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Miniatures on Open Quantum Systems

Jan Derezinski, Vojkan Jaksic, Claude-Alain Pillet

TL;DR

This work consolidates a sequence of C*- and W*-algebraic treatments of open quantum systems into a unified reference, extending foundational topics to include non-equilibrium steady states, entropy production, and linear response. It develops both CCR/CAR algebras for bosonic/fermionic systems and Danov-type representations (Araki–Woods/Wyss), then extends to non-equilibrium via quantum Koopmanism, NESS construction, and spectral/Liouvillean analyses. A core thread is the interplay between equilibrium KMS states and non-equilibrium dynamics, with modular theory providing the backbone for perturbation theory, entropy balance, and formalisms such as the standard Liouvillean. The text also surveys reduced dynamics through Davies-type weak-coupling limits, quantum Langevin dynamics, and open lattice/spin-system models, highlighting positivity of entropy production and Onsager relations in appropriate regimes. Overall, the book offers a self-contained, operator-algebraic reference linking equilibrium notions to the richly structured non-equilibrium behavior of open quantum systems, with broad implications for mathematical physics and quantum statistical mechanics.

Abstract

We presents a unified and concise exposition of key topics in the mathematical theory of open quantum systems, developed within the framework of operator algebras. The manuscript consolidates and extends a series of invited articles originally prepared for the Modern Encyclopedia of Mathematical Physics, combining foundational material with modern perspectives on non-equilibrium quantum statistical mechanics. After introducing the C*- and W*-algebraic formulation of quantum mechanics, the paper reviews quantum dynamical systems, KMS states, and Tomita-Takesaki modular theory, as well as CCR and CAR algebras for bosonic and fermionic systems. Particular emphasis is placed on infinite systems, non-equilibrium steady states, entropy production, and linear response theory. The later sections develop a systematic treatment of small systems coupled to reservoirs, open lattice quantum spin systems, culminating in a detailed discussion of competing notions of quantum entropy production. The presentation highlights structural insights, conceptual clarity, and connections between equilibrium and non-equilibrium phenomena, providing a self-contained reference for researchers and graduate students in mathematical physics.

Miniatures on Open Quantum Systems

TL;DR

This work consolidates a sequence of C*- and W*-algebraic treatments of open quantum systems into a unified reference, extending foundational topics to include non-equilibrium steady states, entropy production, and linear response. It develops both CCR/CAR algebras for bosonic/fermionic systems and Danov-type representations (Araki–Woods/Wyss), then extends to non-equilibrium via quantum Koopmanism, NESS construction, and spectral/Liouvillean analyses. A core thread is the interplay between equilibrium KMS states and non-equilibrium dynamics, with modular theory providing the backbone for perturbation theory, entropy balance, and formalisms such as the standard Liouvillean. The text also surveys reduced dynamics through Davies-type weak-coupling limits, quantum Langevin dynamics, and open lattice/spin-system models, highlighting positivity of entropy production and Onsager relations in appropriate regimes. Overall, the book offers a self-contained, operator-algebraic reference linking equilibrium notions to the richly structured non-equilibrium behavior of open quantum systems, with broad implications for mathematical physics and quantum statistical mechanics.

Abstract

We presents a unified and concise exposition of key topics in the mathematical theory of open quantum systems, developed within the framework of operator algebras. The manuscript consolidates and extends a series of invited articles originally prepared for the Modern Encyclopedia of Mathematical Physics, combining foundational material with modern perspectives on non-equilibrium quantum statistical mechanics. After introducing the C*- and W*-algebraic formulation of quantum mechanics, the paper reviews quantum dynamical systems, KMS states, and Tomita-Takesaki modular theory, as well as CCR and CAR algebras for bosonic and fermionic systems. Particular emphasis is placed on infinite systems, non-equilibrium steady states, entropy production, and linear response theory. The later sections develop a systematic treatment of small systems coupled to reservoirs, open lattice quantum spin systems, culminating in a detailed discussion of competing notions of quantum entropy production. The presentation highlights structural insights, conceptual clarity, and connections between equilibrium and non-equilibrium phenomena, providing a self-contained reference for researchers and graduate students in mathematical physics.
Paper Structure (100 sections, 64 theorems, 348 equations, 1 figure)

This paper contains 100 sections, 64 theorems, 348 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Elements of operator algebras
  3. The $C^\ast$-algebra approach
  4. Why operator algebras?
  5. Examples of $C^\ast$-algebras
  6. Lattice spin systems
  7. CAR algebras
  8. CCR algebras
  9. Quasi-local structure of $\mathfrak{A}(\Gamma,\mathfrak{h})$ and of $\operatorname{CAR}(\mathfrak{h})$
  10. States and the GNS construction
  11. Quantum dynamical systems
  12. $C^\ast$-dynamical systems
  13. $W^\ast$-dynamical systems
  14. Invariant states and Liouvilleans
  15. Perturbation theory
  16. ...and 85 more sections

Key Result

Theorem 2.1

Let $\mathfrak{h}_0$ be a pre-Hilbert space. Up to $\ast$-isomorphisms, there exists a unique unital $C^\ast$-algebra $\operatorname{CAR}(\mathfrak{h}_0)$ satisfying the following two properties:

Figures (1)

  • Figure 1: The cover of Encyclopedia from Springer 2011 web page.

Theorems & Definitions (81)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3: Exponential law for fermions
  • Theorem 2.4
  • Corollary 2.5
  • Theorem 2.6: Exponential law for bosons
  • Theorem 2.7
  • Theorem 2.8
  • Definition 3.1
  • Theorem 3.2
  • ...and 71 more