The modern mathematical foundation of quantum mechanics involves mathematical structures which are not typically presented in a physics curriculum. In particular, topics within functional analysis such as operator theory and spectral theory are of great importance. In this reading course, we will explore the use of functional analysis in quantum mechanics with a focus on mathematical clarity. More abstractly, we will consider how the kinds of mathematical structures that are used in a physical theory are determined by the kinds of statements that theory needs to make and the results of relevant experiments. Topics that might be covered include: Operator Theory, Spectral Theory, Stone-von Neumann Theorem, Wigner’s Theorem, Representation Theory, C*-algebras, Quantization, Measurement, Projection-valued Measures, POVMs, Gleason’s Theorem, Interpretations of Quantum Mechanics, Topos Theory.

Reading:

• An Introduction to the Mathematical Structure of Quantum Mechanics by F. Strocchi

• Mathematical Methods in Quantum Mechanics by Gerald Teschl

Requirements: PHYS 325 (QM 2), some proof-based math course

Recommended: PHYS 329 (Classical Mechanics), MATH 340 (Linear Algebra)