Faddeev Equations for Realistic Three-Nucleon Systems. I. Complete Angular Momentum Reduction and Antisymmetrization of States

A complete angular momentum reduction of the Faddeev equations is carried out for the case of realistic, nonrelativistic three-nucleon systems with (local and/or nonlocal) interactions having general spin, isospin, and velocity dependence. Antisymmetrization of states with respect to particle exchange is properly accounted for by using properties of the permutation group and the isospin formalism. Expressions for the Faddeev equations, in the form of coupled two-variable integral equations, are obtained in two different coupling schemes; $J-j$ [coupling of the (relative orbital plus total spin) angular momentum of a nucleon pair with the total angular momentum of the third nucleon (in the c.m. system) to give the total angular momentum (in the c.m. system)]; and $\mathcal{L}-\mathcal{S}$ [coupling of the total orbital angular momentum (relative orbital angular momentum of a nucleon pair plus the orbital angular momentum of the third nucleon) in the c.m. system with the total spin angular momentum (total spin angular momentum of a nucleon pair plus the spin angular momentum of the third nucleon) to give the total angular momentum in the c.m. system].