# Photons with half integer angular momentum - what's happening?

I have just read this article - what is happening?

Analysing these beams within the theory of quantum mechanics they predicted that the angular momentum of the photon would be half-integer, and devised an experiment to test their prediction. Using a specially constructed device they were able to measure the flow of angular momentum in a beam of light. They were also able, for the first time, to measure the variations in this flow caused by quantum effects. The experiments revealed a tiny shift, one-half of Planck's constant, in the angular momentum of each photon.

Read more at: http://phys.org/news/2016-05-physicists.html#jCp

• See also the abstract. They consider "reduced dimensions" and "identify a new form of total angular momentum, carried by beams of light, comprising an unequal mixture of spin and orbital contributions". – J-T May 17 '16 at 12:49
• – Qmechanic May 17 '16 at 16:21

Nothing is happening. At least, nothing except that a new generalized quantity suggestively called "angular momentum" was defined and subsequently measured. But nothing we know about the usual angular momentum of photons is changed by this in any way.

Standard total angular momentum is $J = L + S$, where $L$ is the orbital and $S$ the spin angular momentum. In three dimensions and usual setups, $L$ and $S$ are not independently conserved quantities - it is only the total $J$ that is conserved. Since $S$ is integral for photons and $L$ is always integer, $J$ is always integral. Additionally, $S$ and $L$ do not separately correspond to actual transformations one can do on light as they do not preserve the transversality of the electromagnetic wave.

All that the paper "There are many ways to spin a photon: Half-quantization of a total optical angular momentum" by Ballantine, Donegan and Eastham does is consider a situation (a light beam) where there is at least one component of $L$ and $S$ that is independently conserved and generates a consistent transformation (one that preserves tranversality), so that a "generalized" angular momentum $J_\gamma = L+\gamma S$ can be defined in that direction. If you choose $\gamma=\frac{1}{2}$, it is obvious that you get half-integer values for $J_{1/2}$.

The significance of this paper (paraphrasing their own words) is firstly that they actually figured out an experimental measurement of $J_{1/2}$ and secondly that this hints at a possible "fermionization" of photons in situations where $J_{1/2}$ is a good operator, i.e. a description of the photonic system by an equivalent fermionic system. However, it must be stressed that this $J_{1/2}$ is not the usual total angular momentum, let alone spin, and hence does not contradict the usual statement of "photon angular momentum comes in integral multiples of $\hbar$". It's a generalization of the usual angular momentum $J_1$ that shows, in some situations, a half-integer quantization.

• according to authors it was not previously known that it could be measured, and was not ever measured in any prior experiment. and it is now established/ known that it can be measured. so yes, literally, "what we know has changed" but maybe not in a restricted sense. moreover, its just a single paper that may open up further research in the area. – vzn May 17 '16 at 15:12
• Hi. Is it correct to say that, although as you mentioned in your post here, the definitions are the same (concerning ang. momentum), the fact that this generalization comes up, experimentally too, is a fact that a new, or not so famous, behavior of light is coming to be studied and expand our understanding and applications? Was this something well known from before and just proved more rigorously by experiment now? Thanks. – Constantine Black May 17 '16 at 19:55
• Also, they don' t choose γ=1/2 . They produce it by demanding a reasonable fact. That is : " However, the field should be unchanged by a complete rotation, implying that l1 and l2 must be integers.", quoting from page 2 of the paper. Thus, we have a total half- integer angular momentum, even though we demanded the field to be invariant after a full rotation. Thanks. – Constantine Black May 17 '16 at 20:03
• @ConstantineBlack: Well, obviously I didn't write down the entire paper here ;) Yes, there is something special about the choice 1/2, and other choices (except the conventional 1) would not be equally good operators for a generalized angular momentum. But it's stilll not ordinary total angular momentum. And yes, this is (as far as I know) the first time this operator is studied and measured. I'm not sure what your point is. If we had known and measured this operator before, there would be nothing to publish here. – ACuriousMind May 18 '16 at 7:49

Just as a supplement to ACuriousMind's answer, it is worth noting that buried in the bottom of their paper they actually show what the "spin 1/2" eigenstates are in terms of the regular basis:

$|j=1/2\rangle=\frac{1}{\sqrt{2}}(|1, -1 \rangle + |0,1\rangle$)

$|j=-1/2\rangle=\frac{1}{\sqrt{2}}(|-1, 1 \rangle + |0,-1\rangle$)

where $|l, \sigma\rangle$ is the angular momentum in the normal $|l,s\rangle$ basis. Written out explicitly, it is clear:

1. That these are eigenstates of $L+S/2$, and
2. That this trick could only work for an integer or half-integer $\gamma$, and
3. That there's nothing too special going on here.

Still, it can still be interesting to frame an old system in a new way. I knew about the possibility of anyons in low dimensions, but I still would not have guessed that the very natural and common reduction in symmetry caused by picking a propagation axis might be sufficient for this effect. However, that might is an important qualifier: since the authors don't actually demonstrate fractional statistics or a procedure to measure them, this remains to be seen.

Edit: Emilio asks for a concrete demonstration of point two:

We want

$(L+\gamma S)(\alpha|l_1,1\rangle+\beta|l_2,-1\rangle)=j(\alpha|l_1,1\rangle+\beta|l_2,-1\rangle)$ .

This is the most general angular momentum superposition possible with a fixed total $j$, since there are only two spin possibilities. Furthermore, we know from beginning QM classes that an eigenstate of $j$ will have all these possible elements with some Clebsch-Gordon coefficients.

Applying the operators:

$((l_1+\gamma)\alpha|l_1,1\rangle+(l_2-\gamma)\beta|l_2,-1\rangle)=j(\alpha|l_1,1\rangle+\beta|l_2,-1\rangle)$

giving the conditions

$l_1+\gamma=j$

$l_2-\gamma=j$,

or $l_2-l_1=2\gamma$, $l_1\neq l_2$ .

Since $l$ are integers, this implies that $\gamma$ must be a half-integer.

• Hello. What do we mean by 3? in your post above? Is this a behavior well known and anticipated or studied? A behavior with no difference with the usual, well- known quantization of light? If you could elaborate a bit more on the no-so-special going on here, I would be thankful. – Constantine Black May 18 '16 at 6:43
• Can you elaborate on the reasons you feel the trick can only work for $2\gamma$ an integer? – Emilio Pisanty May 18 '16 at 12:47
• @EmilioPisanty I have done so – Rococo May 18 '16 at 14:43
• @ConstantineBlack what I mean is that as you can see, these "spin 1/2" states are combinations of two states with normal integer angular momenta. If you measured these states with the normal J=L+S operator, you would get either 1 or 0 for the +1/2 state, and -1 or 0 for the -1/2 state. The authors are certainly not the first people to make angular momenta superpositions of this general type. However, by constructing this alternate operator L+S/2 they have drawn out an interesting property of this simple system that had not been previously noticed. – Rococo May 18 '16 at 14:48