I am learning QFT， and I find it very difficult to understand the spin of photon. Firstly I have some facts listed here:
- We can get the spin of an electron from Dirac equation. The reasons for electrons to have two spin polarizations are due to: The dimension of the physical space time is 4 Why don't we construct a spin 1/4 spinor?. Therefore, if we want to construct a spin space, we need at least four bases vectors. For fermions, we can use two complex bases vectors to construct a spinner(hence two spin polarization states) . This is because, these two could provide four independent variables since complex. This can be easily seen from below:
The word spin comes from the fact that when we try to make a link or map between the spin space and the real physical space, we need, locally, the Lie algrebra, which is responsible for the spinning behaver: SU(2)~SO(3) locally.
2.For photons, things are much the same. However photon polarization expands a 4 dimensional real space! Therefore 4 bases vectors are needed. And hence, it has four polarizations. But, different from fermions, when we map this polarization space with the real physical space, we do not need locally Lie algebras! see below, we can simply align their bases:
My question is :
Since we don't have to use Lie algebra for photons why do we have photon spin? I know that only two polarizations are physical, but how are the other, especially linear polarization case, related to spinning without mathematical structures like Lie algebra?
I also know from student friendly QFT that circular polarization is not spin, so if any answers that say the linear polarization can be treated as the superposition of circular cases, would you please explain more?
following is from student friendly QFT