Reduced form of statistical equilibrium equations and radiative transfer equations

An extension of the vector algebra for irreducible spherical tensor operators (ITOs) is proposed, which involves coupling two ITOs into a third one ( V(kv)=T(kt)circle times U(ku)& lowast;), with the additional condition that one of the operators is a complex conjugate and therefore is not an ITO. The correctness condition is considered for this extension. The new coupling rule treats the density matrices rho(Kl) and all tensors related to light polarization ( J(Kr) or T(K)) equally. This approach has been applied to rewrite the fundamental statistical equilibrium equations and radiation transfer equations relevant to astrophysical polarimetry. The revised equations are more concise and eliminate the need for the nj symbols. Instead, they utilize vector and scalar products of ITOs. All rate coefficients in the equations consist of two independent factors, the reduced matrix element from dipole moments and the product (vector or scalar) of the tensors rho(Kl) and J(Kr) (or T(K)).