28 research outputs found
Quantum group connections
The Ahtekar-Isham C*-algebra known from Loop Quantum Gravity is the algebra
of continuous functions on the space of (generalized) connections with a
compact structure Lie group. The algebra can be constructed by some inductive
techniques from the C*-algebra of continuous functions on the group and a
family of graphs embedded in the manifold underlying the connections. We
generalize the latter construction replacing the commutative C*-algebra of
continuous functions on the group by a non-commutative C*-algebra defining a
compact quantum group.Comment: 40 pages, LaTeX2e, minor mistakes corrected, abstract slightly
change
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
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
Quantization of diffeomorphism invariant theories of connections with a non-compact structure group - an example
A simple diffeomorphism invariant theory of connections with the non-compact
structure group R of real numbers is quantized. The theory is defined on a
four-dimensional 'space-time' by an action resembling closely the self-dual
Plebanski action for general relativity. The space of quantum states is
constructed by means of projective techniques by Kijowski. Except this point
the applied quantization procedure is based on Loop Quantum Gravity methods.Comment: 59 pages, no figures, LaTeX2e, this is a shortened version published
in Comm. Math. Phy
ADM-like Hamiltonian formulation of gravity in the teleparallel geometry
We present a new Hamiltonian formulation of the Teleparallel Equivalent of
General Relativity (TEGR) meant to serve as the departure point for canonical
quantization of the theory. TEGR is considered here as a theory of a cotetrad
field on a spacetime. The Hamiltonian formulation is derived by means of an
ADM-like 3+1 decomposition of the field and without any gauge fixing. A
complete set of constraints on the phase space and their algebra are presented.
The formulation is described in terms of differential forms.Comment: 43 pages, LaTeX2e; the original 73 page paper arXiv:1111.5498v1 was
revised and divided into two parts. The present paper is the first part of
the original one (the second part is available as arXiv:1309.4685
Uniqueness of diffeomorphism invariant states on holonomy-flux algebras
Loop quantum gravity is an approach to quantum gravity that starts from the
Hamiltonian formulation in terms of a connection and its canonical conjugate.
Quantization proceeds in the spirit of Dirac: First one defines an algebra of
basic kinematical observables and represents it through operators on a suitable
Hilbert space. In a second step, one implements the constraints. The main
result of the paper concerns the representation theory of the kinematical
algebra: We show that there is only one cyclic representation invariant under
spatial diffeomorphisms.
While this result is particularly important for loop quantum gravity, we are
rather general: The precise definition of the abstract *-algebra of the basic
kinematical observables we give could be used for any theory in which the
configuration variable is a connection with a compact structure group. The
variables are constructed from the holonomy map and from the fluxes of the
momentum conjugate to the connection. The uniqueness result is relevant for any
such theory invariant under spatial diffeomorphisms or being a part of a
diffeomorphism invariant theory.Comment: 38 pages, one figure. v2: Minor changes, final version, as published
in CM
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
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
Spherically Symmetric Quantum Geometry: Hamiltonian Constraint
Variables adapted to the quantum dynamics of spherically symmetric models are
introduced, which further simplify the spherically symmetric volume operator
and allow an explicit computation of all matrix elements of the Euclidean and
Lorentzian Hamiltonian constraints. The construction fits completely into the
general scheme available in loop quantum gravity for the quantization of the
full theory as well as symmetric models. This then presents a further
consistency check of the whole scheme in inhomogeneous situations, lending
further credence to the physical results obtained so far mainly in homogeneous
models. New applications in particular of the spherically symmetric model in
the context of black hole physics are discussed.Comment: 33 page