Quasi-probability distributions in Loop Quantum Cosmology

TL;DR

This paper develops a family of quasi-probability distributions in phase space tailored for Loop Quantum Cosmology, enabling new analysis tools for quantum states and operator orderings within the LQC framework.

Contribution

It introduces a parametrized family of quasi-probability distributions and Weyl quantization maps for LQC, generalizing the Wigner-Weyl formalism and addressing operator ordering ambiguities.

Findings

Quasi-distributions yield invariant marginal probability densities.

For arbitrary characters, quasi-distributions are positive regardless of ordering.

The framework recovers standard, anti-standard, and Weyl orderings in LQC.

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 Schr\"odinger 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.