Noncommutativity in Effective Loop Quantum Cosmology

Abraham Espinoza-García,Luis Rey Díaz-Barrón,Efraín Torres-Lomas,Sinuhé Pérez-Payán

doi:10.1155/2019/9080218

Abraham Espinoza-García, Luis Rey Díaz-Barrón + Show 2 more

Open Access

https://doi.org/10.1155/2019/9080218

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Abstract

We construct a noncommutative extension of the Loop Quantum Cosmology effective scheme for the flat FLRW model with a free scalar field via a theta deformation. Firstly, a deformation is implemented in the configuration sector, among the holonomy variable and the matter degree of freedom. We show that this type of noncommutativity retains, to some degree, key features of the Loop Quantum Cosmology paradigm for a free field. Secondly, a deformation is implemented in the momentum sector, among the momentum associated with the holonomy variable and the momentum associated with the matter field. We show that in this latter case the scalar field energy density is the same as the one in standard Loop Quantum Cosmology.

Highlights

The idea of noncommutativity in spacetime is not new (1947) [1]
The present investigation aims at constructing a noncommutative effective scheme for the flat FLRW model in the presence of a free scalar field, and establishing whether such noncommutative scheme could be compatible with the Loop Quantum Cosmology (LQC) paradigm, in the sense of retaining key features of the LQC of the flat FLRW model
In earlier treatments of the FLRW with a free scalar field in LQC, the classical dynamics were studied in terms of the proper time and after the quantization of the Hamiltonian constraint the dynamics were describe by an internal time defined by the scalar field [28]

Summary

Introduction

The idea of noncommutativity in spacetime is not new (1947) [1]. It was proposed as an attempt to regularize Quantum Field Theory before the renormalization program was established. The present investigation aims at constructing a noncommutative effective scheme for the flat FLRW model in the presence of a free scalar field, and establishing whether such noncommutative scheme could be compatible with the LQC paradigm, in the sense of retaining key features of the LQC of the flat FLRW model Along this lines, our work could be related to a minisuperspace approximation of a more fundamental noncommutative construction based on the LQG approach, such as the one in [15], signaling possible restrictions on the way noncommutativity is to be incorporated in such general frameworks.

Connection Variables and Effective Dynamics

Deformations in Classical and Quantum Mechanics

Noncommutativity in Flat FLRW Loop Quantum Cosmology

Discussion and Final