Quantum noncomutativity in quantum cosmology

TL;DR

This paper explores a noncommutative quantum cosmology model with a Friedmann-Robertson-Walker universe, finding that the noncommutative parameter affects the energy spectrum but no suitable wavefunction satisfying boundary conditions exists.

Contribution

It introduces a noncommutative extension of a quantum cosmology model and analyzes its quantization, spectrum, and boundary condition issues.

Findings

Discrete energy spectrum depends on noncommutative parameter

Wavefunctions do not satisfy boundary conditions in the noncommutative model

Noncommutativity influences the eigenfunctions and energies

Abstract

In the present work, we study the noncommutative version of a quantum cosmology model. The model has a Friedmann-Robertson-Walker geometry, the matter content is a radiative perfect fluid and the spatial sections have positive constant curvatures. We work in the Schutz's variational formalism. We quantize the model and obtain the appropriate Wheeler-DeWitt equation. In this model the states are bounded. Therefore, we compute the discrete energy spectrum and the corresponding eigenfunctions. The energies depend on a noncommutative parameter ($\theta$). The solutions to the Wheeler-DeWitt equation are function of the scale factor ($a$) and a time variable ($\tau$), associated to the fluid. They also depend on an integer ($n$) and $\theta$. The most general solution ($\Psi(a,\tau)$) to the Wheeler-DeWitt equation is a sum, in the integer $n$, of the solutions mentioned above. We observe that, there is no $\Psi(a,\tau)$ satisfying the appropriate boundary conditions. Therefore, we conclude that it is not possible to obtain a wavefunction satisfying the appropriate boundary conditions for the present model with the considered noncommutativity.