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Spin-orbital composition in relativistic many-fermion systems

The interplay of spins and orbital angular moments of the fermions play an important role for the structure of the many-fermion systems like atoms, nuclei, nucleons (baryons) or mesons. We start our study from the one-fermion eigenstates of angular momentum represented by the spinor spherical harmonics. Afterwards we study the properties of many-fermion states resulting from a multiple angular momentum composition of the one-fermion states, giving the total angular momentum $J=\left\langle L\right\rangle+ \left\langle S\right\rangle $, which is identified with the spin of the composite particle. We demonstrate how the composition rules affect the relativistic interplay between the sums of the spins $\left\langle S\right\rangle $ and orbital angular moments $\left\langle L\right\rangle $ of the constituents, which collectively generate the spin of composite particle. It is suggested that in a relativistic case, when the masses of the constituent fermions are much less than their energy (in the rest frame of the composite particle), then the spin of the composite particle is dominated by the orbital angular moments $\left\langle L\right\rangle $ of the constituents, while $\left\vert \left\langle \mathbb{S}\right\rangle \right\vert \leq$ $J/3$. A special attention is paid to the case $J=1/2$ that is related to the spin of proton generated by the composition of spins and orbital angular moments of the quarks.