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Is there any intrinsic relation between the spin of a particle, and the degree of freedom of it's polarization? does it holds for any particle-wave couple? like EM-photon, gravitational_waves-graviton etc?

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Number of polarization states is strongly connected with helicity (which is spin projection on the momentum direction) of the one-particle representation of the field.

First, note that helicity and spin are not equal to each other. Spin is connected with irreducible massive representations of the Poincare group, and helicity is connected with irreducible massless representations. The spin number s of massive field refer to the $2s + 1$ number of degrees of freedom. You also can describe massive states by helicity, but it doesn't have the main sense in compare with spin (some "physical" explanation is given below).

Then, let's have some info about helicity. Massless particles are characterized by helicity value $\lambda$. Helicity can take on one of $2s + 1$ values $s, s - 1, ... -s$ like the spin. In the massless case helicity of free field is invariant under Lorentz subgroup of continuous transformations, so in general states with different helicity refer to the particles of different grades (in a sense that you can't create some space-time operation which transforms $\lambda_{1}$ state to $\lambda_{2}$ state). But it isn't invariant under operation of spatial reflection, which transform state with $\lambda$ to state with $-\lambda$. So if field theory consist of symmetry under spatial reflections or/and if theory refers to the real field, the one-particle states with helicity $\lambda$, $-\lambda $ will refer to the particles of one grade, and in the other cases it describes the different particles. Helicity, as the spin, refer to the number of degrees of freedom of the field.

For electromagnetic field (i'm not about virtual states) helicity can take one of two values - $1, -1$ (in a reasons of irreducibility condition $\partial_{\mu}A^{\mu} = 0$ and massless of EM field). Field is real and has no Noether charge, so this leads to the particles of one grade with two possible spin projections which are connected with transverse polarizations. Analogical meanings lead to the graviton with two possible states of polarizations.

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No, the number of degrees of freedom is not connected to the spin of a particle. The photon has spin one, with two degrees of freedom for the polarization. The graviton (if it exists) has spin two, but still two degrees of freedom.

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