The ultrahyperfunctional approach to non-commutative quantum field theory

Abstract

In the present paper, we intend 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 ⋆-product, , have the same form as the standard form . We conjecture that the functions 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.

Holomorphic extension theorem for tempered ultrahyperfunctions

“…We now shall introduce another space of holomorphic functions whose elements are analytic in a domain T (C ′ ) which is larger than T (C ′ ; r) and has boundary values in R n . The boundary values so obtained are of importance in the representation of vacuum expectation values in the case of a quantum field theory in non-commutatives spacetimes [18].…”

“…We finish with a generalized version of holomorphic extension theorem à la Martineau. We note that the results obtained here are of interest in the construction and study of quasilocal quantum field theories (where the fields are localizable only in regions greater than a certain scale of nonlocality), since Brüning-Nagamachi [12] have recently shown the importance of tempered ultrahyperfunctions for quantum field theories with a fundamental length. This is the case of a quantum field theory in non-commutative spacetimes [18].…”

On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth

“…in [10,32]. Also in recent times, ultrahyperfunctions have shown to be quite useful in mathematical physics, particularly as a framework for Wightman-type axiomatic formulations of relativistic quantum field theory with a fundamental length [2,12,31,39].…”

Reeh-Schlieder Theorem for Ultrahyperfunctional Wightman Theory

“…From an axiomatic standpoint, a language has been developed which, in principle, ought to enable one to extend the Wightman axioms to this context [6]- [12]. However, the axiomatic approach to local quantum field theory built up by Streater-Wightman [13], Jost [14], Bogoliubov et al [15], Haag [16] and others turned out to be too narrow for theoretical physicists, who are interested in handling situations involving a NCQFT.…”

“…In Ref. [12] has been suggested that tempered ultrahyperfunctions corresponding to tubular radial domains are well adapted for their use in the axiomatic description of NCQFT. The space of tempered ultrahyperfunctions has the advantage of being representable by means of holomorphic functions.…”