Mathematics

Topics: a selection of the following material, based upon the participants' background

• Historical remarks, axioms, Schroedinger’s and Heisenberg’s formulations, difficulties (decoherence, measurement, ..), Bell’s inequalities, alternative theories.
• Kinematics: mapping of states, homeomorphism of observables (theorems of Wigner, Kadison, Segal), continuity. evolution described by one parameter group of unitaries.
• Comparison with Hamiltonian dynamics The problem of quantisation.
• Operators in Hilbert spaces: basic facts.
• Analytic solution for free motion. Propagation (Strichartz) inequalities. Anholonomy.
• Elements of C*-algebras, GNS representation, automorphism and dynamical systems.
• Quantisation: Weyl's system and Weyl's algebra. Representations of Bargmann-Segal, Fock, Berezin.

Undergrad pre-requisites: basics of Quantum Mechanics (familiarity not requried) 

Main references (graduate texts): G. Dell'Antonio, Mathematical Aspects of Quantum Mechanics. Volume 1 (2011). Available online in the form of lecture notes both in English and in Italian. An enlarged version of the English edition is in preparation.

Exam: exposition and discussion in class of an assigned research paper (or of an assigned textbook chapter)