Quantum Cosmology in $f(Q)$ theory

TL;DR

This paper develops a novel quantization approach for $f(Q)$ cosmology using Dirac's method, addressing second class constraints, and explores the impact of matter content on the quantum cosmological model.

Contribution

It introduces a Dirac bracket-based quantization method for $f(Q)$ cosmology with second class constraints, extending the typical Wheeler-DeWitt approach and considering matter effects.

Findings

Quantization differs from Wheeler-DeWitt due to second class constraints.

Matter content significantly alters the quantization process.

Inclusion of fractional Quantum Mechanics for perfect fluid cases.

Abstract

We use Dirac's method for the quantization of constrained systems in order to quantize a spatially flat Friedmann-Lema\^{i}tre-Robertson-Walker spacetime in the context of $f(Q)$ cosmology. When the coincident gauge is considered, the resulting minisuperspace system possesses second class constraints. This distinguishes the quantization process from the typical Wheeler-DeWitt quantization, which is applied for cosmological models where only first class constraints are present (e.g. for models in General Relativity or in $f(R)$ gravity). We introduce the Dirac brackets, find appropriate canonical coordinates and then apply the canonical quantization procedure. We perform this method both in vacuum and in the presence of matter: a minimally coupled scalar field and a perfect fluid with a linear equation of state. We demonstrate that the matter content changes significantly the quantization procedure, with the perfect fluid even requiring to put in use the theory of fractional Quantum Mechanics in which the power of the momentum in the Hamiltonian is associated with the fractal dimension of a L\'evy flight. The results of this analysis can be applied in $f(T)$ teleparallel cosmology, since $f(Q)$ and $f(T)$ theories have the same degrees of freedom and same dynamical constraints in cosmological studies.