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Not only the Dirac operator, but also the spinor bundle of apseudo-Riemannian manifold depends on the underlying metric. This leads totechnical difficulties in the study of problems where many metrics areinvolved, for instance in variational theory. We construct a natural finitedimensional bundle, from which all the metric spinor bundles can be recoveredincluding their extra structure. In the Lorentzian case, we also give someapplications to Einstein-Dirac-Maxwell theory as a variational theory and showhow to coherently define a maximal Cauchy development for this theory.

A Universal Spinor Bundle and the Einstein-Dirac-Maxwell Equation as a Variational Theory