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I am reading a book. It says that spin $s$ describes the symmetry property of a particle. Basically, the number of different wavefunctions which are transformed into linear combination of one another is $2s+1$, under rotation.
Since photon wavefunction is a vector, therefore it has spin $s=1$. So it has $2s+1=3$ components under transformation.
But photon is massless. So it doesn't have rest frame. It always moves with light speed in any frame. As a result, it has a preferred direction which is direction of the momentum of the photon.
What I don't understand is since photon can never be at rest, it doesn't have the rotation symmetry of spin = 1 (seems to me). Indeed, when writing the polarization in density matrix, it is always 2 components. And I thought this corresponds to spin = 1/2 (i.e. 2s+1=2).
So, it seems photon behaves as spin=1/2. What's my misconception here? Is it possible to have longitudinal photon? Thanks a lot!