On Lie derivation of spinors against arbitrary tangent vector fields

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Written by Andras Laszlo

It is well-known since the work of Lichnerowicz and Kosmann in the '70-s, that the notion of Lie derivation along conformal Killing vector fields can be uniquely lifted to the two-spinor bundle over a four dimensional spacetime manifold, via requiring compatibility with the Infeld-Van der Waerden symbol and the preservation of the complex phase of the spinor maximal form. That formula, also called Kosmann Lie derivation, is known to provide a faithful Lie algebra representation of the conformal Killing vector fields within the Lie algebra of the derivation operators on spinor fields. This naturally motivated the question: can this notion be generalized to arbitrary tangent vector fields of the spacetime, which are not necessarily conformal Killing? Lot of discussion appeared in the literature on this and related questions, and are expanded in a large set of papers of Forgach, Manton, Kolar, Michor, Slovak, Godina, Jhangiani, Hurley, Vandyck etc throughout the years. Even nowadays papers appear on the matter. Unfortunately, it seems that the confusion in the literature about the possible generalization of the spinor Lie derivation to arbitrary tangent vector field still persists. Motivated by this, we managed to find a compact answer via the more generic approach of Lie derivations over arbitrary vector bundles. The key finding, usually not mentioned in the literature, is that the Kosmann formula fails to be a Lie algebra homomorphism for tangent vector fields which are not conformal Killing. It shall be shown, this does not mean an interpretational problem, but rather signifies that one needs to allow also a vertical part of the perninent Lie derivation. It will be shown that such notion of spinor Lie derivation is uniquely characterized by preserving the following structures: the vector bundle structrure of the spinor bundle and the solderability of the spinor bundle to the tangent bundle, but not preserving the soldering form, for instance. It also shall be shown that it can serve as a natural unified framework to encode spacetime and internal (gauge) symmetries.