Topics in the mathematics of Quantum Mechanics
(graduate course, SISSA, October-December 2014)

Lecturer: Gianfausto Dell'Antonio

venue and schedule: room A-136, Tuesday 14-16, Friday 9-11
start: Friday 17 October 2014

Table of contents:
  • "Wave mechanics" (de Broglie, Schrödinger) and "matrix mechanics" (Born, Heisenberg, Jordan). Analogy with Classical Mechanics (Dirac, Pauli).
  • Axioms: states and observables (Schroedinger, Heisenberg, von Neumann); measurement. Difficulties: hidden variables? Bell's inequalities. Alternative theories.
  • Construction of kinematics and dynamics. Maps of states. Automorphisms of observables. Unitary dynamics. Generators.
  • Analogy with Hamiltonian dynamics. The problem of quantisation.
  • Operators in a Hilbert space. Quadratic forms (basic facts). An example: analytic solution for the free motion. Propagation inequalities. Quadratic (Dirichlet) form for free motion on a line. Berry phase, adiabatic limit.
  • Elements of C*-algebras. Dynamical systems. GNS representation. An example: systems on a lattice. KMS condition.
  • Quantisation: Weyl's system, Weyl's algebra. Second quantisation. Representations (Bargmann-Segal, Fock, Berezin)
  • Semi-classical limit. Stationary and non-stationary WKB method.

basics of Quantum Mechanics (familiarity not requried).

Literature: G. Dell'Antonio, Lectures on the Mathematics of Quantum Mechanics I (2015)
Italian version:
G. Dell'Antonio, Mathematical Aspects of Quantum Mechanics. Volume 1 (2011).