Physics and geometry from a Lagrangian: Dirac spinors on a generalised frame bundle

We clarify the structure obtained in Hélein and Vey's proposition for a variational principle for the Einstein-Cartan gravitation formulated on a frame bundle, starting from a structure-less differentiable 10-manifold [17]. The obtained structure is locally equivalent to a frame bundle thus we term it “generalised frame bundle”. In the same time, we enrich the model with a Dirac spinor coupled to the Einstein-Cartan spacetime. The obtained variational equations generalise the usual Einstein-Cartan-Dirac field equations in the sense that they are shown to imply the usual field equations when the generalised frame bundle is a standard frame bundle.