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Thesis

# Minkowski, Abraham and the photon momentum

“One recent experiment backs up Minkowski's answer — but still may not prove Abraham wrong.”

One might expect that the momentum of a photon, under almost any conditions, would no longer be a matter of dispute. Most textbooks certainly give that impression. Yet, in fact, a fundamental uncertainty, recognized a century ago, still lingers. Suppose that a photon, having momentum ħk in vacuum, enters a transparent medium with index of refraction n > 1. What is the photon's new momentum? Remarkably, there is still no definitive answer.

In 1908, the German (although Russian-born) physicist Hermann Minkowski derived one possible, yet surprising, answer. In classical terms, Minkowski found the total momentum of the electromagnetic field to be equal to ∫ d3x D × B. In quantum terms, this suggests that the photon momentum should actually increase and take the value nħk. In effect, Minkowski started from Einstein's earlier suggestion that a photon's energy is given by E = hυ. Assuming a velocity c/n and p = h/λ, one finds that p = nħk.

One year later, the German physicist Max Abraham proposed a different answer. Abraham's entire career focused on the classical theory of electromagnetism. In 1902, he proposed an early theory of the electron, and worked hard to find a consistent mathematical description of the reaction force due to radiation emitted by an accelerating charged particle. Abraham argued that the photon inside the medium would have a lower velocity and lower momentum, the medium itself absorbing the difference. In classical terms, Abraham's momentum is ∫ d3x E × H, or in quantum terms p = ħk/n.

Nearly 100 years later there is still no clear answer as to which of these formulae is correct. It's possible, of course, that both could be, yet refer to subtly different situations or interpretations. The past decade has seen renewed interest in this puzzle, stimulated in part by the increasing precision of quantum optics. One recent experiment backs up Minkowski's answer — but still may not prove Abraham wrong.

The idea was to measure the recoil of a Bose–Einstein condensate as photons are reflected from its surface (G. M. Campbell et al. Phys. Rev. Lett. 94, 170403; 2005). Such a set-up offers two advantages. First, extreme accuracy can be achieved through atom interferometry. The researchers used counter-propagating light beams and detected the interference of the two component recoiling condensates. Second, although the index of refraction for a dilute atomic gas is close to one, larger deviations can be obtained for a condensed gas. The interference fringes measured agreed closely with Minkowski's formula.

Even so, Ulf Leonhardt of the University of St. Andrews points out that Abraham's ideas may still have life (Nature 444, 823–824; 2006). Starting from the perspective of general relativity, other theorists have shown that both Abraham's and Minkowski's formulae can be derived, and may apply under different conditions. Still other work argues that these two formulae correspond to essentially different quantities — one (Abraham) to the momentum of the photon itself, and the other (Minkowski) to that of the photon plus the medium in which it lives. To be certain which momentum an experimental set-up measures is non-trivial.

So despite what seems like conceptual simplicity itself, there is actually very little that is simple. And the debate over this matter shows no sign of ending soon.

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Buchanan, M. Minkowski, Abraham and the photon momentum. Nature Phys 3, 73 (2007). https://doi.org/10.1038/nphys519