José M. Velhinho

Hybrid quantization of an inflationary universe

We quantize to completion an inflationary universe with small inhomogeneities in the framework of loop quantum cosmology. The homogeneous setting consists of a massive scalar field propagating in a closed, homogeneous scenario. We provide a complete quantum description of the system employing loop quantization techniques. After introducing small inhomogeneities as scalar perturbations, we identify the true physical degrees of freedom by means of a partial gauge fixing, removing all the local degrees of freedom except the matter perturbations. We finally combine a Fock description for the inhomogeneities with the polymeric quantization of the homogeneous background, providing the quantum Hamiltonian constraint of the composed system. Its solutions are then completely characterized, owing to the suitable choice of quantum constraint, and the physical Hilbert space is constructed. Finally, we consider the analog description for an alternate gauge and, moreover, in terms of gauge-invariant quantities. In the deparametrized model, all these descriptions are unitarily equivalent at the quantum level.

A groupoid approach to spaces of generalized connections

Abstract The quantum completion A of the space of connections in a manifold can be seen as the set of all morphisms from the groupoid of the edges of the manifold to the (compact) gauge group. This algebraic construction generalizes an analogous description of the gauge-invariant quantum configuration space A / G of Ashtekar and Isham, clarifying the relation between the two spaces. We present a description of the groupoid approach which brings the gauge-invariant degrees of freedom to the foreground, thus making the action of the gauge group more transparent.

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.

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. In particular 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.

Quantum Gowdy T3 model: a uniqueness result

Modulo a homogeneous degree of freedom and a global constraint, the linearly polarized Gowdy T 3 cosmologies are equivalent to a free scalar field propagating in a fixed nonstationary background. Recently, a new field parametrization was proposed for the metric of the Gowdy spacetimes such that the associatedscalarfieldevolves in a flat background in (1+1) dimensions with the spatial topology of S 1 , although subject to a time-dependent potential. Introducing a suitable Fock quantization for this scalar field, a quantum theory was constructed for the Gowdy model in which the dynamics is implemented as a unitary transformation. A question that was left open is whether one might adopt a different, nonequivalent Fock representation by selecting a distinct complex structure. The present work proves that the chosen Fock quantization is in fact unique (up to unitary equivalence) if one demands unitary implementation of the dynamics and invariance under the group of S translations. These translations are precisely those generated by the global constraint that remains on the Gowdy model. It is also shown that the proof of uniqueness in the choice of the complex structure can be applied to more general field dynamics than that corresponding to the Gowdy cosmologies.

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.

Unique Fock quantization of scalar cosmological perturbations

This work was supported by the research grants Nos. MICINN/MINECO FIS2011-30145-C03-02, MICINN FIS2008- 06078-C03-03, and CPAN CSD2007-00042 from Spain, and CERN/FP/116373/2010 from Portugal. J.O. acknowledges CSIC for financial support under the grant No. JAE-Pre 08 00791, and M.F.-M. acknowledges CSIC and the European Social Fund for support under the grant No. JAEPre 2010 01544.

A uniqueness criterion for the Fock quantization of scalar fields with time-dependent mass

A major problem in the quantization of fields in curved spacetimes is the ambiguity in the choice of a Fock representation for the canonical commutation relations. There exists infinite number of choices leading to different physical predictions. In stationary scenarios, a common strategy is to select a vacuum (or a family of unitarily equivalent vacua) by requiring invariance under the spacetime symmetries. When stationarity is lost, a natural generalization consists in replacing time invariance by unitarity in the evolution. We prove that when the spatial sections are compact, the criterion of a unitary dynamics, together with the invariance under the spatial isometries, suffices to select a unique family of Fock quantizations for a scalar field with time-dependent mass.

Invariance Properties of Induced Fock Measures¶for U(1) Holonomies

Abstract: We study invariance properties of the measures in the space of generalized U(1) connections associated to Varadarajans r-Fock representations.

Uniqueness of the Fock quantization of scalar fields in spatially flat cosmological spacetimes

We study the Fock quantization of scalar fields in (generically) time dependent scenarios, focusing on the case in which the field propagation occurs in –either a background or effective– spacetime with spatial sections of flat compact topology. The discussion finds important applications in cosmology, like e.g. in the description of test Klein-Gordon fields and scalar perturbations in Friedmann-Robertson-Walker spacetime in the observationally favored flat case. Two types of ambiguities in the quantization are analyzed. First, the infinite ambiguity existing in the choice of a Fock representation for the canonical commutation relations, understandable as the freedom in the choice of inequivalent vacua for a given field. Besides, in cosmological situations, it is customary to scale the fields by time dependent functions, which absorb part of the evolution arising from the spacetime, which is treated classically. This leads to an additional ambiguity, this time in the choice of a canonical pair of field variables. We show that both types of ambiguities are removed by the requirements of (a) invariance of the vacuum under the symmetries of the three-torus, and (b) unitary implementation of the dynamics in the quantum theory. In this way, one arrives at a unique class of unitarily equivalent Fock quantizations for the system. This result provides considerable robustness to the quantum predictions and renders meaningful the confrontation with observation.