For the spin 1 Boson field with isospin t we get for the different

Finally the two-body invariant matrix elements of the bilinear terms

Pi.Pi are obtained from standard recoupling procedures. For example in the case

of spin O Bosons

For the spin 1 Boson field with iso spin t we get for the different multipolarities K

(4.19)

Finally the two-body invariant matrix elements of the bilinear terms

are obtained from standard recoupling procedures. For example in the case of spin O Bosons

For an N body system of relativistic Bosons and Fermions, the center 2 of mass coordinate term R in the expression (4.1) is a sum of many-body operators. In terms of the fields it is of the form (see Appendix, Eq. (A.1))

2

i

where the E's are the free field energies, E1 = (p +

->

'i

(4.21)

The fields have been commuted (anticommuted) so as to be in corresponding pairs with same coordinate X. No sign is introduced by this operation since anticommutation always arises between pairs of Fermion operators. The i(a/at) factors operate on the immediate neighbouring Boson fields.

(

i = 1

2

2

The non-separability character of R originates in the denominator

E). In order to carry out the calculation, we go through an intermediate

2

This way

Ŕ is obtained from the double integration of a sum of separable operators which are functions of the variable Z

=

21+22