On the consequences of twisted Poincare' symmetry upon QFT on Moyal noncommutative spaces

TL;DR

This paper investigates how twisted Poincare' symmetry affects quantum field theory on noncommutative Moyal spaces, revealing that the resulting theory is essentially equivalent to standard QFT in terms of n-point functions and symmetries.

Contribution

It demonstrates that enforcing twisted Poincare' covariance leads to a QFT with unchanged n-point functions, suggesting physical equivalence to the commutative case.

Findings

n-point functions depend only on coordinate differences

QFT on noncommutative space is equivalent to standard QFT

symmetry properties are preserved under twisted Poincare'

Abstract

We explore some general consequences of a consistent formulation of relativistic quantum field theory (QFT) on the Groenewold-Moyal-Weyl noncommutative versions of Minkowski space with covariance under the twisted Poincare' group of Chaichian et al. [12], Wess [44], Koch et al. [31], Oeckl [34]. We argue that a proper enforcement of the latter requires braided commutation relations between any pair of coordinates $\hat x,\hat y$ generating two different copies of the space, or equivalently a $\star$-tensor product $f(x)\star g(y)$ (in the parlance of Aschieri et al. [3]) between any two functions depending on $x,y$. Then all differences $(x-y)^\mu$ behave like their undeformed counterparts. Imposing (minimally adapted) Wightman axioms one finds that the $n$-point functions fulfill the same general properties as on commutative space. Actually, upon computation one finds (at least for scalar fields) that the $n$-point functions remain unchanged as functions of the coordinates' differences both if fields are free and if they interact (we treat interactions via time-ordered perturbation theory). The main, surprising outcome seems a QFT physically equivalent to the undeformed counterpart (to confirm it or not one should however first clarify the relation between $n$-point functions and observables, in particular S-matrix elements). These results are mainly based on a joint work [24] with J. Wess