In this work some issues in the context of Noncommutative Quantum Mechanics (NCQM) are addressed. The main focus is on finding whether symmetries present in Quantum Mechanics still hold in the phase-space noncommutative version. In particular, the issues related with gauge invariance of the electromagnetic field and the weak equivalence principle (WEP) in the context of the gravitational quantum well (GQW) are considered. The question of the Lorentz symmetry and the associated dispersion relation is also examined. Constraints are set on the relevant noncommutative parameters so that gauge invariance and Lorentz invariance holds. In opposition, the WEP is verified to hold in the noncommutative set up, and it is only possible to observe a violation through an anisotropy of the noncommutative parameters.

A quasi-Gaussian quantum superposition of Hořava-Lifshitz (HL) stationary states is built in order to describe the transition of the quantum cosmological problem to the related classical dynamics. The obtained HL phase-space superposed Wigner function and its associated Wigner currents describe the conditions for the matching between classical and quantum phase-space trajectories. The matching quantum superposition parameter is associated to the total energy of the classical trajectory which, at the same time, drives the engendered Wigner function to the classical stationary regime. Through the analysis of the Wigner flows, the quantum fluctuations that distort the classical regime can be quantified as a measure of (non)classicality. Finally, the modifications to the Wigner currents due to the inclusion of perturbative potentials are computed in the HL quantum cosmological context. In particular, the inclusion of a cosmological constant provides complementary information that allows for connecting the age of the Universe with the overall stiff matter density profile.

Astrophysical tests of the stability of dimensionless fundamental couplings, such as the fine-structure constant $\alpha$, are an area of much increased recent activity, following some indications of possible spacetime variations at the few parts per million level. Here we obtain updated constraints on the Bekenstein-Sandvik-Barrow-Magueijo model, which is arguably the simplest model allowing for $\alpha$ variations. Recent accurate spectroscopic measurements allow us to improve previous constraints by about an order of magnitude. We briefly comment on the dependence of the results on the data sample, as well as on the improvements expected from future facilities.Comment: 5 pages, 2 figure

The deformation of the relativistic dispersion relation caused by noncommutative (NC) Quantum Mechanics (QM) is studied using the extended phase-space formalism. The introduction of the additional commutation relations induces Lorentz invariance violation. It is shown that this deformation does not affect the propagation speed of free massless particles. From the deformation of the dispersion relation for massless particles, gamma ray burst data is used to establish an upper bound on the noncommutative parameter, η, namely √ η 10 −12 eV/c. Additionally, a putative metric structure for the noncommutative phase-space is discussed.