Abstract
It is 100 years since Minkowski and Abraham first gave rival expressions for the momentum of light in a material medium. At the single-photon level, these correspond, respectively, either to multiplying or dividing the free-space value () by the refractive index (n). The debate that this work started has continued till the present day, punctuated by the occasional publication of ‘decisive’ experimental demonstrations supporting one or other of these values. We review the compelling arguments made in support of the Minkowski and Abraham forms and are led to the conclusion that both momenta are correct. We explain why two distinct momenta are needed to describe light in a medium and why each appears as the natural, and experimentally observed, momentum in appropriate situations.
1. Introduction: the Abraham–Minkowski dilemma
It has long been appreciated that light has mechanical properties. Indeed, Maxwell (1891) presented a simple calculation of the pressure exerted by sunlight at the surface of the Earth. It was Poynting (1884) who determined that it is the cross-product of the electric and magnetic fields that determines the flux of electromagnetic energy. For light propagation in vacuum, there is no difficulty in also identifying this cross-product with the density of electromagnetic momentum. Within a medium, however, we have a choice to make between the electric and displacement fields (E and D) and the magnetic field and the magnetic induction (H and B). Poynting’s theorem tells us that the flux of energy is E×H, but there are two entirely reasonable and rival forms for the corresponding density of momentum. These are the Minkowski (1908) momentum density, gMin=D×B and the Abraham (1909, 1910) momentum density, gAbr=E×H/c2. The problem of determining which momentum is ‘correct’ is the famous Abraham–Minkowski dilemma. This is not the place to review the large literature devoted to this problem; instead, we recommend to the interested reader the review by Brevik (1979) and the more recent one of Pfeifer et al. (2007).
It is not necessary to quantize the electromagnetic field in order to appreciate the problem, but it is helpful to understand it in terms of the properties of a single photon of angular frequency ω. We can do this by means of a simple scaling argument. The total electromagnetic energy within our volume is simply that of the photon ():





(a) Argument in favour of Abraham
Perhaps the most direct way to calculate the momentum of a photon in a medium is to use the Newtonian idea that the centre of mass (or more precisely the centre of mass-energy) of an isolated system undergoes uniform motion (Einstein 1906). We follow the analysis of Balazs (1953) and apply this idea to a single photon and a block of transparent material initially at rest. We let the photon travel in the z-direction and are then interested in this component of the electromagnetic momentum. The photon has energy and propagates with speed c. If the block has mass M then the total energy is


In order to move the distance Δz while the photon is in the medium, the block must have acquired from the photon the momentum



We have used only the conservation of momentum and the uniform motion of the centre of mass-energy in deriving our result, and it is difficult to see how any component of our derivation could seriously be open to question.
(b) Argument in favour of Minkowski
The first thing to be said in support of the Minkowski momentum is that it is ‘natural’ in that the wavevector in a medium is greater than that in vacuum by the refractive index and hence the Minkowski single-photon momentum (1.3) is simply multiplied by the wavevector in the medium. There are also, however, at least two simple physical arguments in support of the Minkowski momentum.
The first, due to Padgett (2008), is based on single-slit diffraction. A plane wave propagating in the z-direction towards a single slit in the x–y plane will undergo diffraction and produce a characteristic interference pattern in the far field. We can determine the width of the central peak of this pattern by a simple application of the Heisenberg uncertainty principle. If the slit has width Δx then the uncertainty principle requires that the field after the slit has a spread of momenta in the x-direction of . It then follows that the angular spread of the central interference peak will be


Our second argument (Bradshaw et al. in press) is a variant of an idea due to Fermi (1932). Consider an atom of mass m with a transition at angular frequency ω0. Let the atom be in a medium with refractive index n and moving with velocity v away from a source of light with angular frequency ω. The atom can absorb a photon from the beam if the Doppler-shifted frequency matches the transition frequency, so that




These arguments in support of the Minkowski momentum are of a different character from that made in support of the Abraham form, but they are no less convincing for that. Both forms are well supported, therefore, and hence we have a dilemma.
2. Experimental evidence
As theory has presented us with a dilemma, it is reasonable to seek an answer in experiments, and this idea has been pursued on a number of occasions (Jones & Richards 1954; Ashkin & Dziedzic 1973; Walker et al. 1975; Jones & Leslie 1978). The work of Jones, Richards and Leslie confirmed that the force exerted on a mirror submerged in a medium was consistent with each photon in that medium having the Minkowski momentum. The experiment of Ashkin and Dziedzic showed that the action of light on the surface of a liquid was also consistent with the Minkowski momentum, although this interpretation is far from unambiguous (Gordon 1973). The experiments of Walker et al. provide evidence that is no less convincing in favour of the Abraham form. These early experiments and the conclusions derived from them are discussed at greater length in Brevik (1979) and Pfeifer et al. (2007).
The confusing experimental situation has continued, with further experiments seeming to support either the Minkowski or the Abraham momentum. Gibson et al. (1980) exploited the photon drag effect to measure the momentum transfer from far-infrared radiation to free charge carriers in germanium and silicon. In each case the observations were consistent with the Minkowski form of the optical momentum. Campbell et al. (2005) measured the recoil momentum of atoms in a dilute ultra-cold gas, effectively performing the experiment outlined in §1b above. They also found a recoil consistent with the Minkowski form. Most recently, She et al. (2008) measured the displacement of an optical fibre due to light leaving. Their results, although not uncontroversial (Mansuripur 2009), seem to support the Abraham momentum.
There is an angular-momentum version of the Abraham–Minkowski dilemma, with the Abraham angular momentum being that in free space divided by n2 and the Minkowski form being the same as that in free space. An angular version of the argument, given above, in support of the Abraham momentum supports, naturally enough, the Abraham angular momentum (Padgett et al. 2003). An experiment of Kristensen & Woerdman (1994), however, measured the torque on an object in a dielectric medium. The observations gave results in support of the Minkowski angular momentum.
Experimental work has served to confirm that the force exerted by light on an object within a medium is consistent with the Minkowski momentum for the light in that medium. The evidence in support of the Abraham momentum is, perhaps, less convincing, but tends to support the idea that the net effect on a medium due to light passing through it is consistent with the Abraham momentum. Indeed, it could not be otherwise! If the argument advanced in §1a in favour of the Abraham momentum were to be incorrect, then that would bring into question uniform motion of an isolated body as expressed in Newton’s first law of motion. Similarly, a failure of the arguments advanced in §1b in favour of the Minkowski momentum would require us to question the uncertainty principle, momentum conservation and the Doppler effect.
It seems that there is at least some validity to both the Minkowski and the Abraham momenta, and it is for theory to explain the origins of these two momenta and to explain why one of them appears as the ‘correct’ momentum in a specific situation.
3. Electromagnetic force
Our first task is to find a reliable way of determining the momentum exchange between electromagnetic fields and a medium. Using energy and momentum densities has proven to be an unreliable method for this, not least because Minkowski and Abraham have given us different expressions for the electromagnetic energy–momentum tensor. It is safer, therefore, to work with the force exerted on the medium and then to use Newton’s second law of motion to relate this to a rate of change of momentum (Loudon 2002, 2003).
The force exerted on a point dipole, with dipole moment d, is simply


Photon drag occurs in semiconductors and the experiments of interest were performed in silicon and germanium. At the long wavelengths used, these behave, to a good approximation, as free carriers in a background dielectric. We can safely assume that the carriers are responsible for the absorption and make a purely imaginary contribution to the permittivity. The host material is responsible for the real part. We shall assume that the medium is thick enough for our photon to be absorbed.
We consider a single-photon pulse with a narrow band of frequencies centred on ω. The momentum transfer to the medium is readily calculated from the force density formed from equation (3.2) by quantizing the fields and integrating over space and time. We omit the details of this calculation, which can be found in Loudon et al. (2005), and concentrate instead on the results and their physical significance.
The calculated momentum transfer to the free carriers is





We can also apply our method to study a weakly absorbing dielectric with n≈1. If the medium is of finite thickness then this situation coincides with that proposed in §1a in support of the Abraham momentum. We can neglect reflections at the interfaces because n≈1. Our analysis confirms that the momentum transferred to the medium during propagation of a photon through it is


It is clear that the experimental observations to date are consistent with both the Minkowski and Abraham momenta. The momentum transfer to a body (in our case the charge carriers) within a medium is given by the Minkowski momentum. The momentum of a photon travelling through a host dielectric, however, is given by the Abraham momentum. These results have both arisen from applying the same Lorentz force to a simple model dielectric, and confirms the validity of both mechanical arguments proposed in §1a,b. It only remains to explain why there are two momenta and why they appear where they do.
4. The two momenta
We are not often aware of it, but we define our momentum by means of two properties. The first, which would have been familiar to Newton, is the inertial property derived from Newton’s second law of motion. This kinetic momentum is the product of the mass and velocity of a body. The second is most readily appreciated with reference to quantum theory as that associated with de Broglie waves. The canonical momentum for a quantum particle is Planck’s constant divided by its de Broglie wavelength. More formally, the canonical momentum is that derived from the Lagrangian, which is constructed to satisfy the canonical commutation relation

The form of the canonical momentum depends, in fact, on the gauge and the form of the matter–field coupling employed (Power 1964; Craig & Thirunamachandran 1998). In the electric dipole form, most appropriate for our systems, we find that for a single point dipole (Baxter et al. 1993; Leonhardt 2006; Hinds & Barnett 2009)




Clearly, we can identify the Abraham momentum as the kinetic momentum of the light in the medium, while the Minkowski momentum is its canonical momentum.
5. A dilemma resolved: the two natural momenta
We have determined that the Abraham momentum is the kinetic momentum of the light and that the Minkowski momentum is its canonical momentum. It only remains to show how these identifications determine the conditions under which either one of them is the natural momentum.
(a) The Abraham or kinetic momentum
The argument put forward in §1a depends on the displacement of a medium due to the propagation of a photon through it. This displacement is a consequence of a velocity imparted to the medium and hence a kinetic momentum. Global kinetic momentum conservation leads us to the kinetic momentum of the photon, which is, as we have seen, the Abraham momentum.
(b) The Minkowski or canonical momentum
We proposed, in §1b, three arguments in support of the Minkowski momentum: that it is natural, that it was suggested by diffraction and that it is required by momentum conservation for absorption by a moving atom. We address each of these in turn.
It will be recalled that de Broglie identified the wavelength of a quantum body as h/p. This, in turn, led Schrödinger to express the momentum as

Padgett’s diffraction argument is based on the Heisenberg uncertainty principle and hence on the canonical commutation relation (4.1). It is for this reason that the canonical, or Minkowski, momentum is the one that appears. Equivalently, we can simply note that diffraction depends on the wavelength of the light and that this, as noted above, is inversely proportional to the Minkowski momentum.
In order to address the momentum of a body immersed in a dielectric host, we first recall the role of the canonical momentum in generating translations. The commutation relation (4.1) implies that


In the same way we find that the Minkowski momentum generates translations of the electromagnetic field in that

6. Dispersion: a final detail
One remaining issue is the forms of the two momenta in a dispersive medium, in which there are two refractive indices: the phase index



The wavevector has magnitude npω/c, and it is reasonable, therefore, to expect that the canonical or Minkowski momentum should have the value



The resolution of this puzzle lies in the roles of the kinetic and canonical momenta. We can meaningfully evaluate the kinetic or Abraham momentum by taking the expectation value of gAbr. The role of the canonical momentum is to generate translations, however, and its important property, therefore, is the translation relation (5.3) or, equivalently, the commutation relation



7. Conclusion
We have described the two rival momentum densities for the light in a medium and presented, as simply as possible, the compelling physical arguments in favour of each of them. Exisiting experimental evidence strongly supports the Minkowski momentum as that transfered to a body within a host medium. There is also experimental evidence, perhaps not quite so strong, in support of the Abraham momentum as that part of the momentum not transferred to the medium during the passage of the photon through it. Calculations based on the Lorentz force reveal circumstances in which either momentum is the appropriate one and, importantly, verify the validity of the simple arguments made in favour of both the Abraham and Minkowski momenta.
The resolution of the Abraham–Minkowski dilemma lies in the realization that electromagnetism recognizes two distinct momenta, the kinetic momentum and the canonical momentum. The total momentum is conserved, whichever momentum we use, and this leads us to identify, unambiguously, the Abraham and Minkowski momenta, respectively, as the kinetic and canonical momenta for the light.
Finally, we note that a number of momenta have been proposed as rivals to those of Abraham and Minkowski (Brevik 1979), with the aim of thereby resolving the conflict. We may hope that in demonstrating, clearly, the need for and natures of both the Abraham and Minkowski momenta, we may also have removed the need for these and further rival momenta.
Acknowledgements
We are grateful to Les Allen, Mohamed Babiker, Colin Baxter, Ed Hinds, John Jeffers, Peter Milonni and Miles Padgett for many illuminating discussions. We especially thank Miles Padgett for kindly reading the manuscript before submission and for making a number of helpful suggestions. This work was supported by the Engineering and Physical Sciences Research Council, by the Royal Society and by the Wolfson Foundation.
Footnotes
One contribution of 17 to a Theme Issue ‘Personal perspectives in the physical sciences for the Royal Society’s 350th anniversary’.