Relativity and photons

It's a relatively free weekend, so I ended up thinking about bunch of random stuff. Another problem I pondered: is number of photon a Lorentz invariance? That is, do observers in different frame count the same number of photons? From intuition, it seems photon number should be an invariance.

Here is what you calculate when you want to count photon number classically (I am assuming the usual monochromatic plane wave solution):

is the frequency, is the electric field strength. The integral is done over volume in which you are counting photons. Under boost, they transform like this: where . The frequency formula corresponds to just the Doppler effect. The volume would be Lorentz contracted too: When combined, we get that the number of photons in the primed frame is: which is clearly not Lorentz invariance. How do we resolve this paradox? One way we can side-step the problem is to claim that a full understanding of photon needs construction of quantum creation operators. So this attempt to understand photons in a non-quantum context is futile. This does not seems like a satisfying explanation to such a simple paradox.

One problem with counting plane wave is that they extends to infinity. So the total energy and therefore photon number is infinity. So one can count the number of photons within some box, as we did. Since a box must be extended in space, the problem of simultaneity---clearly NOT a Lorentz invariant concept---seeps in.

What we should do instead is to set a plane normal to the direction of boost, which is along the direction of flight, and count photons going through the plane in some time interval. In the rest frame of the detection plane, the volume of "photon fluid" that went through it in time interval is . is the cross section area. So we have:

In the moving frame, because the panel is moving toward the source of the photons, the volume is . So: One easily check that . So , as expected.

I think the moral here is that because of simultaneity, thinking about relativity over space extended over the direction of boost is usually an easy recipe for paradoxes.