This textbook provides a comprehensive introduction to the mathematical foundations of quantum statistical physics. It presents a conceptually profound yet technically accessible path to the C*-algebraic approach to quantum statistical mechanics, demonstrating how key aspects of thermodynamic equilibrium can be derived as simple corollaries of classical results in convex analysis.
Using C*-algebras as examples of ordered vector spaces, this book makes various aspects of C*-algebras and their applications to the mathematical foundations of quantum theory much clearer from both mathematical and physical perspectives. It begins with the simple case of Gibbs states on matrix algebras and gradually progresses to a more general setting that considers the thermodynamic equilibrium of infinitely extended quantum systems. The book also illustrates how first-order phase transitions and spontaneous symmetry breaking can occur, in contrast to the finite-dimensional situation. One of the unique features of this book is its thorough and clear treatment of the theory of equilibrium states of quantum mean-field models.
This work is self-contained and requires only a modest background in analysis, topology, and functional analysis from the reader. It is suitable for both mathematicians and physicists with a specific interest in quantum statistical physics.
Author(s): Jean-Bernard Bru, Walter Alberto de Siqueira Pedra
Series: Latin American Mathematics Series
Publisher: Springer-UFSCar
Year: 2023
Language: English
Pages: 496
City: São Carlos
Preface
Acknowledgments
About the Book
Notation
Contents
1 Ordered Vector Spaces and Positivity
1.1 Basic Notions
1.1.1 Ordered Normed Spaces
2 The Space of Bounded Operators on a Hilbert Space as Ordered Vector Space
2.1 The Positive Cone of Bounded Operators on a Hilbert Space
2.2 Monotone Order Completeness and von Neumann Algebras
2.3 The Lattice of Projectors
2.4 General States of B(H)
2.5 Finite-Dimensional Case: States as Density Matrices
3 Thermodynamic Equilibrium of Finite Quantum Systems
3.1 Gibbs States
3.2 Statical Characterizations of Gibbs States
3.3 Dynamical Characterizations of Gibbs States
4 Elements of C-Algebra Theory
4.1 Basic Notions
4.2 The Spectrum of an Algebra Element
4.3 C-Algebras as -Ordered Vector Spaces
4.4 Ideals and Quotients of C-Algebras
4.5 States
4.5.1 Weak Topology for States in the Separable Case
4.6 C-Algebra Homomorphisms and Representations
4.6.1 The Continuous Functional Calculus
4.6.2 States as Equivalence Classes of Cyclic Representations
4.7 Commutative C-Algebras
4.8 The Universal C-Algebra of a Family of Polynomial Relations
4.8.1 Universal Tensor Products of Unital C-Algebras
4.8.2 The CAR C-Algebras
4.8.3 Self-Dual CAR C-Algebras and Fermionic Quasi-Free States
5 Thermodynamic Equilibrium in Infinite Volume
5.1 Algebraic Framework
5.1.1 Quantum Spin Systems
5.1.2 Lattice Fermion Systems
5.1.3 General Notation Encoding Both Fermion and Quantum Spin Systems
5.2 Interactions of Infinite Spin and Fermion Systems on the Lattice
5.3 The Entropy Density of an Invariant State
5.4 Ergodic States
5.5 Bishop-Phelps Theorem and the Existence of Phase Transitions
5.6 Choquet's Theorem and Existence of Spontaneous Symmetry Breaking
5.7 A Brief Introduction to Hartree-Fock Theory
6 Equilibrium States of Mean-Field Models and Bogoliubov's Approximation Method
6.1 Topological Framework
6.2 Spin and Fermion Mean-Field Models
6.3 Free Energy Density of Mean-Field Models
6.4 Equilibrium States of Mean-Field Models
6.5 Approximating Invariant Interactions
6.6 Purely Attractive Mean-Field Models and Application to the BCS Theory
6.6.1 Purely Attractive Mean-Field Models
6.6.2 Application to the BCS Theory on Lattices
6.7 Thermodynamic Game
6.8 Self-Consistency of Equilibrium States
6.9 From Short-Range to Mean-Field Models
6.9.1 The Short-Range Model
6.9.2 The Mean-Field Model
6.9.3 Thermodynamic Game and Bogoliubov Approximation
6.9.4 The Kac Limit
6.9.5 Historical Observations
6.10 The Generalized Hartree-Fock Theory as a Mean-Field Theory
6.10.1 The Short-Range Model
6.10.2 Restriction to Quasi-Free States
6.10.3 Thermodynamic Game and Bogoliubov Approximation
7 Appendix
7.1 Vector Spaces and Algebras
7.2 Metric Spaces
7.2.1 Basic Notions
7.2.2 Continuous Functions
7.2.3 Metric Vector Spaces and Locally Convex Spaces
7.2.4 Convergent Sequences and Nets
7.2.5 Real-Valued Semicontinuous Functions
7.2.6 Compactness
7.2.7 Uniform Convergence
7.3 Hilbert Spaces
7.3.1 Hilbert Spaces as Generalized Euclidean Spaces
7.3.2 Scalar Products
7.3.3 Orthogonal Decompositions
7.3.4 Hilbert Bases
7.3.5 The Space of Bounded Operators on a Hilbert Space as a -Vector Space
7.3.6 The Spectrum of Bounded Operators on Hilbert Spaces
7.3.7 Weak and Strong Operator Convergence, von Neumann Algebras, and Commutants
7.3.8 Tensor Product Hilbert Spaces
7.4 Riesz Spaces
7.4.1 Basic Definitions and Properties
7.4.2 The Absolute Value in General Vector Lattices
7.4.3 Bands
7.4.4 Bands as Ranges of Positive Projectors
7.4.5 The Order Dual of a Riesz Space
7.4.6 Normed Lattices
7.4.7 Some Remarks on the Relations Between Riesz Spaces and C-Algebras
7.5 Convexity
7.5.1 Basic Notions
7.5.2 The Extreme Boundary of Convex Sets
7.5.3 Barycenters of States of Unital C-Algebras
7.5.4 The γ-Regularization
7.5.5 The Legendre-Fenchel Transform
7.5.6 Subdifferentials and Subgradients
7.5.7 Exposed Points of Convex Sets and the Bishop-Phelps Theorem
References
Index
