A modern guide to {\rm $\theta$}-Poincar\'e

Abstract

Motivated by the recent interest in underground experiments phenomenology, we review the main aspects of one specific non-commutative space-time model, based on the Groenewold-Moyal plane algebra, the $\theta$-Poincar\'e space-time. In the $\theta$-Poincar\'e scenario, the Lorentz co-algebra is deformed introducing a non-commutativity of space-time coordinates. In such a theory, a new quantum field theory in non-commutative space-time can be reformulated. Tackling on several conceptual misunderstanding and technical mistakes in the literature, we will focus on several issues such: $i)$ the construction of fields theories in $\theta$-Poincar\'e; $ii)$ the unitarity of the S-matrix; $iii)$ the violation of locality, $iv)$ the violation of the spin statistic theorem and the Pauli principle; $v)$ the observables for underground experiments.