Lorentz Violation in Astroparticles and Gravitational Waves

Abstract

Lorentz invariance is one of the fundamental continuous symmetries of the laws of nature [...]

Particles in Loop Quantum Gravity formalism: a thermodynamical description

“…Notice that the relation between energy and momentum has a different form from the usual one, as we could naturally expect. In other words, this aspect indicates that the Lorentz symmetry is no longer preserved [54,55] in such a context. Moreover, from the above expression, we can clearly obtain two solutions; nevertheless, accomplishing a Wicklike rotation in the mass term, only one of them is in agreement with our purpose, i.e., of having a real positive definite values, which is…”

Particles in Loop Quantum Gravity Formalism: A Thermodynamical Description

“…Notice that the relation between energy and momentum has a different form from the usual one, as we could naturally expect. In other words, this aspect indicates that the Lorentz symmetry is no longer preserved [ 53,54 ] in such a context. Moreover, from the above expression, we can clearly obtain two solutions; nevertheless, accomplishing a Wick‐like rotation in the mass term, only one of them is in agreement with our purpose, that is, of having a real positive definite values, which is $$\begin{equation} {{\bf k}}= \frac{1}{4} {\left(2 E+\sqrt {{\left(2 E-2 E^2 l_{\rm P}\right)}{}^2+8 m^2}-2 E^2 l_{\rm P}\right)} \end{equation}$$where we have considered $$\alpha = 1$$, and, naturally, we can derive its infinitesimal quantity d k as follows $$\begin{equation} \mathrm{d} {{\bf k}} = \frac{1}{4} {\left(\frac{{\left(2-4 E l_{\rm P}\right)} {\left(2 E-2 E^2 l_{\rm P}\right)}}{\sqrt {{\left(2 E-2 E^2 l_{\rm P}\right)}{}^2+8 m^2}}-4 E l_{\rm P}+2\right)} \mathrm{d}E \end{equation}$$With these above expressions, we are able to perform the integration over the momenta space in order to acquire the accessible states of the system $$\begin{equation} \Omega (E) = \frac{\Gamma }{\pi ^{2}} \int ^{\infty }_{0} \mathrm{d} {{\bf k}} |{{\bf k}}|^{2} \end{equation}$$where Γ is reg...…”