A new approach to collective phenomena in superconductivity models is presented. In this approach nonlinear spinor models are converted into equivalent theories involving fermions and collective bosonic states. For a wide class of nonlinear spinor theories, interactions among fermions and collective states are of the renormalizable kind, and hence these models themselves turn out to be renormalizable. The equivalence of various four-fermion theories to known renormalizable models is pointed out. The original Nambu-Jona-Lasinio model, for instance, is shown to be equivalent to the linear $\sigma$ model. Detailed discussions are given on the origin of massless fields and the local gauge invariance. Implications of Pauli-Gürsey-type symmetry in spinor theories are also expounded.

• Received 26 April 1976

DOI:https://doi.org/10.1103/PhysRevD.14.2755