On Finite Noncommutativity in Quantum Field Theory

TL;DR

This paper explores modifications to the Weyl-Moyal star-product to achieve finite nonlocality in quantum field theory, finding that certain approaches fail and proposing a cutoff-based product as a solution.

Contribution

The paper introduces a novel cutoff-like modification to the Weyl-Moyal star-product to limit nonlocality, addressing the challenge of infinite nonlocality in quantum field theory.

Findings

Exponential derivative modifications do not produce finite nonlocality.

Gaussian damping leads to nonassociative, infinitely nonlocal products.

A cutoff function modification can achieve finite nonlocality, but complicates calculations.

Abstract

We consider various modifications of the Weyl-Moyal star-product, in order to obtain a finite range of nonlocality. The basic requirements are to preserve the commutation relations of the coordinates as well as the associativity of the new product. We show that a modification of the differential representation of the Weyl-Moyal star-product by an exponential function of derivatives will not lead to a finite range of nonlocality. We also modify the integral kernel of the star-product introducing a Gaussian damping, but find a nonassociative product which remains infinitely nonlocal. We are therefore led to propose that the Weyl-Moyal product should be modified by a cutoff like function, in order to remove the infinite nonlocality of the product. We provide such a product, but it appears that one has to abandon the possibility of analytic calculation with the new product.