Integral representation of the star product in Loop Quantum Cosmology
Year: 2020
Type: preprint
Abstract: Guided by recent developments towards the implementation of the techniques of deformation quantization within the Loop Quantum Cosmology (LQC) formalism, in this paper we address the introduction of the integral representation of the star product for LQC. To this end, we consider the Weyl quantization map for cylindrical functions defined on the Bohr compactification of the reals. The integral representation contains all of the common properties that characterize a star product which, in the case under study here, stands for a deformation of the usual pointwise product of cylindrical functions. We also invoke a direct comparison with the integral representation of the Moyal product which may be reproduced from our formulation by judiciously substituting the appropriate characters that identify such representation. Further, we introduce a suitable star commutator that correctly reproduces both the quantum representation of the holonomy-flux algebra for LQC and, in the proper limit, the holonomy-flux classical Poisson algebra emerging in the cosmological setup. Finally, we propose a natural way to obtain the quantum dynamical evolution in LQC in terms of this star commutator for cylindrical functions. We expect that our findings may contribute to a better understanding of certain issues arising within the LQC program.
Source: arXiv (Cornell University)
Authors: Jasel Berra–Montiel, Alberto Molgado
Institution: Autonomous University of San Luis Potosí
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3093313186