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.
Best Answer
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.