# Twisted statistics and the structure of Lie-deformed Minkowski spaces

## TL;DR

This paper develops a covariant framework for noncommutative Minkowski spaces with Lie algebra structures, deriving star products, coproducts, and twist operators to analyze particle statistics at the Planck scale.

## Contribution

It introduces a general covariant approach to Lie-deformed Minkowski spaces, including star products, coproducts, and twist operators, applicable to Planck-scale physics.

## Key findings

- Derived star product and coproduct for Lie algebra deformations.
- Constructed twist, flip operator, and R-matrix for particle statistics.
- Identified a special Lie algebra case used in experimental bounds.

## Abstract

We show that the realizations of noncommutative coordinates that are linear in the Lorentz generators form a closed Lie algebra under certain conditions. The star product and the coproduct for the momentum generators are obtained for these Lie algebras and the corresponding twist satisfies the cocycle and normalization conditions. We also obtain the twisted flip operator and the $\mathcal R$-matrix that define the statistics of particles or quantum fields propagating in these noncommutative spacetimes. The Lie algebra obtained in this work contains a special case which has been used in the literature to put bounds on noncommutative parameters from the experimental limits on Pauli forbidden transitions. The general covariant framework presented here is suitable for analyzing the properties of particles or quantum fields at the Planck scale.