# Self Teaching QFT

I am currently in the process of teaching myself QFT. It is not an easy task. I have armed myself with many of the standard textbooks. However, I am slow learner. I get stuck on a thousand different terms which I don't understand, and a style of 'mathematics speak' which loses me 95% of the time. There are so many gaps in my knowledge that if I were a ship I would be at the bottom of the ocean. What I really need is a book called 'QFT for Morons'. That's a bit harsh but 'QFT for dummies' doesn't quite express it clearly enough and besides there isn't such a book as far as I can find.

Any suggestions of resources or books that might help? What I am really looking for is the one that explains QFT like you were talking to a five year old. Perhaps I am asking the impossible.

I will add a bit more of a specific question here. I have learnt that there are groups and non-abelian Lie groups like SU(2) etc., and then there a lagrangians, as well as a few other things like operators and states. So a really dumb question is how are the lagrangians derived? Are they determined by the group? I am also confused because I understand some of the groups like SU(3) are used for different things in QFT. I feel compelled to appologize for my ignorance but I am determined to make so more progress with this. The problem when one is teaching oneself is you don't know which is the most fruitful path to follow.

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• – HDE 226868 Dec 28 '15 at 21:46
• About the lie algebras, groups, the Physical Mathematics book will help with that. As far as Lagrangians go, most QFT books will give a review of it in the primary chapters, with emphasis on the field theory. Though if you are uncomfortable with Lagrangians altogether, you should find a book on classical mechanics. You have a lot of choices with that. My two favorites that come to mind are Sudarshan and Mukunda, and one by the name of variational principles in classical dynamics and quantum theory. The latter is a cheap dover and is a nice historical introduction. The former is advanced. – Marcus Quinn Rodriguez Tenes Dec 28 '15 at 22:06
• I think Itzuykson is not a good book for this person, he is struggling with for dummies. I have personal experience with Itzuykson as being a difficult book, even in the early chapters. – Marcus Quinn Rodriguez Tenes Dec 28 '15 at 22:19
• Just be easy on yourself, realize that QFT is the culmination of literally hundreds of years of physics and mathematics achievements. Lie groups, algebras, fields, creation and annhilation operators, fourier analysis, this is so much to know! I have heard people estimate that it takes 10 years to get good at this field. – Marcus Quinn Rodriguez Tenes Dec 28 '15 at 22:21
• It is impossible to learn QFT, or any advanced topic, starting with nothing. Why do you think we physicists spend years studying introductory topics in school before tackling something like QFT? We aren't idling away the primes of our lives studying centuries-old stuff just to waste time before getting to real research. There's a bit of hubris in thinking you can learn a subject while avoiding all the years of prep work and many textbooks every professional in the world has found to be necessary. – user10851 Dec 29 '15 at 0:49

Lagrangians are ''derived'' (if they are not simply assumed) by assuming a symmetry group and a collection of fields with given transformation behavior under the Lorentz group and then listing all expressions in these fields that are invariant under the symmetry group. Together with the requirement of renormalizabilty that limits the total degree (but is not always imposed), this fixes everything apart from a number of constants. These can be reduced a bit by linear field transformations, leaving the typical Lagrangians.

For example QED. You assume a vector field $A$ and a spinor field $\psi$. Power counting degrees are 1 for $A$ and $\partial$ and $3/2$ for $\psi$. Renormalizability requires the total degree to be at most four, and the spinor fields must appear in each term an even number of times. This leaves linear combinations of $\partial^rA^s$ $(s>0,r+s\le 4)$, $\psi\psi,\partial\psi\psi, A\psi\psi$, with arbitrary indices attached. Now require Lorentz invariance to figure out the possible indices. The Lagrangian must be a Lorentz scalar. This leaves only a few combinations. The terms with $A$ only give the electromagnetic kinetic energy term $dA^2$, a possible mass term $A^2$, and terms with a factor $\partial\cdot A$. Requiring also $U(1)$ gauge invariance leaves only $dA^2$. Similarly, the terms with $\psi$ only lead to the Dirac kinetic energy term $\bar\psi \gamma\cdot \partial\psi$ and a mass term $\bar\psi\psi$. The only interaction term possible is $\bar\psi \gamma\cdot A\psi$ (wlog since $\psi$ and $A$ commute). This are precisely the terms occurring in QED. The coefficient of the two kinetic terms can be fixed by scaling the two fields, and their sign is determined by the fact that the classical action must lead to a Hamiltonian that is bounded below. Thus QED is completely determineed by its symmetries and its gauge character.

No book on QFT is easy to read but the quantum field theory book by Weinberg is the entrance that explains most of the terms in a clear way. You need as background some thorough exposure to classical mechanics, classical field theory, ordinary quantum mechanics, and Lie algebras. Yoiu can get some of this background through my online book Classical and Quantum Mechanics via Lie algebras, Also see Chapter C4: How to learn theoretical physics of my theoretical physics FAQ for hints how to organize your learning.

• Hi Arnold - could you enlarge a bit on the first paragraph, perhaps by giving an example? I feel like I am close (ish) to understanding what you wrote but I am not quite there yet. Your second paragraph makes very good sense to me and makes me feel a lot better about what I am trying to do. Thanks – Peter Hunt Jan 1 '16 at 20:13
• @PeterHunt: done – Arnold Neumaier Jan 2 '16 at 6:42
• Thanks Arnold. I will need to read your answer carefully a few times. I have now gone back to Robert D. Klauber's Student Friendly QFT. For a person in my position this follows nicely from QFT. I will go and have a look at Weinberg's book again later on. Right now I am reading chapter 3 about the Klein-Gordon equation. For once I do feel more comfortable with the maths aswell as being able to comprehend what is being taught. I think it is about pedagogy. People who study QFT tend to – Peter Hunt Jan 3 '16 at 13:19
• Arnold Neumaier -- Wow, thank you, that is a wonderful explanation. The fact that I don't yet understand all those rules does not diminish how helpful your answer is. Until now I have been mystified by the vague, almost dream-like pedagogy that alway seems to surround this topic. Now that I see the entire construction in outline, I think I understand why the Lagrangian cannot be derived by mathematics alone. It also needs to satisfy the numerous requirements you describe, which are specific to the physics. – Ralph Dratman Apr 15 '17 at 1:22

Before this question gets closed...the most moronic (in a good way, lol) QFT book I know of is McMahon. A lot of complaints on amazon about it not being the real thing, but it does cut to the chase about a lot of things and actually shows you what some of the equations look like and what is going on. It gave me a legitimate appreciation of the Standard Model a la non-abelian gauge theory and the Higgs mechanism. If you want to learn how to make real world calculations, that is going to be a whole different mess. It depends what your motivations are. Try to keep your motivation in mind a lot, otherwise you are going to get lost in a lot of books. Realize a lot of the QFT books out there are written by and for particle physicists who plan on working on the actual experiments and calculations. If you just want the theory, there is a lot you can skip. Knowing what to skip is difficult, but my best advice is try not to get sidetracked into many different books, especially those of an advanced level. I think the Gauge Theories book by Aitchinson and Hey (the newest edition) is one of the gentlest ways. There is another book by Ulrich Mosel, Fields and Quarks that is relatively succinct. You might need some help with the math, and for that I suggest Physical Mathematics by Cahill. And a reference for QM is always useful, Dirac's Principles of QM is topnotch.

• Thank you very much. I look at each of the resources you have mentioned and let you know how I have got on. One book that I was reading which I found helpful was Quantum Field Theory Demystified. I also watched some videos of lectures by Milo deKoch on group theory. I got stuck on Scher's Lema. But I got fairly clear about representations and ireps as well as Lei algebras. I've returned to reading another book which calls itself student friendly. – Peter Hunt Dec 29 '15 at 2:29
• I think your understanding of group theory is probably sufficient enough already for most of QFT. Like I said, you don't always need every topic a given book presents. For example, though you might need to know Lie Algebras for QFT, if you pick up a book on Lie Algebras written for mathematicians, it will go far beyond what you actually need to understand the equations of physics. So I suggest sticking to books and resources centered around applications to physics. That's my reason for the Cahill book. You can find your own way with books you might prefer more. Use google preview. – Marcus Quinn Rodriguez Tenes Dec 29 '15 at 14:37
• Hi Marcus - I realized now that McMahon is one of the books that I have been reading. I agree with everything you said about it. It seems to be the most imediately accesible . I really like that fact that it also has simple problems/questions at the end of each chapter WITH ANSWERS! For a student like myself, struggling on their own this is what one needs in order to get started. I found quite a lot of other resources in the last few days - a lot of concise pdf's from various universities. – Peter Hunt Jan 1 '16 at 20:06

"Quantum field theory for the gifted amateur" by Lancaster and Blundell is good. It would be good for you to know some basic calculus: integration, differentiation, Fourier transforms. It would also be useful to know some special relativity and quantum mechanics at the undergraduate level. If you have that knowledge, Lancaster and Blundell explain what sorts of problems arise with trying to come up with a relativistic version of quantum mechanics, how quantum field theory solves that problem and explanations of mathematical techniques like variational functional differentiation, the variational principle, renormalisation and Feynman diagrams. It also has good a explanation of how to find a Lagrangian: make the simplest guess that includes the features you want to consider.

"Quantum field theory in a nutshell" by Zee is a bit harder but might be worth considering.

• I found the whole book hopelessly deluded in its purpose. To learn QFT, you have to start with a good grasp of classical mechanics, electromagnetism, relativity, quantum mechanics. it would take at least a year of self-study to get through just one of these while holding down a job at the same time. The book you've recommended is likely to be of use to a good physics undergraduate only. – John McVirgo Jan 7 '16 at 0:51