Quantum field theory with a fundamental length: A general mathematical framework

TL;DR

This paper develops a mathematical framework for nonlocal quantum field theory incorporating a fundamental length, demonstrating that such models satisfy Wightman axioms with consistent high-energy bounds and become local as the length approaches zero.

Contribution

It introduces a new mathematical structure for nonlocal QFT with a fundamental length, extending Wightman axioms and analyzing the analyticity domain of vacuum expectation values.

Findings

The analyticity domain is smaller than the usual future tube but has a similar structure in difference variables.

The model satisfies Wightman axioms with an energy bound independent of the number of points.

The theory reduces to local QFT as the fundamental length approaches zero.

Abstract

We review and develop a mathematical framework for nonlocal quantum field theory (QFT) with a fundamental length. As an instructive example, we reexamine the normal ordered Gaussian function of a free field and find the primitive analyticity domain of its n-point vacuum expectation values. This domain is smaller than the usual future tube of local QFT, but we prove that in difference variables, it has the same structure of a tube whose base is the (n-1)-fold product of a Lorentz invariant region. It follows that this model satisfies Wightman-type axioms with an exponential high-energy bound which does not depend on n, contrary to the claims in the literature. In our setting, the Wightman generalized functions are defined on test functions analytic in the complex l-neighborhood of the real space, where l is an n-independent constant playing the role of a fundamental length, and the causality condition is formulated with the use of an analogous function space associated with the light cone. In contrast to the scheme proposed by Bruning and Nagamachi [J. Math. Phys. 45 (2004) 2199] in terms of ultra-hyperfunctions, the presented theory obviously becomes local as l tends to zero.