Ultrahyperfunctional Approach to Non-Commutative Quantum Field Theory

TL;DR

This paper extends the axiomatic framework of non-commutative quantum field theories using tempered ultrahyperfunctions, proving key theorems like CPT and Spin-Statistics within this new approach.

Contribution

It introduces a novel ultrahyperfunctional approach to non-commutative QFT and establishes foundational theorems in this framework.

Findings

Proves CPT theorem for non-commutative QFT using ultrahyperfunctions.

Establishes Spin-Statistics connection in the ultrahyperfunctional setting.

Supports the conjecture that ultrahyperfunctions characterize non-commutative QFT.

Abstract

In the present paper, we intent to enlarge the axiomatic framework of non-commutative quantum field theories (QFT). We consider QFT on non-commutative spacetimes in terms of the tempered ultrahyperfunctions of Sebasti\~ao e Silva corresponding to a convex cone, within the framework formulated by Wightman. Tempered ultrahyperfunctions are representable by means of holomorphic functions. As is well known there are certain advantages to be gained from the representation of distributions in terms of holomorphic functions. In particular, for non-commutative theories the Wightman functions involving the $\star$-product, ${\mathfrak W}^\star_m$, have the same form as the standard form ${\mathfrak W}_m$. We conjecture that the functions ${\mathfrak W}^\star_m$ satisfy a set of properties which actually will characterize a non-commutative QFT in terms of tempered ultrahyperfunctions. In order to support this conjecture, we prove for this setting the validity of some important theorems, of which the CPT theorem and the theorem on the Spin-Statistics connection are the best known. We assume the validity of these theorems for non-commutative QFT in the case of spatial non-commutativity only.