Reconstruction in quantum field theory with a fundamental length

TL;DR

This paper extends the Wightman reconstruction theorem to nonlocal quantum field theories with a fundamental length, addressing the analytic structure, causality, and domain issues of reconstructed quantum fields.

Contribution

It develops a reconstruction framework for nonlocal QFT with a fundamental length, including analytic test functions, causality conditions, and domain analysis.

Findings

Established an analog of Wightman's theorem for nonlocal QFT

Defined test functions analytic in a complex neighborhood of real space

Identified the domain of reconstructed quantum fields in the Hilbert space

Abstract

In this paper, we establish an analog of Wightman's reconstruction theorem for nonlocal quantum field theory with a fundamental length. In our setting, the Wightman generalized functions are defined on test functions analytic in a complex l-neighborhood of the real space and are localizable at scales large compared to l. The causality condition is formulated as continuity of the field commutator in an appropriate topology associated with the light cone. We prove that the relevant function spaces are nuclear and derive the kernel theorems for the corresponding classes of multilinear functionals, which provides the basis for the reconstruction procedure. Special attention is given to the accurate determination of the domain of the reconstructed quantum fields in the Hilbert space of states. We show that the primitive common invariant domain must be suitably extended to implement the (quasi)localizability and causality conditions.