The Statistical Mechanics of Quantum Lattice Systems: A Path Integral Approach
Quantum statistical mechanics plays a major role in many fields such as 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.
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Quantum Mechanics and Stochastic Analysis
Lattice Approximation and Applications
Euclidean Gibbs Measures of Quantum Crystals
Quantum Anharmonic Crystal as a Physical Model
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analytic anharmonic crystal anharmonic potential Banach space Borel called classical cofinal compact construction continuous functions converges convex Corollary correlation function corresponding defined Definition denote described differentiable Dom(r domain eigenvalues elements entire function equation Euclidean Gibbs measures exists ferromagnetic finite Gaussian Gaussian measure given Hamiltonian harmonic oscillator hence Hilbert space holds infinite-dimensional integral interaction introduce kernel latter lattice Lemma limit linear operator Matsubara functions means metric multiplication operators norm obtain parameter particles phase transitions Polish space positive probability measure proof of Theorem Proposition prove quantum anharmonic recall respect right-hand side scalar Section self-adjoint self-adjoint operator Sobolev inequality statement stochastic subset symmetric tempered Euclidean Gibbs theory Thereby topology trace-class translation-invariant uniqueness vector von Neumann algebra weak topology yields zero