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Global observables in statistical mechanics

C. J. F. van de Ven

TL;DR

The paper develops a representation-free $C^*$-algebraic framework to formalize macroscopic observables in statistical mechanics, unifying quantum and classical regimes. It constructs a full C*-product over finite regions, forms a quotient by the vanishing sequences to obtain a global observable algebra, and defines the quasi-local algebra as the thermodynamic limit, then introduces the global observables as the relative commutant, highlighting non-commutativity. It also builds a commutative subalgebra of macroscopic averages via gamma-sequences, and shows that this commutative sector corresponds to tail-measurable quantities in the classical limit. The results connect quantum tail observables to classical tail algebras and lay groundwork for continuous field descriptions across finite and infinite volumes, providing a unified bridge between microscopic and macroscopic thermodynamics.

Abstract

This note presents a canonical construction of global observables -- sometimes referred to in the literature as macroscopic observables or observables at infinity -- in statistical mechanics, providing a unified treatment of both commutative and non-commutative cases. Unlike standard approaches, the framework is formulated directly in the $C^*$-algebraic setting, without relying on any specific representation.

Global observables in statistical mechanics

TL;DR

The paper develops a representation-free -algebraic framework to formalize macroscopic observables in statistical mechanics, unifying quantum and classical regimes. It constructs a full C*-product over finite regions, forms a quotient by the vanishing sequences to obtain a global observable algebra, and defines the quasi-local algebra as the thermodynamic limit, then introduces the global observables as the relative commutant, highlighting non-commutativity. It also builds a commutative subalgebra of macroscopic averages via gamma-sequences, and shows that this commutative sector corresponds to tail-measurable quantities in the classical limit. The results connect quantum tail observables to classical tail algebras and lay groundwork for continuous field descriptions across finite and infinite volumes, providing a unified bridge between microscopic and macroscopic thermodynamics.

Abstract

This note presents a canonical construction of global observables -- sometimes referred to in the literature as macroscopic observables or observables at infinity -- in statistical mechanics, providing a unified treatment of both commutative and non-commutative cases. Unlike standard approaches, the framework is formulated directly in the -algebraic setting, without relying on any specific representation.

Paper Structure

This paper contains 1 section, 4 theorems, 87 equations.

Table of Contents

  1. Introduction

Key Result

Lemma 3

$\mathcal{C}^\infty$ is a norm-closed *-subalgebra of $\prod_\Lambda \mathcal{B}_\Lambda$, and hence a $C^*$-algebra.

Theorems & Definitions (16)

  • Example 1: Quasi-local algebra
  • Example 2
  • Lemma 3
  • proof
  • Definition 4
  • Lemma 5
  • proof
  • Example 6
  • Definition 7
  • Remark 8
  • ...and 6 more