On the Limit of Superposition States

TL;DR

This paper investigates the structure and limits of superposition states on tensor algebras, introducing Schur kernels to describe their correlation functions, and proves the existence and mixing properties of their limits on various graphs.

Contribution

It introduces Schur kernels for describing superposition states and proves the existence and mixing properties of their limits on arbitrary graphs and lattices.

Findings

Existence of limiting superposition states on arbitrary graphs.

Limiting states exhibit mixing and $\alpha$-mixing properties.

Introduction of Schur kernels for correlation function analysis.

Abstract

In this paper, we study the structure of a family of superposition states on tensor algebras. The correlation functions of the considered states are described through a new kind of positive definite kernels valued in the dual of C$^\ast$-algebras, so-called Schur kernels. Mainly, we show the existence of the limiting state of a net of superposition states over an arbitrary locally finite graph. Furthermore, we show that this limiting state enjoys a mixing property and an $\alpha$-mixing property in the case of the multi-dimensional integer lattice $\mathbb{Z}^\nu$.