Registration Deadline: TBA

Instructor: Professor Marco Gualtieri, University of Toronto

Course Date: TBA
Mid-Semester Break: TBA
Lecture Time: TBA
Office Hours: TBA

Registration Fee:

• Students from our Principal Sponsoring & Affiliate Universities: Free
• Other Students: CAD$500

Capacity Limit: TBA

Format: TBA

Course Description

This is a new course on the geometry of quantum mechanics, looking back a century to the beginning of the subject, and using modern geometric tools to express its core ideas. It is aimed at students with an understanding of the basics of manifold theory and complex analysis.

In addition to revisiting all the main concepts of quantum mechanics with a modern toolkit, we will review a line of recent developments in the exact WKB method whereby the perturbative solutions to singular differential equations (such as the Schrodinger equation) are shown to be Borel summable. This would bring us to the research level in the subject (Kontsevich-Soibelman, Nikolaev).

Topics to be covered:

• Groupoids and the Heisenberg approach to Quantum Mechanics (c.f. Connes' Noncommutative Geometry)
• The Schrodinger equation on Riemannian manifolds
• Hilbert spaces, C_ algebras, and Von Neumann algebras (c.f. Witten's recent work on an algebra of observables for de Sitter spacetime)
• Representations of the Lorentz group, elementary particles (c.f. Wigner's paper on irreducible representations)
• The Dirac equation, Spin manifolds, generalized geometry.
• Heisenberg spin chains, the vacuum state (c.f. Penrose Spin Chains)
• The WKB method and its exact version (c.f. text of Kawai-Takei, recent papers of Nikolaev)
• Quantum Field Theory: Atiyah-Segal axioms for topological and conformal field theories

Evaluation will be by a series of assignments which are designed to teach the material through a series of elaborate worked examples.