For the few months, I’ve been trying to learn Quantum Field Theory, working mostly from Zee’s book Quantum Field Theory in a Nutshell and also from John Baez’s course notes. Note that my goal is to actually understand physics, in terms of actual physical objects interacting, not to just understand mathematical constructions. This is going to be the first of what I imagine will be a series of posts where I ask dumb questions and hope that AJ, John Baez or one of our other physcist readers will help me out. For my first question, I’m going to ask about something which has been bothering me since the very start of the book.
Question 1: True or false — there are two kinds of particles in QFT. There are force carrying particles and ordinary particles. The former are described by wave functions on the space of solutions to the classical field equations while the latter are described by wave functions on spacetime.
Sometimes Zee talks as if the answer is “true” and sometimes he talks as if the answer is “false”. I have slowly come to the conclusion that he implies “false” more strongly. Assuming that “false” is the corect answer:
Question 2: What is the classical field whose excited states are electrons?
My best guess is that this is the current J — the vector field on spacetime (four-vector field, in physicist terminology) whose time component is the charge density and whose space component is the “density” of charge x velocity. Following through this idea, I tried to write down a Lagrangian for the evolution of J and ran into trouble. I know the Lagrangian for an individual particle of mass m and charge q, traveling in an electro-magnetic field with potential A (a vector field). If this particle travels a parameterized path y(u) through spacetime, the Lagrangian is
Here is the relativistic inner product . I checked, and this does give the Lorentz force. (I was going to write this up, but John Baez already did.) Based on this, I tried to guess the Lagrangian for J, but something seems to have gone wrong:
Question 3: Is the Lagrangian for the evolution of the J field given by
where the integral is over all spacetime and is the ratio of electron mass to electron charge? If not, what is the right Lagrangian formulation? For that matter, what is the statement of the Lorentz force equation in terms of J?
I feel guilty about composing a post which is all questions but no answers. In partial atonement, here is a list of resources that I found helpful in getting me this far. For a nice introduction to Lagrangians, see the lecture on the “Principle of Least Action” in Volume II of Feynman’s lectures. To learn more about Lagrangians by working examples, see David Morin’s notes. (I am still working from the physical copies that Morin gave me nine years ago, but I think my link goes to identical files.) To convince yourself that those pretty path integrals you’ve heard talked about really do lead to Schroedinger’s equation, see John Baez’s notes from the Quantization and Cohomology Seminar. To convince yourself that solutions of Schroedinger’s equation do (sometimes) look like particles in motion, see John Baez’s Photons, Schmotons page. This final one may only be useful to people working from Zee’s book specifically — Zee starts by presenting a scalar field theory where the force carrying particles are massive but don’t interact with each other. I found it very helpful to discover that the classical analog of this theory is discussed at the end of the lecture on “Electromagnetic Mass” in volume II of Feynman’s lectures (the section entitled “The Nuclear Force Field”).
Thanks in advance for any hints!