# A Poincar\'e Covariant Noncommutative Spacetime?

## TL;DR

This paper introduces a Poincaré covariant noncommutative spacetime derived from a relativistic quantum field theory operator, combining Snyder and Moyal-Weyl noncommutativity, and addresses the soccer-ball problem.

## Contribution

It proposes a new noncommutative spacetime model that is Poincaré invariant and solves the soccer-ball problem using Rieffel deformation techniques.

## Key findings

- The operator generates a Snyder-like noncommutative spacetime with a minimal length.
- The model is Poincaré invariant and covariant under the entire Poincaré group.
- It combines Snyder and Moyal-Weyl noncommutativity types.

## Abstract

We interpret, in the realm of relativistic quantum field theory, the tangential operator given by Coleman, Mandula as an appropriate coordinate operator. The investigation shows that the operator generates a Snyder-like noncommutative spacetime with a minimal length that is given by the mass. By using this operator to define a noncommutative spacetime, we obtain a Poincar\'e invariant noncommutative spacetime and in addition solve the soccer-ball problem. Moreover, from recent progress in deformation theory we extract the idea how to obtain, in a physical and mathematical well-defined manner, an emerging noncommutative spacetime. This is done by a strict deformation quantization known as Rieffel deformation (or warped convolutions). The result is a noncommutative spacetime combining a Snyder and a Moyal-Weyl type of noncommutativity that in addition behaves covariant under transformations of the \textbf{whole} Poincar\'e group.