Conformal invariant self-interacting theory

Abstract

Some time ago Gt~RSE~ (1) investigated the possibility of extending Heisenberg's spinor theory of elementary particles (2) to a conformal invariant theory. Although the kinetic term of the equation is natural ly conformal invariant for the canonical field considered, the interaction terms only exhibit the invariance when fractional powers of the field are involved. From the vantage point of the progress in the spinor theory in recent years, a new starting point for constructing a unified self-interacting theory of elementary particles, based on eonformal invariance, is considered. The basic motivation for requiring an underlying eonformal invariance derives from the circumstance that the substratum of the theory which describes electrodynamic and neutrino phenomena, contained in a unified theory of matter, should possess conformal symmetry. The results of physical interest obtained in the Heisenberg theory at'e a consequence of the noneanonieal quantization (s) of the theory, as implied by the scale invariance (2.4) of the Heisenberg equation of motion for subcanonical dimension of the spinor field (spinor potential (5)). These features manifest themselves physically in the emergence of an attractive core (e) in fermion-antifermion interactions which generate the primary composite bosons in the theory. Furthermore, it has recently been found by DURR and the author (v) that the natural habitat for conformal invariant theories for fields of subcanonical scale dimension resides in higher-order equations, as do Lagrangian theories for such fields (s). With a view to construct a self-interacting theory, invariant under the conformal transformation