ON GENERAL 'SPIN' CONNECTIONS, GRAVITATION AND ELECTRODYNAMICS

It is known that intrinsic spin-vector-field analysis on a manifold provides as beautiful a description of the underlying geometry as does the description in terms of "world-vectors" defined on the manifold. However, requiring that a spinor connection be compatible with a corresponding 4-vector connection still leaves enough additional structure in the former to incorporate a gauge field. The resulting spin-curvature tensor is related to the Riemann tensor as well as an "electromagnetic field tensor" so definable. We advocate that the most general linear combination of dimensionally extended "Euler characteristics", constructed out of the generalized spin-curvature two form, be considered as a candidate for a lagrangian. It turns out to be a "natural" way to construct a unified framework for studying gravitation and electromagnetism. The consistency of the theory signals a rich structure that the underlying manifold must possess. We develop arguments to suggest that the electromagnetic field cannot be associated with amplitude transformation of the local tangent spin space but rather is consistent with the phase transformations.