Groupoid and algebra of the infinite quantum spin chain

TL;DR

This paper explores the algebraic structures underlying infinite quantum spin chains, employing advanced operator algebra techniques to describe their dynamics and applying these methods to the Ising model.

Contribution

It introduces a framework using Dirac-Feynman-Schwinger states and modular theory to analyze infinite quantum spin chains, extending the algebraic approach to quantum field theory.

Findings

Algebras naturally arise in the Schwinger description of infinite spin chains.

The modular theory provides a dynamic framework for these algebras.

Application of the approach to the Ising model demonstrates its effectiveness.

Abstract

It is well known that certain features of a quantum theory cannot be described in the standard picture on a Hilbert space. In particular, this happens when we try to formally frame a quantum field theory, or a thermodynamic system with finite density. This forces us to introduce different types of algebras, more general than the ones we usually encounter in a standard course of quantum mechanics. We show how these algebras naturally arise in the Schwinger description of the quantum mechanics of an infinite spin chain. In particular, we use the machinery of Dirac-Feynman-Schwinger (DFS) states developed in recent works to introduce a dynamics based on the modular theory by Tomita-Takesaki, and consequently we apply this approach to describe the Ising model.