# Prerequisites for ‘Quantum field theory and representation theory: a sketch’ [arXiv:hep-th/0206135]

I'm interested in reading Dr. Peter Woit's article, Quantum field theory and representation theory: a sketch [hep-th/0206135].

What math and physics background would be needed?

(A list of topics from the two fields would be more than sufficient.)

With great appreciation and regards.

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Why two downvotes? The question seems reasonable to me. –  Pete L. Clark Aug 6 '12 at 18:06
Sadiq, it'd help if you could give us a sense of your mathematical experience until now, since the list of prerequisites for quantum field theory ideally amounts to an undergraduate degree in physics, and for representation theory, not much less than one in math. –  Kevin Carlson Aug 6 '12 at 18:18
(Formal) Math: Up to several-variable calculus Physics: First-year physics (Informal) Math: Very basic and sketchy knowledge of real & complex analysis, some differential geometry. Physics: E&M, first-semester QM. –  Sadiq Ahmed Aug 6 '12 at 18:19

There are two very large subjects central to this paper's topic which it looks like you haven't met much so far: (algebraic) topology and group theory. Group theory is a branch of modern algebra. If your multivariable calculus course was quite good, you'll know some linear algebra; otherwise you'll need some knowledge of the material in a book like Sheldon Axler, Linear Algebra Done Right to understand some of the most important groups. Then you'll need to look at, for instance, the first third of Dummit and Foote's Algebra to get some idea of what a group is, and enough James Munkres' Topology and Allan Hatcher's Algebraic Topology to have a decent idea of covering spaces, singular homology, and maybe cohomology.

Most of the math prerequisites are then more along the line of differential topology. I won't suggest a specific text, but you need to know about vector bundles and fiber bundles, and eventually probably about general sheaves over a space. From that last you'll be able to define the group $K_0$ which is the initiation of K-theory, a central topic of the paper you're interested in. You'll also want some general representation theory, for which I can recommend the book of Fulton and Harris. They may also have all the background material specifically on Lie algebras and Lie groups you need. The last large mathematical area that would come in handy is some category theory, since K-theory is going to be developed in terms of functors-for that you can get Steve Awodey's book Category Theory for free from his website.

That pretty well covers the mathematical aspects, and this sort of paper is really much more math than physics. Not being a physicist, I won't guarantee you don't need to find a textbook on quantum field theory proper, but I bet you could get away without it, and moreso the more you're comfortable with the topics I've mentioned here. It probably seems like a lot, but if you want to be able to fully understand work at this level you'll need all of it. If you just want a decent impression, I'd just spend a few weeks reading every Wikipedia article you can on related topics. Good luck!

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Thank you. - I'm more interested in the physics side / implications than in the purely math. -- Essentially, does this line of study lead to a viable approach to unifying gravity with the other fundamental forces? (I know, I know..) –  Sadiq Ahmed Aug 6 '12 at 19:49
A related question on MO –  Sadiq Ahmed Aug 6 '12 at 19:53
The general principle in theoretical physics is that if you don't understand quite a lot of math, you have no chance of understanding the physics. But, I hope you make some progress regardless! –  Kevin Carlson Aug 6 '12 at 19:55
... Any suggestions on where I should start?... –  Sadiq Ahmed Aug 9 '12 at 16:22
The most direct route would be to start reading the paper until you see a word you don't know, find a definition of that word, and recursively find definitions of all the words in the definitions you don't know until you hit bottom somewhere, maybe in an abstract algebra book. Then build your understanding back up, and go back to the paper. –  Kevin Carlson Aug 9 '12 at 22:13