Relativistically invariant representation of arbitrary bilinear combinations of Dirac spinors

It is shown that in the general case a bilinear combination of two arbitrary four-spinors with one of the 16 basic Dirac matrices cannot be expressed in terms of the momentum and spin four-vectors (Pauli–Lubaski vector). A universal general formula obtained in this paper also includes four-vectors corresponding to two three-dimensional unit vectors in the particle rest frame perpendicular to the direction of the particle spin. The transition to vanishing masses is considered. Special expressions that describe binary combinations of spinors in the case of elastic scattering for helicity and “transversal” amplitudes are given.