Tuesday, January 01, 2008

Study Programme

The material for this year's OU course - SMT359 Electromagnetism - arrived in the post just before Christmas and I've begun some preparatory maths revision: vectors; matrices; line integrals; div, grad and curl. I'm using the excellent "Mathematical Methods in the Physical Sciences" by Mary Boas, which I first bought back in 1984.

My onward study plan is rather languid and goes something like this.

2009: the quantum world (SM358). Basic quantum mechanics.

2010: waves, diffusion and variational principles (MS324). Diffusion processes, calculus of variations, Lagrangian formulation of mechanics.

2011: space, time and cosmology (S357). Introduction to special and general relativity.

Then a move to the maths MSc programme, concentrating on mathematical physics modules.

2012: applicable differential geometry (M827). For general relativity.

2013: functional analysis (M826). The OU's nearest thing to a treatment of Hilbert spaces for quantum field theory.

One of the irritating things about the OU's maths MSc programme is that it's almost perversely unsuitable for students of quantum mechanics. I quote from a student review and faculty reply on the M826 website here:


"I didn't enjoy this course at all, and spent most of it having no idea what was going on. I chose it because I'd read (in Gowers' "A Brief Introduction to Mathematics") that Hilbert Spaces are one of the most important things in mathematics. I now more or less know what one is but don't know why they're important. "


"Perhaps the problem here is that M826 is not meant to be a course on 'Hilbert Spaces and its applications'. Indeed, most of the course concentrates on linear spaces that have quite a general topological structure. Only at the very end does the course focus its attention on Hilbert Spaces.

Inevitably there is by then little time to do more than define a Hilbert Space, examine its structure (in cases where it is separable), and characterize its dual spaces. In particular no attempt is made to examine the rich structure of the spaces of linear operators acting on a Hilbert Space, nor is any attempt made to describe the many applications of such spaces (e.g. to quantum mechanics, statistical mechanics, optimisation, partial differential equations, etc.).

Clearly anyone who studies the course hoping to learn all about Hilbert Spaces and its applications could be disappointed.

The work in M826 is quite challenging and requires a good working knowledge of basic set theory, vector spaces, analysis and topological (mainly metric) spaces. It is also necessary to have an aptitude for following proofs and understanding how they relate to the result being proved. Anyone whose preference is for less abstract mathematics may find some of the work difficult to follow.

Perhaps they will have had a change of heart by 2013.