Quantum Theory 2024

Mid-term test

This will be open-book, and held on Wednesday 1 May 09:00 to 23:59. Topics up to week 10 will be covered.

**Announcement**

With effect from 27 Feb, Even week Lecture will be on Tuesday, 3-5pm.

Consultation time

On Wednesday mornings, 10-12, you may look for me in my office BICMR 81-105 for consultations. At other times, please send me an email beforehand.

 

Assignments

(Assignment 1, Due 26 March 3pm)

 

Lecture notes

Notes will be uploaded shortly before or after the week's lectures. As there will be typos, please keep in mind that the repaired notes will be updated after the lectures.

(Week 1) Introduction, classical geometric laws of motion

Regarding affine/fibre bundle structure of Galilean space-time, see Chapter 2 of this article and Chapter 17 of Penrose's book Road to Reality.

(Week 2) Schrodinger equation, special relativity (optional)

(Week 3) Banach/Hilbert spaces, spectrum of bounded operators

(Week 4) Unbounded operators. Formal self-adjointness and SUSY QM

(Week 5) Genuine self-adjoint operators. Aharonov-Bohm effect.

(Week 6) Projection-valued measures and bounded spectral theorem.

(Week 7) Unbounded Spectral Theorem. Stone's Theorem, time evolution, symmetries

(Week 8) Differentiable manifolds. (SUSY QM (skipped))

(Week 9) Tangent and vector bundles, principal bundles, Lie groups

(Week 10) General fibre bundles, reduction of structure group

(Week 11) HOLIDAY, NO LECTURES

(Week 12) Vector fields, Differential forms, Lie algebras

(Week 13)(Week 13-2)(Typos) Lie group Geometry; Connections, curvature, parallel transport, covariant derivatives

(Week 14) Spin and Clifford algebras, Angular momentum

(Week 15) Spin representations, Spin structures, spin connection on spinor bundles, Atiyah-Singer-Dirac operators; Lichnerowicz identity

(Week 16) Quantum Hall effect

 

Assessment

60% final exam. 40% take-home assignments (~2) and mid-term test (~1)

 

 

Assignments

 

 

Some useful references

Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. 1. Functional Analysis. and Vol. II. Fourier analysis, Self-adjointness, Acad. Press, 1980, 1975

Moretti, V.: Spectral Theory and Quantum Mechanics, Springer, 2013

Strocchi, F.: An Introduction to the Mathematical Structure of Quantum Mechanics, World Scientific, 2005

Arveson, W.: A short course on spectral theory, GTM 209, 2002

Baez, J., Muniain, J.P.: Gauge Fields, Knots, and Gravity, World Scientific, 1994

Bleecker, D., Booss-Bavnbek, B.: Index theory with applications to mathematics and physics. International Press, 2013

Choquet-Bruhat, Y., Dewitt-Morette, C.: Analysis, manifolds, and physics. Part I: Basics. North Holland, 1982

Naber, G.: Topology, Geometry, and Gauge Fields. Foundations. Texts in Appl. Math., Springer, 2011

Lee, J.M.: Introduction to smooth manifolds. GTM 218, Springer, 2013

Friedrich, T.: Dirac operators in Riemannian geometry. GSM 25, AMS, 2000

Folland, G.B.: Quantum field theory. A tourist guide for mathematicians. Math. Surveys and Monogr., 149, 2008

 

 

Prerequisites

Linear algebra, real analysis, metric and topological spaces

Basic abstract algebra (especially groups)

Idea of differentiable manifolds

Familiarlity with Hilbert spaces (at least semi-rigorously)

 

Exposure to theoretical physics is helpful, but not compulsory.