We call "covariant" a quantity which transforms according to a (finite-dimensional) representation of the homogeneous Lorenz group. The now familiar transformation in the spin space of amplitudes, which produces covariant analytic amplitudes, is applied to polarization matrices to obtain covariant polarization matrices. The cross section for an arbitrary scattering process involving polarized particles and polarization analyzers is expressed directly in terms of covariants. Whereas the usual formalism represents the polarization of a massive particle in its rest frame, where the polarization matrices form a basis for a reducible representation of the rotation group with an expansion in irreducible multipoles of order 0 to 2s, the covariant polarization matrices form a basis for an irreducible representation of the Lorentz group, denoted by Ds,s for spin s. The polarization of a particle of spin s is represented by a real Lorentz tensor of rank 2s that is completely symmetric and traceless in any pair of indices. The positivity condition and multipole expansion are given in this representation. The case of spin 1 is studied in detail. Covariant polarization matrices are obtained for massless particles and are expressed in terms of a pseudoscalar representing the degree of longitudinal (circular) polarization, a scalar representing the degree of transverse (linear) polarization, and a 4-vector representing the direction of the transverse polarization. The case of photons is studied in detail, and it is seen that the longitudinal and transverse polarization of light correspond, respectively, to the dipole and quadrupole polarization of massive spin-1 particles.
ASJC Scopus subject areas
- Physics and Astronomy(all)