Functorial Aspects of the Space of Generalized Connections

We give a description of the category structure of the space of generalized connections, an extension of the space of connections that plays a central role in loop quantum gravity.Comment: 7 pages. To appear in Proceedings of the Lusofona Workshop on Quantum Gravity and Noncommutative Geometry, Lisbon, July 200

Physical Properties of Quantum Field Theory Measures

Well known methods of measure theory on infinite dimensional spaces are used to study physical properties of measures relevant to quantum field theory. The difference of typical configurations of free massive scalar field theories with different masses is studied. We apply the same methods to study the Ashtekar-Lewandowski (AL) measure on spaces of connections. We prove that the diffeomorphism group acts ergodically, with respect to the AL measure, on the Ashtekar-Isham space of quantum connections modulo gauge transformations. We also prove that a typical, with respect to the AL measure, quantum connection restricted to a (piecewise analytic) curve leads to a parallel transport discontinuous at every point of the curve.Comment: 24 pages, LaTeX, added proof for section 4.2, added reference

Comments on the kinematical structure of loop quantum cosmology

We comment on the presence of spurious observables and on a subtle violation of irreducibility in loop quantum cosmology.Comment: 7 page

The Quantum Configuration Space of Loop Quantum Cosmology

The article gives an account of several aspects of the space known as the Bohr compactification of the line, featuring as the quantum configuration space in loop quantum cosmology, as well as of the corresponding configuration space realization of the so-called polymer representation. Analogies with loop quantum gravity are explored, providing an introduction to (part of) the mathematical structure of loop quantum gravity, in a technically simpler context.Comment: 14 pages. Minor changes, typos corrected, 1 reference added. To appear in Class. Quantum Gra

On the structure of the space of generalized connections

We give a modern account of the construction and structure of the space of generalized connections, an extension of the space of connections that plays a central role in loop quantum gravity.Comment: 30 pages, added references, minor changes. To appear in International Journal of Geometric Methods in Modern Physic

Uniqueness of the Fock quantization of fields with unitary dynamics in nonstationary spacetimes

The Fock quantization of fields propagating in cosmological spacetimes is not uniquely determined because of several reasons. Apart from the ambiguity in the choice of the quantum representation of the canonical commutation relations, there also exists certain freedom in the choice of field: one can scale it arbitrarily absorbing background functions, which are spatially homogeneous but depend on time. Each nontrivial scaling turns out into a different dynamics and, in general, into an inequivalent quantum field theory. In this work we analyze this freedom at the quantum level for a scalar field in a nonstationary, homogeneous spacetime whose spatial sections have $S^3$ topology. A scaling of the configuration variable is introduced as part of a linear, time dependent canonical transformation in phase space. In this context, we prove in full detail a uniqueness result about the Fock quantization requiring that the dynamics be unitary and the spatial symmetries of the field equations have a natural unitary implementation. The main conclusion is that, with those requirements, only one particular canonical transformation is allowed, and thus only one choice of field-momentum pair (up to irrelevant constant scalings). This complements another previous uniqueness result for scalar fields with a time varying mass on $S^3$, which selects a specific equivalence class of Fock representations of the canonical commutation relations under the conditions of a unitary evolution and the invariance of the vacuum under the background symmetries. In total, the combination of these two different statements of uniqueness picks up a unique Fock quantization for the system. We also extend our proof of uniqueness to other compact topologies and spacetime dimensions.Comment: 12 page

Hamiltonian and physical Hilbert space in polymer quantum mechanics

In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity, is considered and shown to be equivalent, in a precise sense, to the standard, experimentally tested, Schroedinger quantum mechanics. The kinematical cornerstone of our framework is the so called polymer representation of the Heisenberg-Weyl (H-W) algebra, which is the starting point of the construction. The dynamics is constructed as a continuum limit of effective theories characterized by a scale, and requires a renormalization of the inner product. The result is a physical Hilbert space in which the continuum Hamiltonian can be represented and that is unitarily equivalent to the Schroedinger representation of quantum mechanics. As a concrete implementation of our formalism, the simple harmonic oscillator is fully developed.Comment: 19 pages, 2 figures. Comments and references added. Version to be published in CQ

Asymptotic Properties of Difference Equations for Isotropic Loop Quantum Cosmology

In loop quantum cosmology, a difference equation for the wave function describes the evolution of a universe model. This is different from the differential equations that arise in Wheeler-DeWitt quantizations, and some aspects of general properties of solutions can appear differently. Properties of particular interest are boundedness and the presence of small-scale oscillations. Continued fraction techniques are used to show in different matter models the presence of special initial conditions leading to bounded solutions, and an explicit expression for these initial values is derived.Comment: 27 pages, 2 figure

Unique Fock quantization of scalar cosmological perturbations

We investigate the ambiguities in the Fock quantization of the scalar perturbations of a Friedmann-Lema\^{i}tre-Robertson-Walker model with a massive scalar field as matter content. We consider the case of compact spatial sections (thus avoiding infrared divergences), with the topology of a three-sphere. After expanding the perturbations in series of eigenfunctions of the Laplace-Beltrami operator, the Hamiltonian of the system is written up to quadratic order in them. We fix the gauge of the local degrees of freedom in two different ways, reaching in both cases the same qualitative results. A canonical transformation, which includes the scaling of the matter field perturbations by the scale factor of the geometry, is performed in order to arrive at a convenient formulation of the system. We then study the quantization of these perturbations in the classical background determined by the homogeneous variables. Based on previous work, we introduce a Fock representation for the perturbations in which: (a) the complex structure is invariant under the isometries of the spatial sections and (b) the field dynamics is implemented as a unitary operator. These two properties select not only a unique unitary equivalence class of representations, but also a preferred field description, picking up a canonical pair of field variables among all those that can be obtained by means of a time-dependent scaling of the matter field (completed into a linear canonical transformation). Finally, we present an equivalent quantization constructed in terms of gauge-invariant quantities. We prove that this quantization can be attained by a mode-by-mode time-dependent linear canonical transformation which admits a unitary implementation, so that it is also uniquely determined.Comment: 19 pages, minor impovementes included, typos correcte