For 100 years physicists have been struggling to reconcile two different formulations describing the momentum of light travelling through a transparent medium. One, put forward by German mathematician Hermann Minkowski in 1908, stipulates that light's momentum increases when it enters a medium, while the other, advanced a year later by the German physicist Max Abraham, instead says that the momentum of light decreases. Now, Stephen Barnett of the University of Strathclyde in the UK has concluded that both formulations are in fact correct, with the difference essentially boiling down to whether one considers the wave or particle nature of light.

It is well known than when light enters a material medium it slows down in proportion to the refractive index, n, of that medium. Minkowski and Abraham wanted to know how light's momentum changes as a result. Abraham calculated that the momentum of a single photon within the light is also reduced by a factor n, a result which agrees with our experience of everyday objects – as their speed drops, so too does their momentum. Indeed, a number of powerful arguments have been put forward over the years in support of this position. Prominent among these has been a simple proof based on Newton's first law of motion and Einstein's equivalence of mass and energy, which considers what happens when a single photon travels through a transparent block and transfers some of its momentum to the block, given that the motion of the system's centre of mass-energy must remain constant.

Minkowski's formulation, on the other hand, seems more natural from the point of view of quantum mechanics. As light slows down inside a medium its wavelength also decreases, but quantum mechanics tell us that shorter wavelengths are associated with higher energies, and therefore higher momenta. In fact, Minkowski's approach suggests that the momentum of a single photon of light increases by a factor n as it passes through a medium. This result can also be supported by strong theoretical arguments, among them one that considers what happens when an atom moving at some speed through a medium absorbs a photon and experiences an electronic transition.

### Fundamental physical principles at stake

As Barnett points out, this problem has kept physicists interested for so long because it appears to put one or more fundamental physical principles at stake – on the one side Newton's first law and Einstein's famous E = mc2 and on the other the notion, familiar from de Broglie waves, that momentum is inversely proportional to wavelength.

These two formulations reflect the fact that in different situations momentum does different things Stephen Barnett, University of Strathclyde

Both formulations have received experimental support, particularly that of Minkowski. For example, in 2005 Wolfgang Ketterle and colleagues at the Massachusetts Institute of Technology reported evidence in favour of Minkowski by transferring momentum from laser beams to matter waves that had been formed from a few million atoms cooled to just above absolute zero. However, in 2008 a group led by Weilong She of Zhongshan University in China passed a laser beam through a tiny filament of silica and found that the filament recoiled as the light exited, indicating, in accordance with Abraham, that the light gained momentum as it left the material.

According to Barnett, however, both formulations are correct. He says that the one put forward by Abraham corresponds to a body's "kinetic momentum" – its mass multiplied by its velocity. Minkowski's momentum, on the other hand, is a body's "canonical momentum" – Planck's constant divided by its de Broglie wavelength. "These two formulations reflect the fact that in different situations momentum does different things," he adds. "In free space they coincide, but not when inside a medium."

### Don't mix up the two

Physicists have known for some years that this distinction might explain the dilemma but have been unable to prove it. That is to say, they have been unable to reconcile the two different formulations with electromagnetic theory. Barnett overcame this problem when he realized that the two approaches cannot be treated in the same way mathematically – that of Abraham requires considering momentum as transferred by individual particles whereas that of Minkowski instead involves the commutation relationship between momentum and position, a wave property. "It is when you mix the two up that you get the problem," he says.

The question is: when is the particle momentum relevant and when is the wave momentum relevant? Ulf Leonhardt, University of St Andrews

This point is underlined by Ulf Leonhardt of the University of St Andrews in the UK, who says that, simply put, Abraham described the momentum of light as a particle whereas Minkowski described the momentum of light as a wave. As such, he agrees that both formulations are correct. However, he does not think that the debate is really over. "The question is: when is the particle momentum relevant and when is the wave momentum relevant? Are there cases when a mixture of wave and particle properties appear?" he asks. "When science answers one question, ten new questions appear."

Barnett is also not entirely satisfied. "We now know that Abraham and Minkowski were both right," he says. "But we don't yet know why nature requires two momenta."

The work is reported in Phys. Rev. Lett. 104 070401.