States and matrix elements in the angular momentum coupled Chapter III III.1 19 19 19 20 22 2880 24 33333 State vectors and wave functions 34 Matrix elements of Fermion- Boson systems 39 FREE FIELD DISCRETIZED EXPANSIONS AND ENERGIES 41 III.4 III.3.3 Multipole expansion and discretization The free field energy for spin 1/2 particles III.5 - Spin 1 field ... With the advent of accelerators for hadrons and electrons in the 0.2-2. GeV range, the development of intermediate energy nuclear physics opens a whole new field of studies. Namely the postulate of the microscopic nonrelativistic theory of nuclei, that is nuclei are made up only of protons and neutrons interacting via two-body potentials, has to be replaced by an explicit description of the mesic degrees of freedom and of the admixture of baryonic resonances in the nuclear medium. In the present work the mathematical tools and the equations needed to compute fully relativistically many-body bound hadronic systems will be written down. As is well known the relativistic description of nuclei entails inescapably the dynamical description of the low energy structure of the nucleons and the mesons themselves. Thus the hadrons with baryonic number 0 or 1 will have to be described in a consistent manner in terms of constituent particles the nature of which (quantum numbers, masses, coupling constants .) consti[1] tutes the various phenomenological models one may want to develop. Since the nuclear physicist is interested in the characteristics (form factors, magnetic and electric multipole moments...) of bound systems, our [2-7] basic tool will be the Lagrangian field theory solved in a time indepen dent formalism in order to obtain relativistic stationary wave functions. The Hamiltonian matrix will be constructed in a truncated Hilbert space on discretized many-body configurations of correct relativistic behaviour, proper statistics and given total angular and isospin quantum numbers. The truncation |