Field Theory of Matter. IV

The relativistic dynamics of $0^{-}$ and $1^{-}$ mesons in the idealization of $U_3$ symmetry is derived from the hypothesis that a compact group of transformations on fundamental fields induces a predominantly local and linear transformation of the phenomenological fields that are associated with particles. The physical picture of phenomenological fields as highly localized functions of fundamental fields implies that the interaction term of the phenomenological Lagrange function can have symmetry properties, expressed by invariance under the compact transformation group, that have no significance for the remainder of the Lagrange function, which describes the propagation of the physical excitations. It is verified that the meson interaction term derived by considering fundamental fermion fields is invariant under the parity-conserving group $U_6\times U_6$. The implied connection between the $\rho \pi \pi$ and $\omega \rho \pi$ coupling constants is well satisfied. There is a brief discussion of the dynamics of fermion-particle triplets, from which it is shown that the invariance of the similarly derived interaction term implies the mass degeneracy of the singlet and octuplet of $1^{-}$ mesons, without relation to $0^{-}$ masses. The triplets are also used to illustrate the derivation of gauge- and relativistically invariant electromagnetic properties. The mass degeneracy of the nine $1^{-}$ mesons, and of nine $2^{+}$ mesons, can be inferred from the commutation properties of bilinear combinations of the fundamental field.