A recent approach (see Phys.org coverage) suggests that the missing piece in the Abraham–Minkowski debate is spin — the intrinsic angular momentum of light.
By projecting momentum onto spin, the two definitions align:
Minkowski momentum is the magnitude of the spin-projected momentum.
Abraham momentum is the expectation value of that vector, directly tied to the Lorentz force on the medium.
Thank you very much for sharing that paper! Its a really nicely written paper, I like their figures a lot.
I think you have slightly misunderstood the paper (either that or I am missing something). In the paper, I think they are abusing the word “spin”. Every single place the paper says “spin”, they don’t actually mean spin (as in, the intrinsic spin angular momentum of light), they actually mean direction. So, when reading the paper try and read it through a mental translator where “left handed spin” translates to “left propagating”.
The spin angular momentum of light is (for a plane wave in vacuum) controlled entirely by its polarization, either left handed circular polarization or right handed. Importantly, this polarization depends on the fact that their are 2 spatial dimensions that are orthogonal to the propegation direction, so that for example the electric field could be expressed as: E = (1, i, 0) in an (x, y, z) basis and z the propegation direction. (Similarly (1, -i, 0) for the other polarization with the opposite spin).
In this paper they define what they call the “left handed” and “right handed” operators in the unnumbered equation immediately under equation (10). However, these operators are NOT left hand polarized and right hand polarized light waves. The operators differ, not by the relative phase of orthogonal electric field components, but by the relative phase of the electric and magnetic fields. This means they are “left travelling” and “right travelling” (IE propagating left or right) light waves. They have confusingly chosen to call these terms “spin”, I think this is because the equation they have derived looks like a Dirac equation, and in the Dirac equation those terms are called spin. But they are not the actual spin angular momentum of the light, they are completely unrelated.
In short, they don’t actually consider real spin at all, they just rename “direction” to “spin”.
They say theyr are in full agreement with Stephen Barnet (option number (1) in my post), that Minkowski’s momentum is the canonical one (to be used in Heisenberg uncertainty type situations) and Abraham’s is the kinetic one (to be used in Newtonian recoil calculations).
Thanks for the thoughtful comment. In this work, “spin” arises from casting Maxwell’s equations in a Dirac-like form, where the photon is treated as a spin-1 particle (unlike the electron’s spin-½ in the Dirac equation). In this framework, spin is a good quantum number, and its precession around momentum encodes polarization dynamics. The key result is straightforward: Minkowski momentum corresponds to the magnitude of the spin-projected momentum, while Abraham momentum is its averaged vector. Rather than being rival definitions, they describe complementary aspects of the same spin-projected structure.
Other works sometimes frame this as “canonical” vs. “kinetic,” but here the physics is more transparent: it’s about magnitude vs. averaged vector, grounded in the photon’s spin-1 character.
A recent approach (see Phys.org coverage) suggests that the missing piece in the Abraham–Minkowski debate is spin — the intrinsic angular momentum of light.
By projecting momentum onto spin, the two definitions align:
Minkowski momentum is the magnitude of the spin-projected momentum.
Abraham momentum is the expectation value of that vector, directly tied to the Lorentz force on the medium.
Details: Phys. Rev. A 112, 033721 (2025) DOI: 10.1103/sxh8-q8tq
Thank you very much for sharing that paper! Its a really nicely written paper, I like their figures a lot.
I think you have slightly misunderstood the paper (either that or I am missing something). In the paper, I think they are abusing the word “spin”. Every single place the paper says “spin”, they don’t actually mean spin (as in, the intrinsic spin angular momentum of light), they actually mean direction. So, when reading the paper try and read it through a mental translator where “left handed spin” translates to “left propagating”.
The spin angular momentum of light is (for a plane wave in vacuum) controlled entirely by its polarization, either left handed circular polarization or right handed. Importantly, this polarization depends on the fact that their are 2 spatial dimensions that are orthogonal to the propegation direction, so that for example the electric field could be expressed as: E = (1, i, 0) in an (x, y, z) basis and z the propegation direction. (Similarly (1, -i, 0) for the other polarization with the opposite spin).
In this paper they define what they call the “left handed” and “right handed” operators in the unnumbered equation immediately under equation (10). However, these operators are NOT left hand polarized and right hand polarized light waves. The operators differ, not by the relative phase of orthogonal electric field components, but by the relative phase of the electric and magnetic fields. This means they are “left travelling” and “right travelling” (IE propagating left or right) light waves. They have confusingly chosen to call these terms “spin”, I think this is because the equation they have derived looks like a Dirac equation, and in the Dirac equation those terms are called spin. But they are not the actual spin angular momentum of the light, they are completely unrelated.
In short, they don’t actually consider real spin at all, they just rename “direction” to “spin”.
They say theyr are in full agreement with Stephen Barnet (option number (1) in my post), that Minkowski’s momentum is the canonical one (to be used in Heisenberg uncertainty type situations) and Abraham’s is the kinetic one (to be used in Newtonian recoil calculations).
Thanks for the thoughtful comment. In this work, “spin” arises from casting Maxwell’s equations in a Dirac-like form, where the photon is treated as a spin-1 particle (unlike the electron’s spin-½ in the Dirac equation). In this framework, spin is a good quantum number, and its precession around momentum encodes polarization dynamics. The key result is straightforward: Minkowski momentum corresponds to the magnitude of the spin-projected momentum, while Abraham momentum is its averaged vector. Rather than being rival definitions, they describe complementary aspects of the same spin-projected structure.
Other works sometimes frame this as “canonical” vs. “kinetic,” but here the physics is more transparent: it’s about magnitude vs. averaged vector, grounded in the photon’s spin-1 character.