Translation Invariance, Commutation Relations and Ultraviolet/Infrared Mixing

TL;DR

This paper demonstrates that ultraviolet/infrared mixing in noncommutative field theories is a general feature of translationally invariant associative products, linking it to the underlying Poisson structure of spacetime.

Contribution

It establishes the universality of UV/IR mixing in noncommutative field theories with translationally invariant products and connects it to the commutator-based phase in nonplanar diagrams.

Findings

UV/IR mixing is a generic feature of translationally invariant associative products.

The phase in nonplanar diagrams is given by the coordinate commutator.

The phase relates to the semiclassical Poisson structure of noncommutative spacetime.

Abstract

We show that the Ultraviolet/Infrared mixing of noncommutative field theories with the Gronewold-Moyal product, whereby some (but not all) ultraviolet divergences become infrared, is a generic feature of translationally invariant associative products. We find, with an explicit calculation that the phase appearing in the nonplanar diagrams is the one given by the commutator of the coordinates, the semiclassical Poisson structure of the non commutative spacetime. We do this with an explicit calculation for represented generic products.