# Relativistic dispersion relation and putative metric structure in noncommutative phase-space

## TL;DR

This paper investigates how noncommutative quantum mechanics deforms relativistic dispersion relations, leading to Lorentz invariance violation, and uses gamma-ray burst data to constrain the noncommutative parameter, also discussing a potential metric structure.

## Contribution

It introduces an extended phase-space formalism to study dispersion relation deformation and derives an upper bound on the noncommutative parameter from astrophysical data.

## Key findings

- Deformation does not alter the speed of massless particles.
- Gamma-ray burst data constrains the noncommutative parameter to $oxed{ ext{less than } 10^{-12} ext{ eV/c}}$.
- A possible metric structure for noncommutative phase-space is proposed.

## Abstract

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, $\eta$, namely $\sqrt{\eta} \lesssim 10^{-12} \,\mathrm{eV/c}$. Additionally, a putative metric structure for the noncommutative phase-space is discussed.