Is the Abraham-Minkowski controversy resolved? A paper was published in 2010 claiming to resolve the Abraham-Minkowski controversy.  Is this paper viewed as definitive by physicists?
 A: The sustained interest in the Abraham-Minkowski debate does not come from any theoretical concern-- theorists on both sides have always thought that they were unquestionably right, and the people on the other side were a bunch of incompetent hacks. The traditional arbiter of any such dispute between theories is experimental evidence, but that evidence has been ambiguous. Some experimental tests give results consistent with the Abraham formula, others with the Minkoski formula (the arxiv paper gives references to these).
The recent paper is interesting not because it shows definitively which side is a bunch of incompetent hacks, but because it provides a framework for understanding how the seemingly contradictory experimental results can both be right. It will settle the question to the extent that this framework for reconciling the irreconcilable holds up under further investigation. Ideally, this would come through not just explaining past experiments, but through predicting some new result that can be verified experimentally.
The paper in question is far too recent for the necessary experiments to have been done. It looks pretty good, but I imagine that a real partisan of one side or the other will be able to find some way to wriggle out, until the predictions have been tested experimentally.
A: good to ask about these historical controversies. It makes sense to review the controversy. In the medium, the density of momentum was (I set $c=\hbar=1$ everywhere)
$$E\times H$$
according to Max Abraham - that's equal to the Poynting vector which determines the flow of energy according to everyone - and
$$D\times B$$
according to Hermann Minkowski who argued that in the medium, the stress-energy tensor is asymmetric. In both cases, one can start with the product of the vacuum fields $E\times B$ and replace one of them by the material "variation" of the field. The two guys differed by whether they replaced the electric or the magnetic factor.
Consequently, a photon according to the first, Abraham form will have momentum
$$p=\omega/n$$
while the second, Minkowski definition will give
$$p=\omega n$$
Now, one may realize that the (complexified) electromagnetic wave depends on the position and the time as
$$\exp(-i\omega t+i k x)$$
where $k=\omega n$. The phase velocity is simply $\omega/k$ and it should be $1/n$, smaller than one, so it's clear that $k=\omega n$. So the wave function shows clearly that the conserved momentum of a single photon agrees with the Minkowski's template. By definition, the momentum is given by the $k$. Barnett agrees that the Minkowski momentum is what generates normal translations - of the vector potential and other fields - by conjugation.
If you try to find out what are the arguments (and interpretations) that support the Abraham form, I think that all of them are incompatible with modern physics. As far as I can say, they're all based on the idea that the momentum of any particle should be equal to the "mass times velocity". This is an extremely shaky assumption that is not really true in this case. In modern physics, the momentum has to be defined by a solid definition - and it is the "quantity conserved, as showed by Noether's theorem, because of the spatial translational symmetry". One can show that this is $m_{total}v$ for ordinary particles in the vacuum but one cannot show this thing for a photon in a material - especially because it is not true.
It is very bizarre for Barnett to say that the "canonical momentum is more subtle than the $mv$ momentun". The "canonical momentum" is the only momentum that is acceptable in modern physics and that may be generalized to new contexts. It's the only Noether-derived momentum. In this case, it's the Minkowski's momentum. The "kinetic momentum", as he calls $mv$, is just an attempt to return to the 17th century as much as he can.
For charged particles, there could be issues about $i\partial$ versus $i\partial-eA$, the velocity operator, but this doesn't occur for $A_i$ in the dielectric because the wave function of the photon cannot be transformed away by any gauge symmetries. Under the U(1) gauge symmetry with an oscillating parameter, $A$ may shift by a constant but it won't change the speed of its oscillations.
The other arguments supporting the Abraham form seem to be bogus, too. For example, the Abraham form is supported by the argument that the center-of-mass-energy continues in a uniform motion as the photon enters the material. However, this is an invalid assumption, too. The uniform motion of center-of-mass only holds - again according to Noether's theorem - in systems which respect the equivalence of inertial frames e.g. the Lorentz (or Galileo) symmetry. This symmetry is explicitly broken by the boundary between the vacuum and the dielectric material, so there is no reason why the corresponding conservation law - the conservation of the velocity of the center-of-mass motion - should hold.
To summarize, Minkowski was simply right while Abraham was wrong. But it's probably OK to "redo" the budget in such a way that a part of the momentum of the photon is attributed to the dielectric material when the photon enters it, and then it is returned back to the photon. In this way, one may justify the Abraham's form - and probably many other forms - but why should one really do it?
Stephen Barnett who wrote the 2010 paper above is a respectable optician but this whole thing was much ado about nothing. If you care about the sociology, I am pretty confident that almost no people in optics will oppose Barnett, and the people from other disciplines may say that this was really a trivial dispute. Minkowski was really right, and by the way, he was a much better physicist than Abraham from all other angles, too. For example, Abraham's model of the electron and the arguments used to support this model were truly painful. That's a very different league than Minkowski who was one of the first people to have understood special relativity.
I don't think that throughout the century, this problem was being studied by top physicists. To make things worse, the experiments are never testing these things "directly", especially because one may always choose different interpretations where the momentum goes when the photon is changing the medium.
Best wishes,
Lubos
P.S.: Let me mention one detail: Minkowski's stress-energy tensor is asymmetric, so if you define the density of the angular momentum purely in terms of this tensor, multiplied by $x$ in the usual way, the angular momentum won't be conserved. However, it's not a problem because this can be compensated by an extra contribution to the angular momentum that comes from the volume and surface density of the spins - including the internal angular momenta of the atoms of the materials. There are many ways how to attribute the conserved quantities (energy, momentum, angular momentum) to regions - only the total has to be conserved. The diverse interpretations will differ about all the functions, but they should ultimately predict the same predictions for the experiments when done right.
A: I've found another recent paper(18 Mar 2011) on an experiment to test the difference.
http://arxiv.org/abs//1103.3559
The trouble is, I can't see how the experiment proves in favor of Minkowski. As far as I can make out, Zhong-Yue. Wang1 , Pin-Yu Wang, Yan-Rong Xu write out some theory, pronounce Minkowski is correct 
e.g."Abraham-
Minkowski controversy is now resolved."
and then do a Cherenkov effect experiment, then write some gobblygook about the soul of physics. All this is written up in Wikipedia as a crucial experiment. 
http://en.wikipedia.org/wiki/Abraham-Minkowski_controversy#Crucial_Experiment
My trouble is, I can't see how it's so?
A: After reading Lobos answer I felt that he is right in supporting Minkowiski version, but on thinking more about it, I changed my mind and found that in fact Abraham formula is the correct one. This is how;
Imagine a steel ball shot into an elastic(rigid) tube. Assume the tube is slightly smaller than the ball, so that the ball makes a bulge in the tube wall as it moves. Assume there are no losses. 
The energy and momentum of the ball before entering and after leaving the tube must be the same. But inside the tube the ball moves slower, so its energy and momentum are less. The deficiency is taken by the stress in the walls of the tube. This energy and momentum is given back to the ball as it leaves. This is very much similar to the ball compressing a spring and loosing speed then given to it back by the spring.This example shows that the momentum of the ball will be smaller inside the medium. This is logical as an increase in momentum and energy require an input, which doesn't seem to be there in the case of a static medium. Note that momentum and energy are conserved all the time. Within matter part of the energy is transferred to the medium, then claimed back on leaving.
Now proceed as Abraham said; pA=mv, and substitute for m from E=mc^2 and for E=hf. We get pA=hf(v/c)/c=hf/nc, where f is frequency, n=c/v the refractive index. This gives Abraham formula.
For Minkowski formula, start with pM=hf/c. In matter of refractive index n=c/v, where v is velocity of radiation in medium. The momentum becomes pM=hf/v. Substitute for v and you get; pM=nhf/c, which is the Minkowski formula, which gives a momentum inside matter that is higher than that in empty space(before entering matter). Since it is not possible to increase momentum without increasing the frequency, clearly this formula is not correct. And we have to conclude that p=hf/v is not a valid formula inside matter. We also conclude that the reactionless drive idea is unfortunately not possible. 
