Mathematical Methods of Physics

There is a sort of mathematical minimum for all the quantitative sciences and engineering: multivariable calculus, linear algebra, probability and statistics, and differential equations. These subjects are now frequently taught in American high schools, but I never saw them before college. Physics requires more. For a long time, much of that has been taught in grab bag courses styled "Mathematical Methods of Physics." In the period just slightly before my grad student days such courses usually concentrated on methods for solving partial differential equations. The classics of that genre were the two volume sets by Courant and Hilbert and Morse and Feshbach. C&H, the original German edition of which dates back to 1930's, is still available in ludicrously overpriced (English language) paperbacks Methods of Mathematical Physics, volume I and Methods of Mathematical Physics, volume II from the blood sucking vampires of Wiley-Interscience. Morse and Feshbach's two volumes Methods of Theoretical Physics are no longer published by the blood sucking vampires of McGraw-Hill, but you can scarf them up for a mere $429 from something called Feshbach publishing - or you try to pry mine from my cold, dead hands.

A new generation of math methods books appeared when I was in school, mostly with a more elementary content and a more modest bulk. One of these was George Arfken's Mathematical Methods for Physicists. My first edition, at 650 pages, was not exactly lightweight, except by comparison with the previously mentioned C&H and M&F. Even though I never used this for a class, at some point I bought the fourth edition, which by then has acquired a squatier body and bulked up to 1000 pages. I notice that the current (sixth) edition has larded up to 1200 pages.

Even with the augmentation, the content is still mostly math dating back a century or two. Chapters on group theory and chaos have been added in a nod to twentieth century physics. Probability makes an appearance too, but mostly it's still about solving those differential equations that come up in Jackson's E&M and Quantum Mechanics courses. All in all, it's a kind of Joy of Cooking for early twentieth century physics - more cookbook than textbook, but I often find it useful.

PDE's are no longer enough math for the physicist, though. The mathematical horizons, especially for theoreticians and particle physicists, have expanded greatly. As a minimum, some topology, differential geometry, some functional analysis and some more algebra seem like mandatory additions for the modern physicist. String theorists and quantum gravitors of all stripes need to know much, much more. So where do you find that stuff?

There are some newer books. Sadri Hassani's Mathematical Physics is an ambitious attempt to address most of the additional new material. He also has an undergraduate version with more elementary content covering a lot of the ground in Arfken's early chapters. I haven't read any of the undergrad book, nor enough of the advanced book to have a solid opinion. Bernard Shutz's Geometrical Methods of Mathematical Physics is a very clear, short, and elementary introduction to manifolds, Lie derivatives and Lie groups, and differential forms. Geometry, Topology and Physics by M. Nakahara has more depth, and the second edition seems to have cooler cover art than my first edition.

There is also a trickle of yet newer books, of which I have no opinion: Peter Szekeres' A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry has an algebraic flavor, and PDEs have been banished. Walter Appel's Mathematics for Physics and Physicists is only a few months old. Mostly functional analysis with a bit of geometry and probability, based on the table of contents. Those things I used to call Green's functions are now Green Functions - ouch!

I would be interested in other's comments on these, other similar books , new and old, as well as relevant math methods war stories.