Effective Lagrangians and Field Algebras with Chiral Symmetry

This paper reviews recent developments of effective Lagrangians and field algebras as means of treating chiral symmetry and partially conserved axial current (PCAC) for the study of elementary particle physics. The techniques employed are developed in considerable detail. As examples, we concentrate primarily on spin 0 and 1, linear and nonlinear realizations of $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ and $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ and some of the significant predictions of the theory are derived. The paper contains an extensive discussion of an effective Lagrangian with nonets of real scalar, pseudoscalar, vector, and axial-vector mesons that illustrate the problems of broken $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$.