Exact Nonperturbative Solution of a Nonlinear Fermion-Field Equation of Motion

Abstract

We construct an infinite-component free neutral fermion field ${\ensuremath{\psi}}^{h}$, of type (\textonehalf{},$j$) or ($j$, \textonehalf{}), carrying compounds lying on a Regge trajectory, as a nonperturbative solution of the non-linear fermion-field equation of motion introduced in a previous paper. The compounds are considered to be strongly interacting. It can be shown that the Reggeization and instability of the compounds lead to a propagator which is much less singular than a classical field with the same spin components but without these features. The solution can be tested in two complementary ways in the equation of motion. First, by applying a compounding technique developed in a previous paper, the corresponding spectral-function equation can be discussed in the limit of high mass-squared values, and its convergence and the necessary scale factor can be determined. The scale factor is related to the role of the effective line width in the scattering theory involved in this technique. Second, by applying the Wick normal-ordering theorem to the infinite-component free-field operators in the equation of motion, an exact nonlinear equation of motion for the field propagator is derived, effectively as a spectral-function equation. A sufficient and exact condition for the existence of this solution is derived and is in fact a linear integral equation for the momentum transform of the propagator. The convergence of this condition of solution is discussed. The condition for the existence of the nonperturbative solution ${\ensuremath{\psi}}^{h}$ is formally exactly the same as that for the existence of the lepton solution ${\ensuremath{\psi}}^{l}$ and is related to the ${\ensuremath{\gamma}}^{5}$ invariance of the primitive field equation of motion (effectively the lepton-field equation of motion), and can be put in a form showing a weak coupling of the two solutions each to a corresponding appropriate massless-fermion propagator and a massive-boson propagator. This form of the exact condition of solution also leads to an interpretation of the lepton solution ${\ensuremath{\psi}}^{l}$, discussed in a previous paper, in terms of a single free massless spin-\textonehalf{} observable fermion field ${\ensuremath{\psi}}^{\ensuremath{\nu}}$, completely satisfying unitarity. ${\ensuremath{\psi}}^{\ensuremath{\nu}}$ couples weakly and effectively nonlocally to ${\ensuremath{\psi}}^{l}$ which has an indefinite metric, and to a massive boson field, both of which are not observable, at least not in the present form of the solution where isosymmetry is not introduced and broken. In this form the Hilbert spaces corresponding to ${\ensuremath{\psi}}^{\ensuremath{\nu}}$ and ${\ensuremath{\psi}}^{h}$ are completely orthogonal.