Thanks to Charlie’s original idea and further insistence, I am taking three graduate level math classes this semester, which is hopefully going to end up in a masters degree in Mathematics. At first, it didn’t look like a necessary or even particularly useful move, but now It’s very clear to me that I should have done this at least a couple years ago. I would even go so far as to suggest that any Physics PhD student should be advised to do the equivalent course work of a Masters degree in Mathematics (or Applied mathematics) along with their normal Physics program. There is so much sophisticated Analysis ( mostly Functional Analysis) involved in advanced theories of Physics from condensed matter physics to QFT, that the few Calculus and Mathematical Physics courses one normally takes as an undergraduate student simply can’t cover the whole spectrum, and whether you can pursue a successful research career in Theoretical Physics may well depend on whether you’d go out of your way to take advanced Analysis courses before it’s too late. In particular I recommend the following book to all future Physicists :
Applied Functional Analysis, by D.H.Griffel
It is a very readable and well-organized book that puts several of the most obscure techniques of Mathematical Physics on a rigorous mathematical foundation. Topics include:
- Distributions and Generalized functions
- Fourier Transforms
- Green’s Functions and their applications in differential equations
- Banach spaces, Function spaces
- Hilbert spaces and the Theory of operators on Hilbert spaces
- Variational methods
It also has several clear and concise reviews of classical analysis in Appendices, which makes it a pretty much self-contained text book given that you already have some training in basic calculus and analysis.Highly recommended.
