Quasi-probability distributions in Loop Quantum Cosmology

Abstract

In this paper, we introduce a complete family of parametrized quasi-probability distributions in phase space and their corresponding Weyl quantization maps with the aim to generalize the recently developed Wigner-Weyl formalism within the Loop Quantum Cosmology program (LQC). In particular, we intend to define those quasi-distributions for states valued on the Bohr compactification of the real line in such a way that they are labeled by a parameter that accounts for the ordering ambiguity corresponding to non-commutative quantum operators. Hence, we notice that the projections of the parametrized quasi-probability distributions result in marginal probability densities which are invariant under any ordering prescription. We also note that, in opposition to the standard Schrödinger representation, for an arbitrary character the quasi-distributions determine a positive function independently of the ordering. Further, by judiciously implementing a parametric-ordered Weyl quantization map for LQG, we are able to recover in a simple manner the relevant cases of the standard, anti-standard, and Weyl symmetric orderings, respectively. We expect that our results may serve to analyze several fundamental aspects within the LQC program, in special those related to coherence, squeezed states, and the convergence of operators, as extensively analyzed in the quantum optics and in the quantum information frameworks.

Star product representation of coherent state path integrals

“…We claim that our developments will be helpful to analyze several inconsistencies related to regularization and continuum limits, encountered in coherent state path integration [14]. From a different perspective, it will also be relevant to implement the techniques described here to establish the star product representation of spin foam models in the loop quantum gravity and loop quantum cosmology framework, since recently, a family s-parametrized quasi-probability distributions and its relation with the Wigner-Weyl representation has been obtained [15], [16], [17]. We intend to dedicate a future publication to address this work in progress.…”

“…Determining this star product representation, by means of quasi-probability distributions, may allow us to analyze several aspects related to some inconsistences associated to regularization techniques and the continuum limit, encountered in coherent path integrals [14]. From another perspective, it will also be relevant to implement the formalism described here in order to establish the star product representation of spin foam integrals, in the context of loop quantum gravity and loop quantum cosmology, since recently, a family of quasi-probability distributions has been determined [15], [16], [17].…”

Tomography in Loop Quantum Cosmology

“…Recently, the polymer representation of quantum mechanics was analyzed from the phase space perspective, obtaining a Wigner quasi-probability function constructed by means of cylindrical functions defined on the Bohr compactification of the real line [24], [25]. Further, in [26], [27] such Wigner function, together with a well-defined star-product, is recovered as a distributional limit of the Schrödinger representation (see also [28], [29] for an analogous description from a different perspective). Nevertheless, as mentioned above, the Wigner function may acquire negative values on certain domains of the phase space.…”

Integral representation of the star product in Loop Quantum Cosmology

“…In order to shed some light on the issues mentioned above, a different approach for the LQG program has been recently proposed [10], [11], [12]. This novel approach is based on the deformation quantization picture of quantum mechanics.…”

“…Our claim is thus that such integral representation for the star product in LQC may allow us to analyze, in a more succinct manner, several aspects related to both the semiclassical limit and the analysis of quantization ambiguities commonly encountered within the LQC framework. We also hope that the proposed star product for LQC, together with the introduction of a family of quasi-probability distributions, as developed in [12], would be advantageous in order to address, from an innovative perspective, the problem associated with the dynamical evolution of quantum states in LQC.…”