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The Statistical Mechanics of Quantum Lattice Systems
EMS Tracts in Mathematics Vol. 8

Sergio Albeverio (University of Bonn, Germany)
Yuri Kondratiev (University of Bielefeld, Germany)
Yuri Kozitsky (Maria Curie-Skłodowska University, Lublin, Poland)
Michael Röckner (University of Bielefeld, Germany)

The Statistical Mechanics of Quantum Lattice Systems

A Path Integral Approach

ISBN print 978-3-03719-070-8, ISBN online 978-3-03719-570-3
DOI 10.4171/070
July 2009, 392 pages, hardcover, 17 x 24 cm.
62.00 Euro

Quantum statistical mechanics plays a major role in many fields such as, for instance, thermodynamics, plasma physics, solid-state physics, and the study of stellar structure. While the theory of quantum harmonic oscillators is relatively simple, the case of anharmonic oscillators, a mathematical model of a localized quantum particle, is more complex and challenging. Moreover, infinite systems of interacting quantum anharmonic oscillators possess interesting ordering properties with respect to quantum stabilization.

This book presents a rigorous approach to the statistical mechanics of such systems, in particular with respect to their actions on a crystal lattice.

The text is addressed to both mathematicians and physicists, especially those who are concerned with the rigorous mathematical background of their results and the kind of problems that arise in quantum statistical mechanics. The reader will find here a concise collection of facts, concepts, and tools relevant for the application of path integrals and other methods based on measure and integration theory to problems of quantum physics, in particular the latest results in the mathematical theory of quantum anharmonic crystals. The methods developed in the book are also applicable to other problems involving infinitely many variables, for example, in biology and economics.

Keywords: Gibbs random field, Dobruchin–Lanford–Ruelle approach, quantum oscillator, Schrödinger operator, KMS state, Green function, algebra of observables, Euclidean Gibbs measure, phase transition, quantum effect, decay of correlations, critical point

Further Information

Review in Zentralblatt MATH 1178.82001

Review in MR 2548038 (2010m:82001)