Noncommutative Field Theory With General Translation Invariant Star Products

TL;DR

This paper computes Green's functions in noncommutative phi^4 theory with general translation invariant star products, revealing their relation to twists and Poincaré invariance at a deformed level.

Contribution

It derives a differential expression for any translation invariant star product and shows these can be expressed via twists, establishing Poincaré invariance in noncommutative field theory.

Findings

Green's functions computed for s-ordered and general translation invariant star products

Any translation invariant star product can be expressed in terms of a twist

Coordinate commutators are invariant under deformed Poincaré transformations

Abstract

We compute the two-point and four-point Green's function of the noncommutative $\phi^{4}$ field theory; first with the s-ordered star products and then with a general translation invariant star product. We derive the differential expression for any translation invariant star product, and with the help of this expression we show that any of these products can be written in terms of a twist. Finally, using the notion of the twisted action of the infinitesimal Poincar\'e transformations, we show that the commutator between the coordinate functions is invariant under Poincar\'e transformations at a deformed level.