I’ve pushed on further in my attempts to learn Quantum Field Theory. (Thank you to everyone who commented on the previous post.) I’ve picked up a second textbook, Ryder’s *Quantum Field Theory*, whose precision balances Zee’s intuition very well. I don’t have so many questions this time, just ideas which I am imperfectly exploring. Let me try to explain what I learned this weekend, which is how to write down a bunch of massive charged spin-zero particles interacting with an electromagnetic field.

When I first learned electro-magnetism, I thought that it was very inelegant that electrons were particles, with particular positions that change according to the Lorentz force law, while light was a field, with an intensity at every position in space that changes according to Maxwell’s equations. I tried to imagine what a field theory would look like for electrons, by imagining an infinite number of charged particles, with infinitesimal charge, all obeying the Lorentz force laws. At first, I thought I would just make a field which, when integrated over any region of space-time, would give the total charge in that region. Later, I realized that I needed to keep track of the momenta as well and imagined a vector-valued field which, integrated over any region of space-time, gave the total momentum in that region. (This history is viewed through the rose-colored classes of hindsight.) If I kept going this way, I would have invented the (charge density, current) four-vector. This field is usually called J.

Still later, I realized that this wouldn’t work either. Here is the reason. Imagine two particle beams right next to each other, with the same particle density and velocity. The particles in the two beams have the same mass, and opposite charges. Then the J-field would be zero, so we couldn’t distinguish it from just a complete absence of charge. From the perspective of Maxwell’s equations, this is true. Two parallel beams of this sort generate no electro-magnetic field. However, from the perspective of the Lorentz force equation, this is **not** true. If our two particle beams pass through a transverse electro-magnetic field they will be separated, one curving to the left and the other to the right. Thus, the future value of the J-field can not be predicted from knowing the present value of the J-field and knowing the electric field. At this point, I sort of gave up on the project, figuring that all you could do was to imagine a probability density on the state-space of an electron.

It turns out that there is a much nicer answer. It doesn’t even require any complicated math; it could have been a bonus chapter in the second volume of Feynmann’s lectures.

Continue reading →