The controversy touches our analysis of the photon phase shift in the real photon vacuum polarization diagram. We blogged last week and claimed that a photon which goes through a glass pane pushes that pane a little bit forward. This is because the speed of the center of mass must stay constant.
Let the refractive index be n = 2. Abraham claims that the photon inside the glass has lost half of its momentum to the glass pane. Minkowski, strangely, claims that the momentum of the photon has doubled!
The strange claim of Minkowski derives from the photon wave function which inside the glass looks roughly like this:
exp(-i (E t - 2 p x))
(we have set c = 1 in vacuum, h = 1). The energy E is the same as when the photon was flying in vacuum. The coefficient 2 is there because the speed of light is just half in the glass (n = 2). Thus the wavelength is just a half, and the phase of the wave function must rotate faster when we move along the x axis. If the wave would exist in empty space, then 2 p would really be the momentum.
What would an experiment say? We believe in the conservation of the speed of the center of mass. Then it has to be that the light pulse has lost half of its momentum to the glass pane when the light pulse is inside. The light pulse could be a short laser flash.
Suppose that we have something inside the glass pane which absorbs the light pulse. That something will absorb the rest of the momentum in the light pulse.
Does the same hold for an individual photon? We may imagine that the photon has a very large energy. Then it should be no problem to measure separately the momentum which it loses to the glass pane when it enters, and the momentum which it loses in absorption.
Thus, Abraham is right. This also shows that the energy-momentum relation for an interacting photon is not the same as for a free photon. For a free photon, |p| = E, but inside the glass, |p| = E / n.
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