C*-Algebras and Mathematical Foundations of Quantum Statistical Mechanics

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.

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

Auteur
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.

Contenu
Preface.- Ordered vector spaces and positivity.- The space of bounded operators on a Hilbert space as ordered vector space.- Thermodynamic equilibrium of finite quantum systems.- Elements of C*-algebra.- Thermodynamic equilibrium in infinite volume.- Equilibrium states of mean-field models and Bogolioubov's approximation method.- Appendix.- References.- Index.