

Lecturer:
Alessandro Michelangeli Exercise sessions: Harold Bustos venue and schedule: Mo + Tue, 9:1511:00, room A136; Tue 31 Jan in A132 start: 16 January 2017 duration: 40 hours (2 cycles); partial credits are possible office hours: Tue, 15:0016:00, office A724 Synopsis: This course presents the mathematical framework of Quantum Statistical Physics, with special emphasis on the formalism and the problems for infinite systems. For reasons that are both conceptual (the primary role in the theory is given to the [local] observables) and technical (the infinite tensor product of a Hilbert space loses separability and is difficult to control), the appropriate language for Quantum Statistical Physics is rather that of the C*algebras of observables, the positive linear functional on which give the states of the system. The customary Hilbertspace picture is recovered in a suitable representation. At this algebraic level one formalises naturally the notion of quantum dynamics, equilibrium, return to equilibrium, phase transitions, locality, etc. The main facts and problems of the theory will be discussed in concrete in application to infinite quantum spin systems. Topics:
Prerequisites: the preceeding courses of Introduction to C*algebras and applications (van den Dungen) and Mathematical Quantum Mechanics I (Michelangeli) provide relevant background and are warmly recommended, albeit not strictly needed. A first (undergraduatelike) exposition to the general framework of Quantum Mechanics, as well as an amount of basic knowledge of functional analysis (Hilbert spaces and operators, L^p spaces, distributions, Fourier transform) will be given for granted or recapped along the way. The course is also designed to intersect the Analysis, MathPhys, and Quantum seminar. Exam: by one of the following procedures:
Literature: Attal, Joye, and Pillet (Eds.), "Open Quantum Systems I. The Hamiltonian approach", LNM Springer (2006) Bratteli and Robinson, "Operator Algebras and Quantum Statistical Mechanics" 2nd ed., vol III Springer (1987) Dell'Antonio, "Lectures on the Mathematics of Quantum Mechanics I and II", Springer (2015) Dereziński and Gérard, "Mathematics of Quantization and Quantum Fields", Cambridge (2013) Emch, "Algebraic Methods in Statistical Mechanics and Quantum Field Theory", WileyInterscience (1972) Haag, "Local Quantum Physics. Fields, Particles, Algebras.", 2nd ed., Springer (2012) Ruelle, "Statistical Mechanics. Rigorous Results.", World Scientific (1999) Ruelle, "Thermodynamic formalism", Cambridge (2004) Sewell, "Quantum Theory of Collective Phenomena", Dover (2014) Simon, "Statistical Mechanics of Lattice Gases", Princeton (1993) Strocchi, "An Introdution to the Mathematical Structure of Quantum Mechanics", World Scientific (2008) Strocchi, "Symmetry breaking", Springer (2008) Thirring, "Quantum Mathematical Physics. Atoms, Molecules, and Large Systems.", Springer (2002) 

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