leright said:
OK, I want to self study classical EM in greater depth than my electromagnetics course, and I plan on working through Jackson (I am currently an undergrad). I want to get into the relativistic treatment of the subject. However, to properly tackle this, I feel my mathematics background is lacking.
I have of course had u. phys 1 and 2, and electromagnetics (textbook I used was elements of electromagnetics by N.O Sadiku). The mathematics I have had includes, calc 1, calc 2, calc 3, diffEQ, Adv. eng math (complex analysis), and prob/stats. I, surprisingly, have never taken a linear algebra course.
Okay. Some partial differential equations would help if you're going to attack Jackson. You probably learned the meat of what you need to know from your undergraduate (presumably Griffiths-level) E&M course, so you may or may not need to dig too deeply into the formalism. The first thing that will be mathematically more advanced in Jackson's text is the introduction of Green's functions. In E&M you can think of Green's functions as the potential from a point source, from which you can build up the potential of a charge distribution. In math, they are the solutions to PDEs that prodce delta functions. In QFT they are the propagators that tell you how a particle travels from one point in spacetime to another. They're pretty important--I'm not sure if a formal study of them from a mathematical point of view is necessary, but it couldn't hurt.
I should also warn you that there are a lot of details in Jackson that aren't necessarily prerequisite for the course of study you're proposing. In fact, you might want to intersperse a little bit of advanced mechanics in your self study. It helps to see the Lagrangian and Hamiltonian formalism of a classical field theory and how to manipulate some slightly more formal things, such as poisson brackets and such. (And if you can geometrically quantize a classical field theory into a quantum field theory, you're automatically a rockstar in my book.) Some recommendations: Marion/Thornton will do the trick, though if you really like the subject, you might want to dabble with some of Arnold's Mathematical
Methods of Classical Mechanics (the latter is somewhat advanced for physics students).
So, that brings me to my next point. Before I finally take a linear alg course, what book would you recommend for self study in linear alg?
You're probably not missing too much. You learn a lot of linear algebra in quantum mechanics, but conversely quantum mechanics is a piece of cake if you have a strong linear algebra background. A basic introduction can be found in Bretscher's text. A more advanced text is Apostol (which I think is multivariable calculus/linear algebra in a unified approach). I believe the latter talks about some more formal things for quantum mechanics ("hermitian operators" or "self adjoint operators" are the key words that you want to look into when studying sections).
Also, I want to self study tensor analysis. What book would you recommend for this?
Tensor analysis is usually first introduced to physicists in general relativity (GR). You'll pick up some in Jackson. You might want to try to read the first few chapters of Sean Carroll's GR book. Or find the GR book which 'speaks' to you. (Everyone has their own ideal textbook.) You can look into D'Invierno, Schwarz, or even Misner Thorne Wheeler (if you're really perverse and like antequated language).
I also would like another go around in vector analysis, just to make sure I know the subject like the back of my hand. I have already studied the subject from Stewart's calculus. What book would you recommend for a more in depth treatment of the subject?
Maybe the Apostol book mentioned above. But hey, you'll learn/review more vector calculus by going through Jackson and making sure you understand every step and do problems. I wouldn't recommend spending too much time reviewing vector analysis math textbooks unless you feel particularly weak in certain topics.
Are there any other fields of mathematics I should self study? After tackling classical E&M I would like to self study relativity (beyond what is introduced in Jackson), Quantum Mechanics, QED, and finally QFT, all at the ugrad level. What mathematics should I be comfortable with to properly tackle these subjects.
You mentioned you had complex analysis, so that's very good (you don't really use it until QFT and it can sneak up on you if you're lazy). You should also have a good background in Fourier analysis. This is usually taught in a PDE course, though some schools have separate Fourier analysis courses. I have no good textbooks to recommend to learn this. You learned some in Griffiths. The only PDE book I'm really familiar with is Stewart, which I didn't think was particularly illuminating (but it will teach you about Green's functions and Fourier transforms). Perhaps a mathematical physics book (Boas or Arfken).
For Quantum Mechanics the standard "first book" is Griffiths. It's exceptionally clear and well written. You'll want to go a little more advanced than this, however, and Shankar is very good for self-study. This will get you caught up to just about graduate-level quantum. There are other 'classic' texts to read select topics from: Sakurai, Schiff, Merzbacher. But don't get too bogged down going through entire chapters that you've read in other texts already.
For quantum field theory, first read a particle physics book such as Griffiths' Elementary Particles or Halzen & Martin or Perkins. Learn how to calculate Feynman diagrams, even if it's somewhat mysterious at the time. I would then suggest a combination of the pink oxford press book ("A Modern Introduction to QFT" or something like that) and Zee's Quantum Field Theory in a Nutshell. After this, you can easily graduate to texts such as Greiner (Field Quantization, Quantum Electrodynamics) and Peskin & Schroeder.
Best of luck with your self study.
-F
