  The relevance of the contracted SU(4) group as a symmetry group of the pion
nucleon scattering amplitudes in the large $N_c$ limit of QCD raises the
problem on the construction of effective Lagrangians for SU(4) supermultiplets.
In the present study we suggest effective Lagrangians for selfconjugate
representations of SU(4) in exploiting isomorphism between so(6) and ist
universal covering su(4). The model can be viewed as an extension of the linear
$\sigma$ model with SO(6) symmetry in place of SO(4) and generalizes the
concept of the linear wave equations for particles with arbitrary spin. We show
that the vector representation of SU(4) reduces on the SO(4) level to a
complexified quaternion. Its real part gives rise to the standard linear
$\sigma$ model with a hedgehog configuration for the pion field, whereas the
imaginary part describes vector meson degrees of freedom via purely transversal
$\rho$ mesons for which a helical field configuration is predicted. As a
minimal model, baryonic states are suggested to appear as solitons of that
quaternion.
