Green's Functions for Translation Invariant Star Products

TL;DR

This paper derives Green functions for scalar field theories with translation invariant star products, analyzing one-loop corrections and demonstrating that UV/IR mixing arises solely from noncommutative components.

Contribution

It provides explicit formulas for Green functions in scalar theories with generic translation invariant star products, clarifying the role of noncommutativity in UV/IR mixing.

Findings

UV/IR mixing occurs only with noncommuting variables.

Explicit one-loop corrections for two and four point functions are derived.

Commutative parts of the star product do not exhibit UV/IR mixing.

Abstract

We calculate the Green functions for a scalar field theory with quartic interactions for which the fields are multiplied with a generic translation invariant star product. Our analysis involves both noncommutative products, for which there is the canonical commutation relation among coordinates, and nonlocal commutative products. We give explicit expressions for the one-loop corrections to the two and four point functions. We find that the phenomenon of ultraviolet/infrared mixing is always a consequence of the presence of noncommuting variables. The commutative part of the product does not have the mixing.