A rather nonstandard quantum representation of the canonical commutation relations of quantum mechanics systems, known as the polymer representation, has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schrödinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schrödinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schrödinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle, and a simple cosmological model.

The basic framework for a systematic construction of a quantum theory of Riemannian geometry was introduced recently. The quantum versions of Riemannian structures - such as triad and area operators - exhibit a non-commutativity. At first sight, this feature is surprising because it implies that the framework does not admit a triad representation. To better understand this property and to reconcile it with intuition, we analyse its origin in detail. In particular, a careful study of the underlying phase space is made and the feature is traced back to the classical theory; there is no anomaly associated with quantization. We also indicate why the uncertainties associated with this non-commutativity become negligible in the semiclassical regime.

Quantum gauge theory in the connection representation uses functions of holonomies as configuration observables. Physical observables (gauge and diffeomorphism invariant) are represented in the Hilbert space of physical states; physical states are gauge and diffeomorphism invariant distributions on the space of functions of the holonomies of the edges of a certain family of graphs. Then a family of graphs embedded in the space manifold (satisfying certain properties) induces a representation of the algebra of physical observables. We construct a quantum model from the set of piecewise linear graphs on a piecewise linear manifold, and another manifestly combinatorial model from graphs defined on a sequence of increasingly refined simplicial complexes. Even though the two models are different at the kinematical level, they provide unitarily equivalent representations of the algebra of physical observables in separable Hilbert spaces of physical states (their s-knot basis is countable). Hence, the combinatorial framework is compatible with the usual interpretation of quantum field theory.

Topological gravity is the reduction of general relativity to flat space-times. A lattice model describing topological gravity is developed starting from a Hamiltonian lattice version of $B\w F$ theory. The extra symmetries not present in gravity that kill the local degrees of freedom in $B\wedge F$ theory are removed. The remaining symmetries preserve the geometrical character of the lattice. Using self-dual variables, the conditions that guarantee the geometricity of the lattice become reality conditions. The local part of the remaining symmetry generators, that respect the geometricity-reality conditions, has the form of Ashtekar's constraints for GR. Only after constraining the initial data to flat lattices and considering the non-local (plus local) part of the constraints does the algebra of the symmetry generators close. A strategy to extend the model for non-flat connections and quantization are discussed.

We present an implementation of Wilson's renormalization group and a continuum limit tailored for loop quantization. The dynamics of loop quantized theories is constructed as a continuum limit of dynamics of effective theories. After presenting the general formalism we show as first explicit example the 2d Ising field theory. It is an interacting relativistic quantum field theory with local degrees of freedom quantized by loop quantization techniques.