Poincaré gauge invariant spinor theory and the gravitational field. II

Abstract

The nonlinear equation for an abstract noncanonical 2-component Weyl spinor field — as used with the inclusion of internal symmetries in Heisenberg's nonlinear spinor theory of elementary particles — which is invariant under scale, phase, and Poincaré transformations is modified in such a way as to become invariant under spacetime dependent phase gauge and Poincaré gauge transformations. In such an equation a phase gauge field B _m , six Lorentz gauge fields A[κλ]m and four translation gauge fields g_μm have to be introduced. It is demonstrated that all these fields can be identified as certain combinations of the Weyl spinor field, and hence should be considered in a rough sense as ‘bound states’ of this spinor field. In particular the ‘electromagnetic field’ B_m and the ‘gravitational field’ g μm appear as S-states and P-states of a spinor-antispinor system. The noncanonical property and the operator character of the spinor field is essential for this result. The relation between the translation gauge field and the spinor field involves a fundamental length. In a classical geometrical interpretation this relation leads to Einstein's equation of gravitation without cosmological term in a Riemannian space without torsion if the fundamental length is identified with Planck's length. It is shown that this equation is covariant under the larger symmetry group of phase gauge and Poincaré gauge transformations. The modified nonlinear equation constructed solely from a single 2-component Weyl field hence seems to incorporate in an extremely compact way ‘electromagnetic’ and ‘gravitational’ interaction in addition to non-mass-zero interactions. In this equation no arbitrary dimensionless constants enter. The considerations can be generalized to Dirac spinor fields and to spinor fields involving additional interior degress of freedom.