (the part I) we
outlined those results. Here, we present the technical details.Comment: 51 pages, LaTeX, no figures, revised versio
Relation between polymer and Fock excitations
To bridge the gap between background independent, non-perturbative quantum
gravity and low energy physics described by perturbative field theory in
Minkowski space-time, Minkowskian Fock states are located, analyzed and used in
the background independent framework. This approach to the analysis of
semi-classical issues is motivated by recent results of Varadarajan. As in that
work, we use the simpler U(1) example to illustrate our constructions but, in
contrast to that work, formulate the theory in such a way that it can be
extended to full general relativity.Comment: Clarifying remarks and three references added. To appear in CQ
Background Independent Quantum Gravity: A Status Report
The goal of this article is to present an introduction to loop quantum
gravity -a background independent, non-perturbative approach to the problem of
unification of general relativity and quantum physics, based on a quantum
theory of geometry. Our presentation is pedagogical. Thus, in addition to
providing a bird's eye view of the present status of the subject, the article
should also serve as a vehicle to enter the field and explore it in detail. To
aid non-experts, very little is assumed beyond elements of general relativity,
gauge theories and quantum field theory. While the article is essentially
self-contained, the emphasis is on communicating the underlying ideas and the
significance of results rather than on presenting systematic derivations and
detailed proofs. (These can be found in the listed references.) The subject can
be approached in different ways. We have chosen one which is deeply rooted in
well established physics and also has sufficient mathematical precision to
ensure that there are no hidden infinities. In order to keep the article to a
reasonable size, and to avoid overwhelming non-experts, we have had to leave
out several interesting topics, results and viewpoints; this is meant to be an
introduction to the subject rather than an exhaustive review of it.Comment: 125 pages, 5 figures (eps format), the final version published in CQ
Representations of the Weyl Algebra in Quantum Geometry
The Weyl algebra A of continuous functions and exponentiated fluxes,
introduced by Ashtekar, Lewandowski and others, in quantum geometry is studied.
It is shown that, in the piecewise analytic category, every regular
representation of A having a cyclic and diffeomorphism invariant vector, is
already unitarily equivalent to the fundamental representation. Additional
assumptions concern the dimension of the underlying analytic manifold (at least
three), the finite wide triangulizability of surfaces in it to be used for the
fluxes and the naturality of the action of diffeomorphisms -- but neither any
domain properties of the represented Weyl operators nor the requirement that
the diffeomorphisms act by pull-backs. For this, the general behaviour of
C*-algebras generated by continuous functions and pull-backs of homeomorphisms,
as well as the properties of stratified analytic diffeomorphisms are studied.
Additionally, the paper includes also a short and direct proof of the
irreducibility of A.Comment: 71 pages, 1 figure, LaTeX. Changes v2 to v3: previous results
unchanged; some addings: inclusion of gauge transforms, several comments,
Subsects. 1.5, 3.7, 3.8; comparison with LOST paper moved to Introduction;
Def. 2.5 modified; some typos corrected; Refs. updated. Article now as
accepted by Commun. Math. Phy
Entropy calculation for a toy black hole
In this note we carry out the counting of states for a black hole in loop
quantum gravity, however assuming an equidistant area spectrum. We find that
this toy-model is exactly solvable, and we show that its behavior is very
similar to that of the correct model. Thus this toy-model can be used as a nice
and simplifying `laboratory' for questions about the full theory.Comment: 18 pages, 4 figures. v2: Corrected mistake in bibliography, added
appendix with further result