I present and integrated approach to tetrad gravity and Maxwell-Dirac fields, which is not singular in the case when the tetrad is degenerate. All fields equations are linear. The geometric data consist only in a complex vector bundle $S\to M$ with $2$-dimensional fibres, where $M$ is a real $4$-manifold. All needed structures are functorially derived from $S$. Any considered object which is not a functorial construction is assumed to be a dynamical field. 

1991 MSC: 81R20, 53C07, 81R25, 83C60, 83C22 

1996 PACS: 03.65.Pm, 03.50.-z, 04.20.Fy, 03.50.De 

Keywords: two-spinors, connections, tetrad gravity, Maxwell-Dirac fields.

In the classical two-spinor formalism the correspondence between spacetime and spinor indices is usually considered as `adapted' to the spacetime metric, which represents the gravitational field. A somewhat different attitude consists in viewing that correspondence as the gravitational field itself. Namely, if $S$ is a complex vector bundle with $2$-dimensional fibres, one can formulate a `tetrad-gravity' approach in which the tetrad is valued in the Hermitian subspace $H$ of $S\otimes\bar S$. We obtain an integrated formulation of Einstein-Cartan-Maxwell-Dirac fields by assuming {\it minimal geometric data\/}: all the needed geometric structures are derived functorially from $S$, without fixing any extra background field; if we consider an object which is not functorially determined from $S$, then it is a dynamical field of the theory. So, the tetrad is a map $\Theta:TM\to H$; a linear connection $C$ on $S$ naturally splits into the gravitational and electromagnetic parts; the Dirac field $\psi$ is a section of $W:=S\oplus\bar S^*$ (there is a natural Clifford map $H\to{\rm End}W$). Even coupling factors naturally arise as covariantly constant sections of (tensor) powers of a real line bundle $L$, where $L^2:=L\otimes L$ is defined to be the Hermitian subbundle of $\wedge^2 S\otimes\wedge^2\bar S$. This $L$ can be seen as the `space of length units' (a spacetime object tensorialized by some power of $L$ might be called a `conformal density'. In general I will not assume a given $\epsilon\in\wedge^2S^*$, or a given Hermitian metric on the fibres of $S$ (a global fixed $\epsilon$ implies that $\wedge^2 S$ is a trivializable bundle, thus excluding the possibility of describing topological effects). 

If we translate {\it directly\/} the standard Einstein-Cartan-Maxwell-Dirac theory in terms of the above said fields $\Theta$, $C$ and $\psi$, then we are forced to use $\Theta^{-1}$ (in the standard formulation, the inverse $g^\#$ of the spacetime metric is essential). However, it has been suggested that a theory which remains non-singular also when $\Theta$ is degenerate may be interesting in view of quantization: it would allow metric signature change, and the degenerate case could be seen as a geometrical basis for a stochastical approach to quantum theory. From the mathematical point of view, the non-singularity requirement constitues a selection criterion which leads to a particularly simple formulation. 

The novelties of this paper, with respect to the previous one, are essentially the following:
now, also the treatment of the electromagnetic field satisfies the non-singularity criterion and is well integrated with the rest---as a consequence, all Lagrangians and equations are first-order; the study of the electromagnetic gauge in this context is new; the description of two-spinor connections is improved, leading to simpler expressions of the field equations, and in particular of the generalized Dirac equation; some conventions have been changed in order to have a description more consistent and readily comparable with standard approaches. \end