The Statistical Mechanics of Quantum Lattice Systems: A Path Integral ApproachEuropean Mathematical Society, 2009 - 379 Seiten 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. |
Inhalt
Introduction | 1 |
Quantum Effects | 10 |
Quantum Mechanics and Stochastic Analysis | 15 |
Lattice Approximation and Applications | 155 |
Euclidean Gibbs Measures of Quantum Crystals | 191 |
Quantum Anharmonic Crystal as a Physical Model | 249 |
Thermodynamic Pressure | 262 |
Phase Transitions | 280 |
Bibliography | 352 |
| 377 | |
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The Statistical Mechanics of Quantum Lattice Systems: A Path Integral Approach Keine Leseprobe verfügbar - 2009 |
Häufige Begriffe und Wortgruppen
algebra anharmonic anharmonic potential Banach space Borel bounded called CB,A classical compact condition constructed continuous functions converges convex Corollary correlation function corresponding defined Definition denote Dom(T domain eigenvalues elements estimate Euclidean Gibbs measures exists ferromagnetic finite function f Gaussian given Hamiltonian harmonic oscillator hence Hhar Hilbert space Hölder continuous inequality integral interaction kernel l'EA l'EAC l'Es L²(R lattice Lemma Lfin linear operator Matsubara functions means metric multiplication operators n}neN norm o-algebra obeys obtain parameter particle phase transitions Polish space positive probability measure proof of Theorem Proposition prove QB,A quantum anharmonic respect scalar Section self-adjoint self-adjoint operator statement subset tempered Euclidean Gibbs topology translation-invariant vector von Neumann algebra weak topology yields zero ΚΕΛ ΜΕΛ Σ Σ