Possibility of Formulation of a Theory of Strongly Interacting Fermions

A fermion field is investigated with the interaction Lagrangian density equal to g(ψ̅ O_jψ)·(ψ̅ O_jψ). 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 Λ and/or if the momentum transferred in a collision of two particles exceeds λ. The relation between Λ and λ is such as to make the quantity λ²/ln(Λ/λ) arbitrarily small as λ→∞ and Λ→∞. This choice of the limiting procedure considerably simplifies the investigation of the theory. It is shown that in the limit λ→∞, Λ→∞, the physical interaction between particles vanishes in all types of four-fermion interactions. The case of two interacting fields ψ and χ with different isotopic spin is also considered. Going over to the local theory, the physical interaction vanishes in this case as well.

This result shows that in the cases considered no strongly interacting fermion theory can be constructed. In the case of the weak interaction, although no logically consistent theory can be built up, there does exist the perturbation theory, as in electrodynamics, which is valid for sufficiently small energies.