Ultrahyperfunctional Approach to Non-Commutative Quantum Field Theory

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 Sebastião 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, ${\mathfrak{W}}_m^{\star }$, have the same form as the standard form ${\mathfrak{W}}_m$. We conjecture that the functions ${\mathfrak{W}}_m^{\star }$ 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.

Published in "Journal of Physics A"

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Cited by in corpus (6)

• On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth
• The Edge of the Wedge Theorem for Tempered Ultrahyperfunctions
• The Edge of the Wedge Theorem for Tempered Ultrahyperfunctions II. A Generalized Version
• A Uniqueness Theorem and Its Application to Field-Theoretical Models with a Fundamental Length
• Holomorphic Extension Theorem for Tempered Ultrahyperfunctions
• Paley-Wiener-Schwartz Theorem and Microlocal Analysis in Theory of Tempered Ultrahyperfunctions