How Does the Mathematics in Quantum Mechanics and General Relativity Differ?

[nlsherrill]

I usually don't do threads like this, but I wanted to get some opinions for either people in the field, or people who have at least taken graduate level classes in GR or QM.

What are the main math topics in each of these fields? Is there much overlap at all or are the two topics pretty separate? I have taken modern physics, and am currently in spacetime physics(intro GR course), but so far all I have seen is a little bit of calculus and differential equations in both courses. Where does it really separate?

I am a math major as well as physics, and I have a ton of free math electives to choose from. Which courses out of this list would be most applicable to senior level QM or an introductory graduate course in GR?

http://www2.acs.ncsu.edu/reg_records/crs_cat/MA.html

I have to take QM 1 and 2, and want to take GR my last semester in undergrad. I have about a year before then, so if there's anyway I could take some math courses that could prepare me or even put me ahead of the math in these topics, then that would be great.

thx

 

[LBloom]

I'm not sure what courses in that list are specifically for undergraduates or graduate students, but I can give you a good idea about the math you'll need for these topics in general.

For GR you can learn all the math you need without cracking open a math book. Schutz and Carroll give a good introduction to the mathematics needed. I'm more familiar with Carroll to be honest. In general GR requires a knowledge of topics like tensors, Riemannian geometry, differential geometry, and manifolds. You'll obviously also need to know linear algebra.

For QM, you need to be comfortable solving differential equations and linear algebra. Group theory is also useful when you learn about the importance of SU(2) for spin. QFT has a lot of math you need to learn, especially functional integrals when they come up in the path integral approach.

Once more I should add a caveat that the math you need for the course you usually learn while taking the course. Also QFT isn't mathematically rigorous. There's a reason people still study the mathematical foundations and questions about Yang Mills theories are worth 1 million dollars! My advice is to download the draft of Srednicki's book. The introduction has a list of equations you should know before taking QFT
http://web.physics.ucsb.edu/~mark/qft.html

Differential equations, linear algebra, and abstract algebra are very important for QM and QFT and I'm sure there's some things I'm forgetting...oh complex analysis is also needed for QFT. It's helpful to know about residues and contour integrals. Lie groups and algebras will also come up all the time, but that's not typically taught at the undergraduate level, although once again you probably won't need the whole formal framework.

 

[nlsherrill]

LBloom,

Once the course numbers get to 500, they are considered graduate level. That's fine for me to take those. I am in real analysis right now and have taken intro grad level applied PDE's