Help David learn Quantum Field Theory II

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.

A few caveats before I go on. First of all, electrons have spin, and properly describing the electron theory would require going into this. I’m going to dodge the issue by talking about a hypothetical massive charged spin-zero particle instead. This particle will be callen a zeron until someone in the comments tells me the correct name. Secondly, I can describe this theory, but I can not motivate it. Any suggestions of how to do the latter are highly welcome! Finally, both of my books set c and \hbar=1. I find it convenient not to do this, because I like to take limits as c goes to infinity and \hbar to zero. I’ve reintroduced these constants below in a way which is consistant, but I don’t know whether or not it is standard.

Ok, the zeron field will be described by a field with two components, called \psi=(u,v). Often it will be convenient to think of \psi as taking complex values and write \psi=u+vi for some real-valued functions u and v. However, this is solely a convenience. We are still working with smooth functions, not analytic ones, and in particular our space-time coordinates are still real. Space-time is not assuming a Kahler structure, nor are we doing analytic continuation of our solutions. Perhaps I am beating a dead unicorn, ruling out exotic scenarios that you would never consider, but this confused me so I want to be clear about it.

The charge density is given by
\rho=u (dv/dt) - (du/dt) v.
Similarly, the x-component of current is given by
J_x/c=u (dv/dx)- (du/dx) v
and so forth.
Maxwell’s equations are the same as they ever were, namely
(1/c^2) \partial A_t/\partial t+\left( \partial A_x/\partial x+\cdots \right)=0 and
\nabla^2 A - (1/c^4) \partial^2 A/(\partial t)^2= J,
where A is the four-vector-potential.
(This isn’t what you think Maxwell’s equations say? You need to read about the vector potential!)
Finally, the field \psi evolves by the Klein-Gordan equation:
(1/c^2) (\partial_t +i \beta A_t)^2 \psi - \left( (\partial_x + i \beta A_x)^2+\cdots \right) \psi = - \alpha^2 \psi.
Note the curious mix of linearity and nonlinearity — Maxwell’s equation is linear in A but quadratic in \psi, while the Klein-Gordan equation is the reverse. We will discuss \alpha and \beta below; for now think of them as constants to be determined experimentally.

The constant \alpha, which has the units 1/distance, has the value m_z c/\hbar, where m_z is the mass of the zeron. Conceptually, however, this is backwards. Only \alpha will appear in our formulas, not \hbar or m_z. In the absence of an electro-magnetic field, \psi has wave solutions whose period T and wavelength \lambda are related by
1/(c^2 T^2) - 1/\lambda^2=\alpha^2.
One could imagine physicists on another planet measuring the wavelength of \psi waves (perhaps by a double slit experiment, or by the Bohm-Aharonov effect) and measuring these waves’ velocity. They could find the relation above without ever discovering \hbar or quantum mechanics. Only much later, they would discover that these fields were quantized, in lumps with mass \hbar \alpha/c. In the posts that I have planned so far, I will not discuss why this quantization occurs. (Similarly, \beta=q_z/c \hbar, where q_z is the charge on the zeron.)

Suppose that \psi_1=C e^{i(t/T+x/\lambda)} is one of these wave solutions. Then it is easy to compute that \rho is identically C^2/T throughout space-time and J, which points solely in the x-direction, has magnitude C^2/\lambda. If \psi_2=C e^{-i(t/T+x/\lambda)} is the comp[lex conjugate solution, then the charge density and current acquire the opposite sign. Adding these two solutions together, we can obtain the case of two parallel particle beams discussed in the introduction: \psi=\psi_1+\psi_2=2 \cos(t/T+x/\lambda). Notice that the J-field is indeed zero, but the \psi field is not. (Since the expression for J in terms of \psi is not linear, this has to be checked by hand; it is not automatic.)

This post is getting long, so I’ll cut it off here. In the follow-up post, I’ll explain how the Klein-Gordan equation relates to Schrödinger’s equation and, in a more vague manner, how it relates to the Lorentz force equation that I started out discussing.


3 thoughts on “Help David learn Quantum Field Theory II

  1. In the above you’ve simplified QFT by going to the spin-0 (scalar) case. There’s another way of simplifying QFT, and that is to use qubits as the quantum states, but keep the physically correct spin structure. This is quite recent, (see Quantum Electrodynamics of qubits), and much more elegant than the usual in that it means that Feynman diagrams can be calculated to all orders in perturbation theory without having to cancel infinities. What’s more, they explain how to do this in a short, 23 page paper that is written at a very easy to understand level.

    The above paper is for “condensed matter” situations where a qubit is a good representation of a quantum state. My own interest is in elementary particles, but qubits can also be used there. I’ve been blogging the applications recently.

    The difference between these two introductions is that in the scalar case they ignore the spin structure of the particles, while in the qubit case they are ignoring the Lorentz structure of spacetime. I think you will find that the second case is a lot more elegant and much easier to understand. The traditional approach to QFT requires a lot of complicated calculus and a little bit of algebra. The qubit version reverses this.

    Ryder is an excellent text for the traditional approach, but if I were a grad student looking to learn the essence of the subject quickly, or to find something publishable, I woudl take a quick look at the qubit theory.

  2. Ah, that’s what those QFT people are on about.

    There’s a typo in the second-to-last paragraph: “comp[lex”.

    I’m really interested in what will come out of not using natural units and keeping all the h-bar’s and c’s together. Thanks.

  3. I hope I am right in thinking that you are describing here the classical Maxwell theory as a U(1) gauge theory where the gauge field is A and the matter field is the electron field \psi (ignoring for the moment the spin degrees of freedom of \psi). It should be possible then to see in some simple way the fundamental electron-photon interaction, where the photon hits the electron and changes its phase in the u-v plane. Of course, (electron + photon ) —> electron cannot really happen by energy conservation, but it can for “virtual” photons. Perhaps we can see this in some way from the classical gauge theory?

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