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The main goal of this work is to explore the symmetries and develop the dynamics associated with a 3D Abelian

The now quasi-hundred-year-old General Relativity as a theory of gravitation, despite its tremendous successes in accounting for predicting phenomena, still lacks a quantum version. Previous perturbative attempts have shown the nonrenormalizability of the theory [

By contrast, the lower-dimensional gravitation theories are much more easy to handle, since they can be described as topological gauge theories, when not coupled to matter [

The purpose of this paper is to present the loop quantization of a background independent theory of the

The model and its gauge invariances are presented in Section

The field content of the model is a

The most general action, invariant under the gauge transformations (

The parameter

It turns out that this action (

In order to check the invariances of the action (up to boundary terms), as well as for all the manipulations involving partial integrations, it is useful to remember that the covariant derivative

In the present theory, like in the topological theories of the Chern-Simons or

We apply here the canonical formalism of Dirac [

Noting that the velocities do not appear in any of (

We are left with the five constraints

Since this Hamiltonian is entirely made of constraints—a characteristic of theories with general covariance—the stability of our five constraints

The constraints

In order to construct a scalar product defined by an appropriate integration measure in configuration space, we first restrict the space of wave functionals to the set of functions of finite numbers of holonomies of the connection

With this scalar product in hands we dispose of a norm so one can define a Hilbert space

An orthonormal basis of

Let us now turn to the constraint

Closed graph

The vectors of

The last constraint to be imposed is the curvature constraint

The condition of null curvature means that,

Let us begin with the case where the topology of

The next case is that with the topology of

Two charge network graphs with the singular point

Applying the charge conservation condition as in the previous case shows that any charge network graph with the point

The generalization to a plane with

One remarks that diffeomorphism invariance, which in the classical theory is a consequence of its gauge invariances, is explicit in the quantum theory constructed here, once all constraints are fulfilled. Note that the states of the (nonseparable) kinematical Hilbert space, which still do not obey the curvature constraint

It follows from the above discussion that no nontrivial observables do exist in the case of a trivial topology such as that of

What we have shown, using the Dirac canonical scheme together with the LQG quantization procedure, is that the three-dimensional Abelian

Two main conclusions can be drawn. First, the model we have presented is a simple example of how restrictive is the assumption of background invariance. It eliminates from the action an infinity of terms which otherwise would be present if one only postulates

The second main conclusion is that we have succeeded to implement the loop quantization scheme up to the construction of the physical Hilbert space and its observables. The implementation has turned out to be rather simple, in contrast to the difficulties which one encounters in 4-dimensional gravity [

Similar achievements for 3-dimensional theories, non-Abelian but without coupling with matter, may be found in [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Brazil, for Diego C. M. Mendonça and in part by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil, for Olivier Piguet.