Possibility of Formulation of a Theory of Strongly Interacting Fermions

A fermion field is investigated with the interaction Lagrangian density equal to $g\left(\overline{\psi }{O}_j\psi \right)·\left(\overline{\psi }{O}_j\psi \right)$. This point interaction is considered as a limit of an extended one, where it is supposed that the interaction vanishes if the momentum of a particle exceeds $\Lambda$ and/or if the momentum transferred in a collision of two particles exceeds $\lambda$. The relation between $\Lambda$ and $\lambda$ is such as to make the quantity $\frac{{\lambda }^2}{\ln \left(\frac{\Lambda }{\lambda }\right)}$ arbitrarily small as $\lambda \to \infty$ and $\Lambda \to \infty$. This choice of the limiting procedure considerably simplifies the investigation of the theory. It is shown that in the limit $\lambda \to \infty$, $\Lambda \to \infty$, the physical interaction between particles vanishes in all types of four-fermion interactions. The case of two interacting fields $\psi$ and $\chi$ with different isotopic spin is also considered. Going over to the local theory, the physical interaction vanishes in this case as well.