Kimball A. Milton

I still remember the first time I came across Schwinger’s take on QFT. Before reading Schwinger, I had tried to read (skimming) few of QFT books out there to get the general ideas. Reading at the TOCs alone, they always felt like a beast, with lots of chapters even if their page numbers were not always that much for physics textbooks…I also read quite a bit of standard introductory textbooks in the field (pun intended), but I never seem to get myself a "click" on what was it all about. Maybe it was from how I just read and never actually get myself wet with pen and paper, or it was that my QM needed more refining, or both, anyway I had never seem able to make sense of what is it all about (I did learn bits about canonical quantization of field by imposing commutation relation on them, and bits about creation and annihilation operator). Then I thought most of my problems seem to be in how there seem to be a whole loads of seemingly unrelated maths in them, so the core structure of QFT didn't seem to correspond to specific math fields, like how differential geometry was to GR, or because there seemed nothing in math for QFT as explained in intro texts (e.g., perturbative QFT). So I set out to read real math books, and there does seem something interesting like algebraic QFT that used sheaves of observables, that felt more intuitive (it looked like to be Heisenberg formalism but on local patches of space).

So what's Quantum Field Theory all about? Was it like QM but with general field as its states? What's the differences between QFT and textbook QM in its math methods? In normal quantum mechanics, you've surely heard about the uncertainty relation between position and momentum. This is inspired from the classical view that particles are dots, which have properties such as a position and a velocity or momentum (their trajectories are lines), and then applying quantization to them (so now the particle is in many places "at the same time"). Now, in reality particles such as the photon are not dots. A more complete theory of photons describes them as electromagnetic waves/fields, with a given frequency, polarization and so on. So if you want to describe a photon in a quantum way, you have to quantize this theory, the field theory, not the antiquated dot photon theory. This is where QFT comes in. There is no longer a dot with position and momentum to impose an uncertainty on, but you have the photon's electromagnetic field. So you impose the uncertainty relation on, or quantize, the field itself: the field now doesn't have a definite value until it is measured, there is a superposition of many field configurations, just like in the dot case there were many positions superposed. The photon now is described by the quantized field, or quantum field.

Now the photon is not the only particle out there: there are electrons, neutrinos, and so on. Each has its own field that has to be quantized too. In general there are 5 types of fields which you deal with in QFT, classified by their "spins". In particle physics you see only 3 of them: spin 0 (Higgs), spin 1/2 (matter, like electron or quarks) and spin 1 (photon, gluon, etc.) fields. If you are interested in unification with gravity and supersymmetry, there are two other fields: spin 3/2 (gravitino) and 2 (graviton), which have never been observed, but are theoretically allowable. There don't exist higher spin fields because of a theorem that says they're inconsistent. So these are the basic building blocks of QFT, the 5 types of (quantum) fields.

Now, in general you're interested in the interactions between these particles (what happens when two electrons collide, or when a photon collides with an electron?). So you make the quantum fields interact, but the interactions make the whole thing very messy, so messy in fact that finding an exact solution is often impossible. Because it is impossible, you use many different mathematical frameworks and tricks, like Perturbation theory with its Feynman diagrams, Non-perturbative methods when perturbation theory fails, seek for symmetries that hopefully simplify your calculations, and so on. However, this is just the beginning: QFT is really an enormous beast, and it is a very general framework to compute particle interactions, full of brilliant physical insight and mathematical structures.

On the differences with normal QM: In philosophy, they are at first sight similar: the field quantization is akin to the dot particle quantization, the QFT perturbative theory is similar to the QM perturbation theory and so on. However, the mathematical methods and structure of QFT are much, much richer and technical than normal QM, and the connections with other areas of physics such as string theory, quantum information, and condensed matter and so on have filled it with insights which have no normal QM analogs. Furthermore, QFT is still under intensive research, and new structures are discovered each decade.

In QM, you quantize a dot's position and momentum. In QFT you quantize a field. There are many fields so you care about how they interact which is messy and very mathy. QFT is much more mathematically rich than regular QM, but philosophically the principle is similar.

As I wrote elsewhere, I don’t “believe” in QFT. But I can surely appreciate Schwinger’s work as masterful.… (mer)