Coproduct and star product in field theories on Lie-algebra non-commutative space-times

TL;DR

This paper introduces a star product approach to field theory on $kappa$-Minkowski non-commutative space-time, clarifying symmetry properties of interaction vertices and defining planar and non-planar Feynman diagrams.

Contribution

It presents a novel star product framework for $kappa$-Minkowski space-time and corrects previous assumptions about vertex symmetry in such theories.

Findings

Interaction vertices are symmetric under particle exchange despite non-symmetric coproducts.

The paper introduces the concepts of planar and non-planar diagrams in $kappa$-Minkowski field theories.

The approach aligns $kappa$-Minkowski with canonical non-commutative field theory techniques.

Abstract

We propose a new approach to field theory on $\kappa$-Minkowski non-commutative space-time, a popular example of Lie-algebra space-time. Our proposal is essentially based on the introduction of a star product, a technique which is proving to be very fruitful in analogous studies of canonical non-commutative space-times, such as the ones recently found to play a role in the description of certain string-theory backgrounds. We find to be incorrect the expectation, previously reported in the literature, that the lack of symmetry of the $\kappa$-Poincare' coproduct should lead to interaction vertices that are not symmetric under exchanges of the momenta of identical particles entering the relevant processes. We show that in $\kappa$-Minkowski the coproduct and the star product must indeed treat momenta in a non-symmetric way, but the overall structure of interaction vertices is symmetric under exchange of identical particles. We also show that in $\kappa$-Minkowski field theories it is convenient to introduce the concepts of "planar" and "non-planar" Feynman loop-diagrams, again in close analogy with the corresponding concepts previously introduced in the study of field theories in canonical non-commutative space-times.