Test Functions Space in Noncommutative Quantum Field Theory

TL;DR

This paper establishes that in noncommutative quantum field theory, the test functions form a Gel'fand-Shilov space with specific properties, which is crucial for rigorous mathematical formulation.

Contribution

It proves that the test function space in NC QFT is a Gel'fand-Shilov space $S^{eta}$ with $eta < 1/2$, clarifying the mathematical framework.

Findings

Test functions are in Gel'fand-Shilov space $S^{eta}$ with $eta < 1/2$.

Noncommutative Wightman functions are generalized distributions or hyperfunctions.

The result aids rigorous formulation of NC QFT in the Wightman framework.

Abstract

It is proven that the $\star$-product of field operators implies that the space of test functions in the Wightman approach to noncommutative quantum field theory is one of the Gel'fand-Shilov spaces $S^{\beta}$ with $\beta < 1/2$. This class of test functions smears the noncommutative Wightman functions, which are in this case generalized distributions, sometimes called hyperfunctions. The existence and determination of the class of the test function spaces in NC QFT is important for any rigorous treatment in the Wightman approach.