Point interactions in one dimension and holonomic quantum fields

TL;DR

This paper introduces quantum fields linked to delta-interactions in one dimension, connecting them to holonomic quantum fields, and explores their form factors, correlation functions, and relations to determinants of Schrödinger operators.

Contribution

It develops a novel framework for quantum fields associated with point interactions, relating them to holonomic fields and determinants of Schrödinger operators.

Findings

Form factors and correlation functions computed.

Determinants coincide with tau functions.

Fields form an infinite-dimensional representation of SL(2,R).

Abstract

We introduce and study a family of quantum fields, associated to delta-interactions in one dimension. These fields are analogous to holonomic quantum fields of M. Sato, T. Miwa and M. Jimbo. Corresponding field operators belong to an infinite-dimensional representation of the group $SL(2,\Rb)$ in the Fock space of ordinary harmonic oscillator. We compute form factors of such fields and their correlation functions, which are related to the determinants of Schroedinger operators with a finite number of point interactions. It is also shown that these determinants coincide with tau functions, obtained through the trivialization of the $\mathrm{det}^*$-bundle over a Grassmannian associated to a family of Schroedinger operators.