Mathematical Methods in
Quantum Statistical Physics
(graduate course, SISSA -- January-March 2017)


Lecturer: Alessandro Michelangeli
Exercise sessions: Harold Bustos

venue and schedule: Mo + Tue, 9:15-11:00, room A-136; Tue 31 Jan in A-132
start: 16 January 2017
duration: 40 hours (2 cycles); partial credits are possible
office hours: Tue, 15:00-16:00, office A-724


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 Hilbert-space 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:
  1. Motivations and scope
    - Hilbert space for infinitely many particles, difficulties of infinite tensor products
    - algebra of observables and C*-algebraic formulation of quantum mechanics
    - inequivalent representations for infinite systems
    - quasi-local structure of quantum observables of infinite systems
  2. Recap on C*-algebras and von Neumann algebras
    - basic definitions and structure of C*-algebras
    - functional calculus
    - states, representations, GNS
    - definition and elementary properties of von Neumann Algebras
    - normal states and the predual
  3. Continuous quantum systems
    - the CAR and CCR relations
    - the CAR and CCR algebras
    - states and representations
    - the ideal Fermi gas
    - the ideal Bose gas
  4. KMS states
     - the KMS condition
    - the set of KMS states
    - the set of ground states
  5. Stability and equilibrium
    - stability of KMS states
    - stability and the KMS condition
    - passive systems
  6. Quantum spin systems
    - kinematical and dynamical description
    - the Gibbs condition for equilibrium
    - the maximum entropy principle
    - translationally invariant states
    - uniqueness/non-uniqueness of KMS states
    - ground states

 Pre-requisites: 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 (undergraduate-like) 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, Math-Phys, and Quantum seminar.


Exam: by one of the following procedures:
  • an open seminar where to discuss a research paper or other material related with the course, previously decided together with the instructor (intermediate discussions with the instructor are recommended before delivering the seminar)
  • a short essay (~10 pages) on themes previously agreed with the instructor
  • an oral examination
  • a 90' written test
  • a take-home exam (exercises to solve at home and to present to the instructor).
The examination panel will be formed by: H. Bustos, L. Dabrowski, G. Dell'Antonio, K. van den Duden, A. Michelangeli


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 I-II 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", Wiley-Interscience (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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