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I recently got this answer to a question I asked about some naive intuitions of mine regarding the angular momentum carried by a photon, which seems plausible to me as far as I understand it, though I have to admit its in parts a bit above my head.

For my purpose here I would boil it down to: The polarization of a photon can be described by 2 dimensional a Hilbertspace and is connected to the amount of angular momentum the photon carries around the axis of propagation. When intending to talk about angular momentum around any other axis, carried by the photon, phrases like “infinite-dimensional irreducible representation of the Poincaré group“ fell and my take away is that you have to turn to hard relativistic QED for that and the naive mapping to classical intuition just doesn’t work anymore.

However I also found this rather intuitive answer, in which a procedure is described, that from my understanding I would break down as follows: When talking about the angular momentum a photon could transfer to an atom around an axis not pointing towards propagation direction of the photon: First develop the state of your atom as a superposition of states with quantization axis pointing into the propagation direction of the photon. Then calculate the angular momentum transfer your photon can excite depending on its polarization state. Then transform back to a superposition of states with quantization axis in the direction you are interested in. Depending on how the angular momentum of the atom changed around that axis, one can now make an argument about the angular momentum, the photon carried around that axis.

If we follow that approach, as far as I can tell, we would effectively map the algebraic structure of the angular momentum-states of our non-relativisticly described atom (meaning the description with $SU(2)$) onto the photon, which would be a direct contradiction to the first answer.

So my questions would be: How to resolve this contradiction? Is that practical approach just an approximation and, if yes, when is it a good approximation and when does it break down?

Qmechanic
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Zaph
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ACM is correct. You should learn and understand Wigner's classification, which will undoubtedly provide a tremendous amount of clarity.

Note that Jagerber is not actually himself certain of what to do.

Instead, the practically obvious way to deal with the situation, is to find the direction of the photon, express the atom's spin in the direction of the photon's momentum, and then consider what each helicity eigenstate does to the atom. For a photon of arbitrary helicity state, it will then have to be a quantum superposition of the two helicity eigenstates, which you can then figure out the atom's new spin state in the direction of the photon's momentum. Finally, you can rotate the atom's new spin state back to the original axes if you want.

I'd even want to decompose the plane wave into photonic spherical harmonics, i.e. multipole expansion, but that is way overkill.

naturallyInconsistent
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