Mathematical Prerequisites for QFT

Question

I am curious about which areas of mathematics one should be comfortable with before learning QFT. I am familiar with the "learn-it-as-you-go" approach often advocated in physics, but would like to know how to avoid that in learning QFT. Naming of specific textbooks is appreciated.

(For the sake of this discussion, "learning QFT" can be taken to mean "learning at the level of/from Peskin and Schroeder's text.")

In particular, I would like to know what, in addition to the material in Rudin's Real and Complex Analysis, one should know. Would this be sufficient? How about Folland's Real Analysis?

This post imported from StackExchange Physics at 2014-09-12 10:01 (UCT), posted by SE-user Optional

Comments

@KyleKanos Not what I had in mind. I'm only looking for the names of books that point to the knowledge required; I'm not looking for textbooks from which to learn. I think the last paragraph clarified this.

This post imported from StackExchange Physics at 2014-09-12 10:01 (UCT), posted by SE-user Optional

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Real and complex analysis will get you almost nowhere in QFT. You have to have, at the very least, a good idea of Hilbert spaces, operators, group theory, Lie algebras... and I am not even sure that gets you much past the introduction.

This post imported from StackExchange Physics at 2014-09-12 10:01 (UCT), posted by SE-user CuriousOne

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@CuriousOne Some aspects of Hilbert Spaces and operators are covered in the textbooks I mentioned... hence why I asked about what else, in addition to the information in those texts, is necessary.

This post imported from StackExchange Physics at 2014-09-12 10:01 (UCT), posted by SE-user Optional

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@KyleKanos That question only relates to group theory. I was looking for a much more general answer, as well as a specific one relating to Rudin and Folland.

This post imported from StackExchange Physics at 2014-09-12 10:01 (UCT), posted by SE-user Optional

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@Optional: OK, then you got a start on it. Books about real and complex analysis didn't use to cover those topics in the past, hence my skepticism.

This post imported from StackExchange Physics at 2014-09-12 10:01 (UCT), posted by SE-user CuriousOne

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@CuriousOne Can you point to math texts that cover the other topics you mention? Maybe by perusing Rudin's or Folland's table of contents?

This post imported from StackExchange Physics at 2014-09-12 10:01 (UCT), posted by SE-user Optional

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@Optional: My last math class was over 30 years ago. If I remember correctly, the last math book I looked into was Choquet-Bruhat's "Analysis, manifolds and physics", and I don't think that's a good way to get started...

This post imported from StackExchange Physics at 2014-09-12 10:01 (UCT), posted by SE-user CuriousOne

Answer 1

In order to learn or understand introductory QFT you need to understand, vector algebra, vector calculous, complex numbers, the mathematics of operators acting on a Hilbert space, the mathematics of special relativity and some elements of complex analysis, series expansions, Fourier transformations, group theory etc.

I think the best way one to proceed is by physics and not mathematics textbooks. If you understand most topics in non-relativistic quantum mechanics, special relativity and classical mechanics, the QFT textbooks will take you further and complement any lacking skills (usually). In any case, you start learning QFT and when you find a mathematical subtlety you dont understand you stop and cover it. Learning from pure math textbooks, to me, is not the most helpful way to learn QFT. You say you would like to avoid this way, but I think that you would need a really long time by studying only "pre-requisites" and then going into QFT.

Answer 2

You can use as reference work the series of books "Methods of Modern Mathematical Physics" by M.Reed and B.Simon, especially the first two volumes (the first in Functional Analysis and the next on Self-Adjoint Operators) and for a newer look the series by E.Zeidler Quantum Field Theory (three out of the projected six volumes have yet appeared).

Both these works combine very closely the Mathematics and Physics Parts.

Comments

Thanks for the references.  In fact, Reed and Simon's first volume is named by Folland in his QFT book as covering, together with his own Real Analysis text, just most of the mathematical prerequisites.

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Zeidler's book (vol.1) is a thick but excellent synopsis of the required background, with lots of references for deepening one's knowledge.

Answer 3

It depends on whether you want the physical or mathematical exposition to QFT. @conformal_gk already gives a very nice list of prerequisites for the physical exposition; here are some references for the math (http://math.ucr.edu/home/baez/books.html#quantum_field_theory):

• Robin Ticciati, Quantum Field Theory for Mathematicians, Cambridge University Press, Cambridge, 1999.
• Richard Borcherds and Alex Barnard, Lectures On Quantum Field Theory.
• Charles Nash, Differential Topology and Quantum Field Theory, Academic Press, 1991 (this book talks about ``Witten stuff'' (my terminology!))
• This book might be a bit too advanced, but see @UrsSchreiber's book (with Hisham Sati) http://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory.

• Lastly, see Mathematics for theoretical physics, Jean Claude Dutailly. Available online at http://arxiv.org/abs/1209.5665 , 787pp.

Comments

@ SDevalapurkar arxiv tells me that your last reference does not exist. Can you provide a direct link to the pdf? 

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@conformal_gk It should work now.