We present an axiomatic approach to Dirac's equation in General Relativity based on intrinsically covariant geometric structures. Structure groups and the related principal bundle formulation can be recovered by studying the automorphisms of the theory. Various aspects can be most neatly understood in this context, and a number of questions can be most properly addressed (specifically in view of formulation of QFT on a curved background). In particular we clearify the fact that the usual spinor structure can be weakened while retaining all essential physical aspects of the theory. 

1991 MSC: 15A66, 53A50, 53B35, 53C07, 53C15, 81R20, 81R25 

Keywords: spinors, quantum mechanics on a curved background, connections, Dirac equation, Weyl equations. 

In the physics literature, Dirac's equation is usually introduced and studied on flat Minkowski spacetime, while spinors have beeen extensively used in the context of classical General Relativity. Generalization of the quantum theory to curved spacetime is not so popular, perhaps because formulation of QFT on a curved background encounters severe difficulties and even paradoxes. On the other hand, there exists a rich mathematical literature about spinor structures and the Dirac equation on curved spacetime and general Riemannian manifolds, based essentially on the language of groups and principal bundles. 

Despite the aboundance of the available literature, the non-expert reader who whishes to understand the basic aspects of the relativistic physics of $1/2$-spin particles---and is little interested in mathematical generalizations---will be puzzled by the not quite clear and, sometimes, misleading presentations found in the textbooks. Even the initiate may have some difficulties in stating clearly all the precise relations between the various objects appearing in the theory: which is to be viewed as fundamental and which as a derived object; what does assuming a given object precisely imply from the algebraic and dynamical point of view; why a charged spinor is not exactly a ``spinor with charge'' (there is a subtle involvement of the group ${\bf Z}_2$), and so on. 

This presentation is intended as a setting-up of the fundamental mathematical concepts needed for the Dirac equation in General Relativity. Our strategy is to deal with a set of `minimal geometric data', namely we present a formulation containing no {\it distinguished\/} or {\it chosen\/} object devoid of a precise physical interpretation. This is to be contrasted with usual matricial formulations, which tend to mix different levels and different questions. Our language is essentially that of vector bundles and connections on vector bundles. The principal bundle approach is recovered a-posteriori, the symmetry group being the group of automorphisms of the assumed structures. Actually we recognize that principal bundle techniques are invaluable for many purposes, but we also observe that several essential features of the theory of connections, which are commonly attributed to principal bundles, can be formulated with greater generality---and even simplicity---at a more basic level. 

At the algebraic level the fundamental objects are the complex vector bundle $W$ of `$4$-spinors' over general relativistic spacetime $M$, a Dirac map\footnote{}{ This yields what is also called a `module of Clifford algebras'.} $\gamma:TM\to{\rm End}(W)$ and a Hermitian $2$-form $k$ on $W$, with signature $(2,2)$, commuting with $\gamma$ (this is essentially the `Dirac adjoint' map usually denoted by $\psi\mapsto\bar\psi$). The group of automorphisms turns out to be a kind of complexified Spin group, which in mathematical works is often denoted by Spin$^c$ (here is the involvement of the group ${\bf Z}_2$). Namely this structure is weaker than the usually assumed spinor structure, but we argue that it is completely sufficient for describing all physical facts. Assuming a proper spinor structure amounts to fixing a `charge conjugation' $\cal C$ or, equivalently, a symplectic form $\epsilon$ of a certain type on $W$. At the purely algebraic level there is no stringent motivation to regard any one of the three objects $k$, $\cal C$ and $\epsilon$ as more fundamental than the others: if one is fixed, the others are determined up to some factor. But a factor which may change from point to point is a physical field; the connections which preserve our Spin$^c$ structure contain the electromagnetic potential in a natural way, with correct gauge transformations. Fixing $\cal C$ or $\epsilon$ yields a {\it global\/} $1$-form $A$, which is too much. So, by assuming given $\cal C$ or $\epsilon$ we would get unnecessary extra-structure. Similarly, fixing a {\it positive\/} Hermitian metric on $W$ is equivalent to fixing an observer, so this is unnecessary extra-structure too. We also stress that the $2$-spinor approach turns out to be completely equivalent to our weakened $4$-spinor approach; in particular note that in $W=S\oplus\bar S^*$, where $S$ is the $2$-spinor bundle, $k$ is just a natural contraction. 

A language which is not based on principal bundles in an essential way, besides being suitable for the kind of clarification we seek, may suggest generalizations which in the principal bundle context are hardly attainable. For example one could allow for $k$ to become a dynamical rather than a fixed and static object. This would mean that the gauge group itself can be a dynamical variable and need not be constant throughout spacetime. In such a case, standard principal bundle techniques will apply only in some regions. 

Similarly, it is easy to generalise our formulation in such a way to allow discussion of spacetimes with degenerate metric of non-constant rank and no requirement for the existence of Spin$^c$ structure [a17]. Such a formulation of the dynamics may prove to be necessary in those approaches that strive to include quantum fluctuations of the metric. The ``classical" approach is not suitable for this purpose. 

Even if an expert may find that we presented no essentially new mathematical results, our consistent scheme is not at all a trivial consequence of known facts. The paper can be seen as a self-consistent alternative introduction to the Dirac equation of $1/2$-spin particles. The reader is supposed to have some familiarity only with the very basic concepts concerning (real and complex) vector bundles and connections, which are briefly recalled in the first section. 

Acknowledgements. This research has been supported by Italian MURST (national and local funds). Thanks are due to Andrzej Trautman and Marco Modugno for stimulating discussion.