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Two-spinor tetrad and Lie derivatives of Einstein-Cartan-Dirac fields. (English) Zbl 1463.53025

Summary: An integrated approach to Lie derivatives of spinors, spinor connections and the gravitational field is presented, in the context of a previously proposed, partly original formulation of a theory of Einstein-Cartan-Maxwell-Dirac fields based on “minimal geometric data”: the needed underlying structure is determined, via geometric constructions, from the unique assumption of a complex vector bundle \(S\rightarrowtail M\) with \(2\)-dimensional fibers, called a \(2\)-spinor bundle. Any further considered object is assumed to be a dynamical field; these include the gravitational field, which is jointly represented by a soldering form (the tetrad) relating the tangent space \(\text{T}M\) to the \(2\)-spinor bundle, and a connection of the latter (spinor connection). The Lie derivatives of objects of all considered types, with respect to a vector field \({\scriptstyle X}\colon M\rightarrow\text{T}M\), turn out to be well-defined without making any special assumption about \({\scriptstyle X}\), and fulfill natural mutual relations.

MSC:

53B05 Linear and affine connections
58A32 Natural bundles
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
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