Algebraic spinors on Clifford manifolds

Abstract

A Clifford manifold of n dimensions is defined by the fundamental relation {e?(x), e?(x)} = 2g??(x)1 between the n frame field components {e?(x)} and the metric matrix {g??(x)}. At any point x, the tangent space, orthonormal frames and the spin group are defined in terms of the frame field. Different types of field are classified in terms of their properties under the general linear coordinate transformation group on the manifold, and under spin group transformations. Connections for different types of field are determined by their covariance properties under these two groups. The bivector spin connection is then uniquely determined by the 'uniformity assumption' for Clifford algebraic grades. A key result is established, that the frame field is necessarily covariantly constant on a Clifford manifold, with both vector and spin connections. 'Spin elements' are formed by contracting the frame field with Riemannian vector fields, and possess a 'two-sided' commutator covariant derivative. A set of Riemannian fields orthonormal with respect to the manifold defines an orthonormal set of spin elements in the tangent space, from which idempotents can be constructed. If S is an asymptotically flat (n ? 1)-dimensional submanifold on which a constant idempotent is defined in terms of a constant spin frame, parallel transport along geodesics from each point of S defines a unique position-dependent extension of the idempotent in a patch P of the manifold. In an earlier model which describes the electroweak interactions of leptons, with a simplification of the Glashow Lagrangian, the 'right-hand' part of the two-sided spin connection gives rise to new gravitational terms. The nature of these new terms is discussed.