Spin structures are not necessary to construct and make use of physically useful fields on pseudo-Riemannian manifolds. All that is needed is a smooth choice of one component of the space of isotropic tangent elements of maximal degree at each point on the manifold. Assuming M has this property, we may define four fundamental types of field on M. Using the algebraic operations possible with these fields the concepts of covariant differentiation, curvature and the gradient operator may be developed easily. The fact that these fields are related in a natural way to certain vector valued forms can be used to relate the methods used here to the equations of structure on M and to show how exterior differentiation and the gradient operator are connected. The Maxwell, Einstein, Lorentz and Dirac equations can be simply expressed in terms of these fields and operators. Finally, Stokes theorem combined with these equations yields integral identities for the fields of the Maxwell and Dirac theories.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics