John Baez: A Gaussian  Schmotons  John Baez: A Rather

John Baez: A Six-Step Program

I suggested:

Let's work out the QFT description of an electromagnetic plane wave, and see how many photons it has in it (on average).

and Michael Weiss wrote:

I thought we did the ``how many photons on average'' part already: A²/w per unit volume.  [Final reminder: w is the best we can do for omega, in HTML.]

We figured that out by a combination of black magic and seat-of-the-pants intuition. Namely, we took the classical formula for the energy density of the electromagnetic field, and the quantum formula for the energy of a photon of a given frequency, combined them, and used that to guess the photon density of a plane wave. This is known as ``semiclassical'' (i.e., half-assed) reasoning. It'll probably give the right answer here, but it's not as satisfying as doing it all properly using quantum field theory.

So let's work it out using quantum field theory! It's always crucial, when doing an involved calculation, to know the answer ahead of time. Now that we know the answer to ``what's the photon density of the quantum-field-theoretic description of an electromagnetic plane wave'', we are in a good position to derive the answer in a more careful way.

This should be fun, because it's really not so obvious to me why the answer is what it is. In redoing the computation carefully, we can try to make it obvious why the answer is A²/w (if it really is.)

For example, when I start picturing an electromagnetic plane wave, I say, OK, that's A e^{i (k. x-w t)}. And I picture stacked planes, and I remember that this is really 4d spacetime and (k,w) are really the coordinates of a 1-form. Hmm, what about A? Well, there's a vector field on each of the stacked planes. Gee, I should really somehow drag in the 2-forminess of (E,B) and that fact that it's the differential of the vector potential, and the U(1) gauge and all that, but I can't picture any of that stuff yet. Also there's something wrong with saying that A e^{i (k. x-w t)} is a vector. I think I complexified somewhere without realizing it.

Let's see, I think you are trying to do too many things at once here. You are trying to understand classical electromagnetism while simultaneously trying to understand coherent states. I suggest the following game plan:

1.

understand coherent states for a harmonic oscillator with one degree of freedom.

2.

understand coherent states for a harmonic oscillator with n degrees of freedom.

3.

understand coherent states for a free massless scalar field in one dimensional space.

4.

understand coherent states for a free massless scalar field in 3 dimensional space.

5.

undestand how the equation of a free massless scalar field is related to that of a free massless spin-1 field, i.e. electromagnetism.

6.

understand coherent states for the electromagnetic field.

This may seem to multiply our difficulties by six, but I really think it will make the problem much easier. Break it down into bite-sized pieces! Steps 1-4 are a nice gentle ramp, and then mixing in the answer to 5 it should be a snap to get 6.

3/10/1998