spin-1 particles. In this case, the single-particle density matrix, being a reduced description parameter, can be written as a linear combination of the unit $3\times 3$ matrix $I_{\alpha\beta}$ and the Gell-Mann linearly independent traceless Hermitian matrices $\lambda^{a}_{\alpha\beta}$ (see Appenix): $$\label{eq:2.4}
f_{\alpha\beta}({\bf p})=f^{0}({\bf p})I_{\alpha\beta}+f^{a}({\bf p})\lambda_{\alpha\beta}^{a}, \quad a=1,\dots,8.$$ The scalar $f^{0}({\bf p})$ and vectorial $f^{a}({\bf p})$ coefficients are given by $$\label{eq:2.5}
f^{0}={1\over 3}{\rm Tr}\,f({\bf p}), \quad f^{a}({\bf p})={1\over 2}{\rm Tr}\,f({\bf p})\lambda^a.$$ In contrast to spin-1/2 systems, we see that three components of spin are insufficient to describe the many-body states of the system. Indeed, according to Eqs. (\[eq:2.4\]), (\[eq:2.5\]), the states are specified by eight independent parameters determined by the generators $\lambda^{a}$ of the SU(3) group.

To clarify the physical meaning of eight parameters associated with internal symmetry, consider the realization of spin-1 operators in the vector (Cartesian) basis $|x\rangle$, $|y\rangle$, $|z\rangle$ instead of the usual canonical (irreducible) basis $|S,m\rangle$ with $S=1$ and $m=-1,0,1$. These two are related by (see e.g. [@Papa]): $$|x\rangle=\sqrt{1/2}\left(|1,1\rangle+|1,-1\rangle\right), \quad |y\rangle=-i|1,0\rangle,
\quad |z\rangle=i\sqrt{1/2}\left(|1,1\rangle-|1,-1\rangle\right).$$ In the vector basis we have $$\label{eq:2.6}
\langle i|k\rangle=\delta_{ik}, \quad S^{i}|k\rangle=i\varepsilon_{ikl}|l\rangle,$$ so that $S^{i}$ meet the usual commutation relations for spin operators, $$[S^{i},S^{k}]=i\varepsilon_{ikl}S^{l}.$$ From Eq. (\[eq:2.6\]), one finds the matrix elements for the corresponding spin 