II.5 - STATES AND MATRIX ELEMENTS IN THE ANGULAR MOMENTUM COUPLED FOCK REPRE- Symmetrization or antisymmetrization of many-particle states of good angular momentum is carried out by fractional parentage coefficients. In field theory on the other hand the observable quantities are given in terms of field operators expressed with annihilation and creation operators well adapted to a single Slater determinant representation. Thus we must now give the tools to cast the fields into a form suitable to the angular momentum representation. The creation and annihilation operators of section II.1.2 obey the commutation or anticommutation relations In order to as (2.62) (2.63) go to a coupled scheme we note that this expression may be written with n 1 for Fermions and +1 for Bosons. After summation over the magnetic quantum numbers, with the restriction m = m' {}+ in eq. (2.63) is here of course to distinguish it from the coupling brackets [ ]. Taking away the restriction m = m' and coupling to an angular momentum I #0 with of course n (-) 2j = 1. This commutator or anticommutator relation in angular momentum representation shall be represented graphically by figure 2.12, where the square boxes are the usual crossing phases and the slash in the first one represents the factor n (Recall that the overlap box, Fig. 2.5a, implies a Kronecker factor 10.) The basis configurational state vectors for n particles, defined in Eq. (1.20), in the occupation number representation, shall be constructed and used in the invariant form In the R representation the corresponding invariant wave function is The factor I ensures normalization as shown by the norm diagram of figure 2.13 (x) entering ..... The norm √ is equal to unity if the single particle wave function in the field (x) ≈ a + can be read off the diagram of figure 2.14. We have introduced the crosshatched box for the termination of the amplitudes having a value 1/Î according is assumed to be normalized. This expression |