# Spin of photons and angular momentum of classical radiation field. Are they connected?

Photons, the quanta of the radiation field, carry spin $$-$$ an intrinsic angular momentum. Classical radiation also has angular momentum having the expression $$\vec{J}=\frac{1}{\mu_0c^2}\int dV \left(\vec{r}\times\left(\vec{E}\times\vec{B}\right)\right).$$

Is it possible that the angular momentum of the classical radiation field emerges as the cumulative effect of spins of the photons? If yes, how to show this? I asked this question to various people in my University but I did not get a very definite answer let alone a rigorous one.

## 2 Answers

They are the same. This is the subject of my favorite classic physics paper: Richard Beth, Mechanical detection and measurement of the angular momentum of Light, Physical Review 50 115 (1936).

Beth constructed a torsion pendulum by suspending a half-wave plate from a fiber, and illuminated the device from below with bright circularly-polarized light. Above this he mounted a fixed quarter-wave plate and a mirror. In the quantum-mechanical picture, each circularly-polarized photon traveling through the half-wave plate exchanges angular momentum $$2\hbar$$ with the pendulum on the way up, has its spin returned to the original orientation during its two trips through the fixed quarter-wave plate, and exchanges another $$2\hbar$$ with the half-wave plate on the way down. By toggling a circular polarizer outside of the vacuum chamber at the pendulum’s resonant frequency, Beth used the angular momentum from this light to cause this macroscopic object to twist.

The angular momentum stored in the classical electromagnetic field predicts the same pendulum motion as the quantum-mechanical explanation; the analysis occupies a substantial fraction of Beth’s paper.

• Note that a current hot topic is constructing light fields where photons have “orbital angular momentum”; that’s beyond the scope of my answer. – rob Dec 10 '20 at 16:33
• Are you answering this question: physics.stackexchange.com/questions/599693/… – mithusengupta123 Dec 10 '20 at 16:39
• I started writing this answer before you asked that question. To my mind the other is a duplicate. – rob Dec 10 '20 at 16:44
• But in this one, I am asking how could we show if angular momentum really arises from spin, mathematically. Anyway, thanks for the answer though :-) – mithusengupta123 Dec 10 '20 at 16:55
• Angular momentum is fungible. Spin is interesting in QED because it appears unexpectedly when you try to construct all of the systems that are invariant under Lorentz transformations. But in a big composite system, you can’t distinguish between spin angular momentum and orbital angular momentum. This is a big issue in nuclear physics: even the simplest composite nucleus, $^2$H, has a ground-state wavefunction with both $L=0$ and $L=2$ contributions, so you can’t talk about “the spins of the nucleons” in a deuteron even if you polarize it. ... – rob Dec 10 '20 at 17:20

There is no expression for spin in the standard theory of electromagnetism. The subject is in fact avoided. For example the only reference to electromagnetic spin in Jackson Classical Electrodynamics is in problem 7.27. The main text does not mention it.

The total angular momentum, including spin, is correctly given by your equation, which has the form of angular momentum. So what is happening? Jackson's p7.27 gives a clue. The Noether AM distribution belonging to the standard, gauge invariant Lagrangian contains a spin and an orbital AM density contribution. However, only their sum is conserved. Also the energy-momentum distribution is asymmetric. None of the Noether current distributions are gauge invariant. The solution to keep everything gauge invariant is to replace the Noether energy-momentum (EM) distribution with the Belinfante distribution. This leads to your expression of AM. The spin contribution is no longer explicit.

I published a paper that proposes an alternative. It also discusses a paradox that arises when you try to understand Beth's experiment, mentioned by @rob, with your expression for $$J$$. My solution is incompatible with the principle of gauge invariance. Instead it accounts for any physical situation with a unique electromagnetic potential.

• I think, from skimming your paper, that your paradox is that the torque in the Beth experiment is non-local when analyzed using classical fields. Is that the gist? That’s interesting. – rob Dec 10 '20 at 16:37
• @rob Perhaps the term "nonlocal" could describe the situation. There is no angular momentum in a plane circularly polarised wave in standard theory, as the Poynting vector is perpendicular to the propagation direction. So how could there be a torque? What happens is that behind the reflective disk hanging on the torsion balance the wave is no longer plane. At the edge of the shadow region an angular momentum appears that exactly accounts for the torque. – my2cts Dec 10 '20 at 16:45
• Very interesting. I have some follow-up questions, but they are too complex for a comment and I have work to do before I can turn them into proper standalone questions. – rob Dec 10 '20 at 16:50
• @my2cts " ... to replace the Noether EM distribution with the Belinfante distribution." What do you mean by Noether EM distribution? The stress-energy tensor? If you could use the standard symbols, in your answer, it will be easier for me to follow it. Thanks – mithusengupta123 Dec 10 '20 at 16:53
• @rob I am most interested in your feedback. – my2cts Dec 10 '20 at 17:02