Spin potential for relativistic particles with spin 1/2

We analyze algebraic structure of a relativistic semi-classical Wigner function of massive particles with spin and show that it consistently includes information about the spin density matrix both in two-dimensional spin and four-dimensional spinor spaces. This result is subsequently used to explore various forms of equilibrium functions that differ by specific incorporation of spin potential. We argue that a scalar spin potential should be momentum dependent, while a tensor one may be a function of space-time coordinates only. This allows for the use of the tensor form in local thermodynamic relations. We furthermore show how scalar and tensor forms can be linked to each other.