Photon spin and total angular momentum

Question

I'm studying the photon spin and I assume that the angular momentum of spin is equal to 1 (from QED). In my book it's written, related to photons, that it doesn't make sense distinguishing between spin and angular momentum. Can you explain me why?

Answer 1

There's a number of things that your book could mean by that phrase, and it's impossible to tell exactly what the text entails without the precise wording. However, there's a number of salient points to keep in mind.

• Spin is a type of angular momentum. Angular momentum, in its very essence is the conserved quantity that corresponds, via Noether's theorem, to rotational invariance: in other words, if the hamiltonian of a system is rotationally invariant then angular momentum is conserved, and angular momentum acts as the generator of rotation transformations. For particles with spin, it is the spin that acts as the rotation generators, so that alone should seal the deal, but we also know that it can be exchanged into the more usual mechanical angular momentum (via the Einstein-de Haas effect).

  The same goes for photons ─ their spin acts as the generator for the rotations of the internal degrees of freedom of the electromagnetic field, i.e. the vector aspects of the EM field's polarization, and it can equally well act mechanically (a tool known as an optical spanner) to transfer angular momentum to material particles.

• On the other hand, spin is not the only type of angular momentum that light can hold. Instead, just like matter, light can hold orbital angular momentum, which comes from how its linear momentum density is distributed in space and therefore from how its wavefronts and spatial dependence are laid out. And, as in the link above, optical spanners can also be used to translate it to mechanical angular motion.

• That said, there is a fundamental issue in trying to split the total angular momentum of light $\mathbf J$ into spin and orbital components $\mathbf J = \mathbf L + \mathbf S$. There's a ton of subtlety involved if you want to do the maths right, mostly to do with the gauge-freedom aspects of QED (with which you can start e.g. here), but the core idea is that you cannot rotate the polarization of light arbitrarily and keep a straight toe to the Maxwell equations: if you have a wave that's linearly polarized along $x$ propagating along $z$ and you do a 90° turn about the $y$ axis, then the wave will no longer be transverse and it will break the Gauss law.

  This ultimately means that it is hard to give a fully bullet-proof definition of the spin angular momentum of a photon, but there's plenty of definitions that (while not bulletproof) are plenty good for an overwhelming majority of practical purposes.

Finally, if you want a comprehensive yet readable introduction to the subject of the angular momentum of light, I would recommend this PhD thesis:

R.P. Cameron. On the angular momentum of light. PhD thesis, University of Glasgow (2014).