Math Textbooks to Read Before Starting QFT

[tas_dogu]

So, I am a 4th year physics student (undergrad), and I will do my master, then hopefully my PhD on theoretical physics. In my university, we take 1 year QM/Classical Electrodynamics/Classical Mechanics in terms of physics then the second year we pick the classes we want to take, I will take QFT I and II for sure, I haven't decided on anything else yet.

I want to keep a relatively high GPA as I did in undergrad (currently 3.91) so I decided to take a gap year to study the masters courses by myself, I have done this before, and it worked like wonders to me, I have already read Jackson's, Goldstein and Shankar, which I assume is a good thing, my uni uses Sakurai's books for grad school QM, so I am going to be reading that next.

Here's my question, after I read Sakurai, I want to get a good foundation for math I will need to study QFT (and maybe string theory), because to me, if I understand the math behind it completely I can really learn the intuition and physics behind things. So, which books should I read? List anything that is relevant to QFT. Thanks in advance!

 

[shinobi20]

Nice question, this issue also baffles me. I've taken QFT before, it seems to me that there is no particular book/resource that deals specifically with the "math" of QFT. It's apparent to me that the math used in QFT is quite diverse and there is no obvious structure that tells you, e.g. it's calculus, it's complex analysis, it's functional analysis, etc. In general, it involves some Fourier analysis and distributions, complex analysis, group theory (particularly Lie groups), some tensor analysis. Based on my experience, I haven't seen these compiled in a single book, I mean of course, writing a book that contains the whole of each subject is impossible.

However, I'd also like to know if there is a book that at least touches upon the important topics of each subject that is really used in QFT.

 

[tas_dogu]

Yeah, I haven't come by any book that actually touches up on all subjects mentioned, that's why i decided to read multiple books on certain subjects but i really can't decide on which books to read, all my tensor analysis knowledge comes from Boas' book on math for physics, for topology, I've seen people mention Munkres' book and Hatcher's book also Nakahara's book but i don't think I really need to read 3 books for Topology needed in QFT, do I? Another example for this is Differential geometry, people told me to read Tapp, Axler and Spivak(Whole 5 volumes!). I know that these are very important topics and the more I know the better but realistically, I am not going to read all these in my entire Ms period. I want some books to get me up to speed and i will properly learn the details and more in depth stuff as i go along. Does that make sense? So if anybody knows books to get me up to speed on these subjects, i would appreciate if you post the names here. Thanks in advance again

 

[WWGD]

Well, a good chunk of Math itself is covered in Advanced Calculus classes, with Calculus meant in the broader sense, like some Linear Algebra connected with Calculus , re derivatives as linear maps, Jacobians, etc.
There are good quality free books available online.

 

[romsofia]

Not a textbook on the math, but the appendix of this book gives you enough to see what topics you need to cover a little more in depth: "Introduction to Quantum Fields in Classical Backgrounds" by V. F. Mukhanov and S. Winitzki (which they uploaded a draft for free https://uwaterloo.ca/physics-of-inf...-information-lab/files/uploads/files/text.pdf)

It's only 15 pages or so, but it should help. The book is also pretty fun IMO.

 

[haushofer]

I once wrote some personal notes on QFT about stuff I was bothered with; the basics of i-epsilon prescriptions, contour integration, Planck units and renormalization; things like that. Somehow you'd say somebody wrote a textbook with this kind of techniques used in QFT. But apparently it's trivial for many people :P

 

[romsofia]

But apparently it's trivial for many people :P

As I've gotten older, I realize some people are really good at just accepting that's how something works, and using it accordingly. I think these people can easily accept things as definitions.

When I'm presented with a definition, I often struggle to motivate why something is a definition, and this leads me to rarely see anything as trivial. This is also why I never pursued pure math courses after my first few experiences with them LOL, I would overthink proofs. So, you may be like me where you look for the motivation behind a definition vs just accepting a definition and focusing on how to apply it.

Both types of people are needed in research though. Sometimes I'll get stuck on some reasoning that's not really relevant to the problem. The others sometimes won't be able to see more abstract connections since they never cared for the motivations.

 

[haushofer]

As I've gotten older, I realize some people are really good at just accepting that's how something works, and using it accordingly. I think these people can easily accept things as definitions.

When I'm presented with a definition, I often struggle to motivate why something is a definition, and this leads me to rarely see anything as trivial. This is also why I never pursued pure math courses after my first few experiences with them LOL, I would overthink proofs. So, you may be like me where you look for the motivation behind a definition vs just accepting a definition and focusing on how to apply it.

Both types of people are needed in research though. Sometimes I'll get stuck on some reasoning that's not really relevant to the problem. The others sometimes won't be able to see more abstract connections since they never cared for the motivations.

Yeah, I recognize that. When I followed a course on QFT, I just wanted to be able to calculate some amplitudes and understand renormalization. Only after that course the questions kicked in: why are spacetime suddenly mere labels? Don't we have a position operator anymore? What's the ontology of virtual particles? What's the meaning of those divergent sums in your amplitudes? How does one calculate with path integrals (i.e. how would one write code in Python to calculate numerical values with it)? Etc. etc.

I have the same experience with the notion of general covariance in General Relativity.

 

[mathwonk]

When I was a young researcher (in pure mathematics), incompletely trained (by my own fault), I learned to use information I did not understand. I was happy to find out that was possible, as it helped me make research progress in less time. There was some loss of mental security since I had to base the confidence in the correctness of my work, on the correctness of work I took for granted but did not fully understand. I.e. in this approach one is working from statements one has read, rather than an understanding one possesses.

Now that I am old and retired, I enjoy going back and revisiting those concepts, which have not persisted in memory. I am now surprised that I once had enough temporary grasp to actually use them profitably. But my goals have changed: once I wanted to compete in a race to solve problems and obtain funding; now I want to understand and savor the beauty of the ideas. With that understanding, I may also better help train young people to run their own race.

By way of advice, when trying to utilize concepts one has not studied thoroughly, it is very helpful to at least discuss them with someone who has. Apologies, since I do not know anything about the current topic QFT, but I hope this might be relevant to the recent remarks on methods of research.

 

[Demystifier]

But my goals have changed: once I wanted to compete in a race to solve problems and obtain funding; now I want to understand and savor the beauty of the ideas.

That's a big problem in most fundamental research nowadays. The goal of fundamental research should be the increase in understanding, not the increase in the number of published papers. Unfortunately, this is often not the case. Sorry for the off topic!