On "full" twisted Poincare' symmetry and QFT on Moyal-Weyl spaces

TL;DR

This paper demonstrates that enforcing full twisted Poincare' covariance in quantum field theory on Moyal-Weyl spaces results in theories that are practically indistinguishable from standard theories, with noncommutativity effects being unobservable.

Contribution

It shows that full twisted Poincare' symmetry allows for QFT formulations where noncommutativity does not alter observable physics, preserving key structures like causality and correlation functions.

Findings

Coordinate differences commute and are Poincare' invariant.

Standard QFT notions are retained despite noncommutativity.

Noncommutative effects act as unobservable translations.

Abstract

We explore some general consequences of a proper, full enforcement of the "twisted Poincare'" covariance of Chaichian et al. [14], Wess [50], Koch et al. [34], Oeckl [41] upon many-particle quantum mechanics and field quantization on a Moyal-Weyl noncommutative space(time). This entails the associated braided tensor product with an involutive braiding (or $\star$-tensor product in the parlance of Aschieri et al. [3,4]) prescription for any coordinates pair of $x,y$ generating two different copies of the space(time); the associated nontrivial commutation relations between them imply that $x-y$ is central and its Poincar\'e transformation properties remain undeformed. As a consequence, in QFT (even with space-time noncommutativity) one can reproduce notions (like space-like separation, time- and normal-ordering, Wightman or Green's functions, etc), impose constraints (Wightman axioms), and construct free or interacting theories which essentially coincide with the undeformed ones, since the only observable quantities involve coordinate differences. In other words, one may thus well realize QM and QFT's where the effect of space(time) noncommutativity amounts to a practically unobservable common noncommutative translation of all reference frames.