On Araki's extension of the Jordan-Wigner transformation

TL;DR

This paper reviews and elaborates on Araki's extension of the Jordan-Wigner transformation, connecting quantum spin systems and Fermi gases in one dimension, enabling rigorous analysis without thermodynamic limits.

Contribution

It provides a detailed, self-contained exposition of Araki's extension, including a concrete realization of the crossed product extension, facilitating rigorous study of the XY chain.

Findings

Rigorous formulation of Araki's extension

Concrete realization of the crossed product extension

Enhanced understanding of quantum spin and Fermi gas connection

Abstract

In his seminal paper [1], Araki introduced an elegant extension of the Jordan-Wigner transformation which establishes a precise connection between quantum spin systems and Fermi lattice gases in one dimension in the so-called infinite system idealization of quantum statistical mechanics. His extension allows in particular for the rigorous study of numerous aspects of the prominent XY chain over the two sided infinite discrete line without having to resort to a thermodynamic limit procedure at an intermediate or at the final stage. We rigorously review and elaborate this extension from scratch which makes the paper rather self-contained. In the course of the construction, we also present a simple and concrete realization of Araki's crossed product extension.