Abstract [to be compiled by Plain TeX]

We show that a complex vector bundle $S\to M$, where $M$ is a $4$-dimensional real manifold and the fibres of $S$ are $2$-dimensional, yields in a natural way {\it all\/} structures which are needed in order to formulate a (classical) theory of Einstein-Cartan-Maxwell-Dirac fields. Namely, all needed bundles and their fibre structures follow from functorial constructions with no further assumptions, while any considered object which is not a functorial construction is taken to be a field: the vierbein is a section $M\to T^*M\otimes H$ (where $H$ is the Hermitian subbundle of $S\otimes\bar S$, carrying a distinguished conformal Lorentz metric), a connection on $S$ naturally splits into the gravitational and electromagnetic parts, and the Dirac field is a section of $S\oplus\bar S^*$. The conformal factor is a section of a real line bundle $L$ (again functorially derived from $S$) and turns out to be covariantly constant as a consequence of the field equations. Moreover, even coupling constants arise as covariantly constant sections of powers of $L$. 

In the above said context we also discuss to what extend one can give a formulation which is not singular in the case of a degenerate vierbein. It turns out that the electromagnetic field is the only obstruction to such an extension.