Physics-Math or Math-Math

[the_kid]

Hi all,

I have a question regarding what I should about my math education going forward. I'm currently a sophomore, aiming toward physics graduate school (probably in HEP-T). My school offers a graduate physics course on mathematical methods. The description is below:

"Survey of mathematical techniques useful in physics. Includes vector and tensor analysis, group theory, complex analysis (residue calculus, method of steepest descent), differential equations and Green's functions, and selected advanced topics."

Should I take this or should I take the full-fledged math courses (Vector Analysis, Complex Analysis, Abstract Algebra, etc.)? Obviously, the latter would clutter my schedule and make it more difficult to take many advanced physics courses. I'm wondering where the trade-off leaves me. Will grad schools look down upon me for taking the grad mathematical methods in physics vs. the real math courses? What should I do?

Thanks!

 

[the_kid]

Also, I should note I've taken the full calculus sequence, linear algebra, differential equations, and real analysis thus far.

 

[the_kid]

Surely someone must have an opinion on this! What makes the most sense for someone in this situation?

 

[Fredrik]

I think the "methods" course is sufficient for a theoretical physicists, but not for a mathematical physicist (someone expected to do rigorous proofs). As a grad student in theoretical physics, you will probably be required to study some of the math books on your own, but not to enroll in math courses with math students. However, different universities probably have different attitudes about these things, so don't take my word for it.

If you are considering "real" math courses, I would recommend real analysis, topology and differential geometry. The first two will give you some of the prerequisites that you will need if you ever decide to study functional analysis, and increase your mathematical maturity a lot. (For many courses on functional analysis, measure/integration theory is also a prerequisite). Differential geometry is necessary to understand GR well. I don't think a course on abstract algebra will be very useful. You will need the basic definitions (group, ring, field, ordered field, vector space, homomorphism, isomorphism) from a book on abstract algebra, but you can learn those by studying the book on your own for a few hours. Complex analysis is more useful than abstract algebra, but not so useful that you can't do without it.

 

[the_kid]

Thank you the reply, Fredrik. What do you think about math courses such as Lie Algebra and Representation Theory. I know they both come up in certain aspects of theoretical physics, but I'm not sure if it's worth taking a full course on one or both of them. I have the opportunity to do so next year, so please let me know what you think. Thanks again!

 

[Fredrik]

You need those things to really understand the particle concept in QM. If you take a look at chapter 2 of Weinberg's QFT book, you will get an idea. So it can't be bad idea to take those courses, if you're going to continue with something that involves particles. However, it's also a topic that can be studied independently. The book by Brian Hall is very readable. (I think the title is "Lie groups, Lie algebras and representations"). I should also mention that some courses on Lie groups and algebras require that you know some differential geometry. Hall's book is based on the observation that you can avoid differential geometry completely if you focus on matrix groups.

You have almost certainly already studied something about representations of lie algebras without knowing it. The argument that finds all the possible values of the spin quantum numbers is also finds all the irreducible representations of the Lie algebra su(2).