Mathematical pre-requisites for Relativistic QFT

[hawkingfan]

Hello,

I plan on continuing to study physics, mathematics and Earth science. (independently) What are the mathematical pre-requisites for learning relativistic quantum field theory as smoothly as possible. On MIT's opencourseware, it indicates that a class on advanced ODEs is enough, but we all know that they usually teach the rest of the stuff in the physics class itself. I'm more comfortable with learning the mathematical theories independently of the class.
So far, I have utmost confidence in my knowledge of Newtonian physics, (getting there with Lagrangian and Hamiltonian), electromagnetism, real analysis, calculus (single and multivariable), and linear algebra. I'm still getting there with ODEs but after those, I'm moving onto Fourier analysis, PDEs, topology and differental geometry. (needed for geodesy, seismology, quantum theory and general relativity) The plan is to start at basic Quantum Physics I and end at Relativistic Quantum Field Theory III. (see the opencourseware listings at http://ocw.mit.edu/OcwWeb/Physics/index.htm)

Am I leaving out any mathematical course that would allow for this to go as smoothly as possible? Appropriate books are no problem for me to get my hands on so don't feel the need to leave anything out. Here's the link to MIT's math section. Can you guys help me to pick out the appropriate mathematical pre-requisites?

 

[muppet]

A decent grasp of basic group theory is necessary to some extent. Otherwise, your proposed maths courses seem to cover virtually all physics I've ever come across. However, I wouldn't underestimate how long that little lot will probably take you to get to grips with

 

[Fredrik]

For an introductory class, you already have everything you need. (You won't need the classes about differential equations[1]. Some complex analysis could help, but only a little[2]). For a more advanced class, you need to know some stuff about groups and representations of groups[3], and it would be even better if you know about Lie groups, Lie algebras and their representations[4].

To understand algebraic QFT or constructive QFT (stuff that won't be covered in the QFT class(es) you intend to take), you would need to take classes in advanced analysis[5], integration theory[6], functional analysis[7] and differential geometry[8].
1. It's useful to understand how to find solutions of certain partial differential equations using the "separation of variables" trick. If you understand why writing $\psi(x,t)=T(t)u(x)$ in the Schrödinger equation gives you two separate equations, one of which is the energy eigenvalue equation, you know enough already.

2. It might help you understand an occasional comment here and there that you could almost certainly skip anyway without affecting the grade you're getting.

3. Study the definition of "group" and "representation", find out what group homomorphisms and isomorphisms are, and what the notation G/H means (where H is a normal subgroup of G).

4. Most presentations of the subject of Lie Groups and Lie algebras require that you understand differential geometry first. I know one good book that focuses on teaching only what you can understand without differential geometry (which turns out to be almost everything): "Lie groups, Lie algebras and representations", by Brian C Hall. The book "Modern differential geometry for physicists" by Chris Isham is an excellent introduction to both differential geometry and the basics of Lie group/algebra theory. You might want to get both. If you only get one, get Isham.

5. E.g. "Principles of mathematical analysis" by Walter Rudin.

6. Any book about Lebesgue integrals and that kind of stuff will do. "Foundations of modern analysis" by Avner Friedman contains all you need, but don't get that one unless you like your math books to be just a long sequence of definitions, theorems and proofs with no explanations. (It's actually a very good book if you like that style).

7. Friedman contains the basics. I'm currently reading "Functional analysis: spectral theory" by V.S. Sunder, and I like it a lot, but "Introductory functional analysis with applications" by Erwin Kreyszig may be a better place to start. If you don't believe you need this stuff, try reading some of this.

8. Get Isham's book. It's awesome.

 

[hawkingfan]

OK, thank you guys. It's good to know that I've only left out the subject of group theory and Lie algebras. I will get those books and in 17 - 30 weeks, I'll get started on Quantum Field Theory after learning the math and basic quantum theory. I have nothing but time on my hands and I do independent readings for 10 hours every weekday. (I'm not in college) Thanks for the book recommendations. I will get them all. (especially Isham's) I actually have some of those from the list with me right now.

 

[arunma]

I'm taking first semester QFT right now. Being a math major, I usually feel comfortable with any math they throw at me. But honestly I've never heavily used anything beyond sophomore level calculus and differential equations/linear algebra. All the math you need to know beyond this, they teach you on the fly. For example, they often use Fourier transforms in QFT to derive stuff, but you never end up having to manually do one. As long as you get the basic principle, you're OK. So I wouldn't bother spending too much time on the math.

That's how they teach QFT at my school anyway. Maybe at other posters' schools they teach a version that involves group theory. There's a group theory class in my department (i.e. the physics department offers their own class taught by physicists), but it's not a prereq for QFT.

 

[muppet]

I don't think you need to know a lot about groups, but the basics helps, and often fairly early on. For example, representations of the Lorentz algebra are covered in chapter 3 in Peskin and Schroeder, and in chapter 2 in Weinberg. Also, if you want to look a little beyond the formalism of QFT, then the standard model is constructed using symmetry groups.
One omission I only just noticed from the OP- complex analysis. In a single complex variable will do fine. But in a book like P+S you'll see propagators quite early on, which feature contour integrals in a prominent way.