There is a family story that I wrote to Patrick Moore, noted TV Astronomer, when I was ten years old asking him about time-dilation in Special Relativity. He is alleged to have written back stating that the theory was rather new and difficult to understand. This is absolutely true, although we have lost the letter.
I went to Warwick University to become a theoretical physicist. I was completely up for mathematically hitting Quantum Mechanics and Relativity: cosmology was my great interest. Instead, I found myself measuring the heat loss of fluids running through pipes. I despaired, and became a refugee in Philosophy and Politics, joined the International Marxist Group and duly got chucked out of Warwick at the end of my second year.
When I was in the sixth form at Bristol Grammar doing Maths and Physics, my much-respected Maths teacher told us that mathematicians cared about the structure and integrity of the mathematics; physicists had a utilitarian, toolbox approach to maths. They just grabbed any techniques which got them the right answer, and never cared about whether they were violating the applicability constraints of the techniques - that is, if they even knew they existed!
I cared, because my motivation for studying physics was to understand how the universe worked and was put together. The mathematical structures were a proxy for the universe itself. A cookbook approach was hopeless, because it violated explanatory power in favour of a black-box preoccupation with getting the "right answers" via mathematical hacks. In particular, I had real problems with the usage of the real numbers.
Mathematical theories couched in the form of functions over R**4 (three spatial dimensions + a time dimension) "explained" the universe in terms of the dynamics of qualities referenced to a four-dimensional real coordinate system. (Quantum mechanics uses complex infinite dimensional configuration spaces, which does not at all alter the point following).
Such coordinate systems are ontologically prior to any phenomena. However, the universe is intrinsic - it doesn't depend on us existing (or methodologically shouldn't!). The coordinate systems are an artefact. So such theories could never model the real universe, but could only correlate observations made of it. My other concern was the denseness and completeness of the real numbers. Too many point actual infinities. This seemed unrealistic - space-time surely couldn't be like that!.
However, if you try to do calculus on restricted sets such as the Rationals (Q), it doesn't work, so I felt completely confused. I found it impossible to communicate these difficulties, and they did not appear in the literature back in the 1970s.
These days, however, I'm a little more sophisticated about mathematical physics, and recently restarted studying physics with the Open University. My objective is to spend as long as it takes to get my head around general relativity and quantum field theory, and to this end I've mapped out a study schedule as follows.
2008: electromagnetism (SMT359). Covers Maxwell's equations and some applications. (Done).
2009: the quantum world (SM358). Basic quantum mechanics. (Done).
Then a move to the maths MSc programme, concentrating on mathematical physics modules including:
- applicable differential geometry (M827). For general relativity.
- 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.
Student "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. "
Faculty: "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.