Overview
- Provides a comprehensive introduction to the mathematical foundations of quantum statistical physics
- Bridges a gap between the mathematics and physics communities around the quantum many-body problem
- Offers a technically-friendly approach, making the topics available to a broader audience
Part of the book series: Latin American Mathematics Series (LAMS)
Part of the book sub series: Latin American Mathematics Series – UFSCar subseries (LAMSUFSC)
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Table of contents (7 chapters)
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Front Matter
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Back Matter
Keywords
- quantum statistical mechanics
- mathematical foundations of quantum theory
- phase transitions
- quantum mean-field models
- quantum lattice systems
- interacting fermions
- C*-algebras
- convex analysis
- finite quantum systems
- Bogolioubov’s approximation method
- Gibbs states
- Bishop-Phelps theorem
- Choquet’s theorem
- Hartree-Fock theory
- operator algebras
- functional analysis
- Hilbert space
About this book
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.
Authors and Affiliations
About the authors
Jean-Bernard Bru is a (Ikerbasque) Professor at the University of the Basque Country (UPV/EHU) and BCAM – Basque Center for Applied Mathematics. He obtained his Ph.D. degree in 1999 at the center of theoretical physics of Aix-Marseille University, France. The bulk of his research covers a scope from the mathematical analysis of many-body problems to operator algebras, stochastic processes, evolution equations, convex and functional analysis, to name a few.
Walter Alberto de Siqueira Pedra is a full professor at the Mathematics Department of the Institute of Mathematics and Computer Sciences of the University of São Paulo, Brazil, and an external scientific member of the BCAM – Basque Center for Applied Mathematics (Bilbao). He obtained his Ph.D. degree in 2006 at the University of Leipzig with summa cum laude distinction, having done graduate studies in mathematical physics at the Mathematics Department of the ETH Zurich and the Max Planck Institute for Mathematics in the Sciences (Leipzig). His main research interests concern mathematical aspects of interacting fermions, in particular constructive methods and applications of operator algebras and convex analysis.Bibliographic Information
Book Title: C*-Algebras and Mathematical Foundations of Quantum Statistical Mechanics
Book Subtitle: An Introduction
Authors: Jean-Bernard Bru, Walter Alberto de Siqueira Pedra
Series Title: Latin American Mathematics Series
DOI: https://doi.org/10.1007/978-3-031-28949-1
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
Hardcover ISBN: 978-3-031-28948-4Published: 18 June 2023
Softcover ISBN: 978-3-031-28951-4Due: 18 July 2023
eBook ISBN: 978-3-031-28949-1Published: 16 June 2023
Series ISSN: 2524-6755
Series E-ISSN: 2524-6763
Edition Number: 1
Number of Pages: XXVII, 477
Topics: Mathematical Physics, Mathematical Methods in Physics, Theoretical and Computational Chemistry, Quantum Physics, Functional Analysis