Tom Lancaster

After reading a Schwinger biography I was in the mood for a more technical book on QFT and I found this one by Blundell and Lancaster. Okay, yes, you need to get through a part of tedious and unintuitive learning. After that and some further practice and read, QFT will become more intuitive to you though. This is the background you need:

1 - To know about the first 3 kinds of free fields (Klein-Gordon 0, Dirac 1/2, Maxwell 1). This is usually on the first chapter of books. Klein-Gordon is intuitive and the easiest, it described the Higgs field. Maxwell you already know, describes the photon. The only tricky one is the Dirac field, but you get used to it. Math required: Spinors and the Clifford algebra for the Dirac field.

2 - Learn the Quantum Electrodynamics Lagrangean, which studies the interaction of matter to light. So it is has a free matter (Dirac) term, a free photon (Maxwell) term, and an interaction term. Then learn about gauge symmetry, which is straightforward if you know the QED Lagrangean.

3 - How to quantize QED. Covariant quantization is very tedious. Learn the Path Integral method better, it's much faster and clear. Math required: Functional calculus.

4 - Since by quantising QED you have in principle a complete dynamics of your system, you can use the theory to study how particles interact. For this you need scattering theory, which is based on a simple assumption: your initial particles come from very far away (at r->infinity) so you assume they don't interact at the beginning (i.e. they are initially free waves). Then they get closer, interact according to your theory, particles can be created or destroyed so you get a new set of final free particles that fly off to infinity. You don't know what will happen, what the final state will be, but you can calculate the probability of each outcome, and for this you use a perturbation series. The first order term is the tree amplitude, it is the easiest to compute. You use Feynman diagrams to help you. There are many algebraic tricks here, nothing abstract though.

5 -The second order term is the loop amplitude, and it has the problem that (naively) it blows off to infinity. To correct this and make the amplitude finite, you need a mathematical technique (or better, set of techniques) called renormalization. There are higher order terms in the series, but these are more tedious and you don't need to compute them. With the loop amplitude calculation you get the general idea.

This is pretty much the whole idea behind a big chunk of QFT. Define your Lagrangean which has the dynamics (the particles and interactions) of your system. Quantize the Lagrangean with the path integral. Compute the scattering of particles using it, use renormalization if you calculate loop amplitudes. What changes is that there are more complex Lagrangeans with more sophisticated quantization and renormalization methods. The Yang-Mills Lagrangean, which is a generalization of the QED one, is the general one used in particle physics. However, the YM Lagrangean only allows massless carrier particles, but the W and Z bosons have masses, so you introduce the Higgs spin 0 field to give masses to the bosons and other particles through the Higgs mechanism. With the Yang Mills Lagrangean and the Higgs mechanism, you have pretty much the whole Standard Model covered, enough for most applications. If you're more theoretical though, you will want to learn non-perturbative QFT (which deals with global effects of a system), and that is a whole other beast, in my opinion much more complex than perturbative QFT.

After this, QFT is really a tool used in very different systems, it bifurcates a lot. You can study QFT in curved backgrounds (entering into Hawking radiation, black holes, etc.), in condensed matter physics, in Supersymmetric theories and Supergravity, in String Theory, in holography and so on, each of which have their own mathematical techniques and physical intuition. At this point you have to specialize, although there is quite a bit of cross-breeding between subjects. I think it's safe to say none knows every "branch" of QFT at depth (and you don't need to). It's just too vast, so you pick what you can and work from there.

Tom Lancaster and Stephen J. Blundell's book unfortunately (or fortunately, depending on which side of the barricade you are) belongs only to the "shut-up-and-calculate" school. That means no ontological insights. If you’re only interested in the technical side of QFT this the book for you.… (mere)