The dynamics of the Einstein-Dirac system. I: A principal bundle formulation of the theory and its canonical analysis
A program for the global analysis of the solutions of the source-free gravitational field equations of Einstein had been developed in considerable detail by \textit{A. Fischer}, \textit{J. Marsden}, and \textit{V. Moncrief} [Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 33, 147-194 (1980; Zbl 0454.53044)]; in the paper under review the groundwork is laid for an extension of this program to Einstein-Dirac fields (that is, metric gravity coupled to an anticommuting spin \(1/2\) field). This development is carried out in terms of a principal O(3,1) bundle formulation of the Einstein-Dirac fields. The latter include the space time metric, represented as a frame field, together with an anticommuting Dirac spinor. A precise gauge-invariant definition of a symmetry of the Einstein-Dirac field is formulated together with a systematic analysis of gauge choice and gauge transformations. A \((3+1)\) analysis of the Einstein-Dirac fields is carried out and it is shown that the Einstein- Dirac theory admits a consistent Hamiltonian formulation. Deformations and perturbations of the Einstein-Dirac fields are examined in terms of parametrized curves in the space of all fields. Finally, solution symmetries are related to the kernel of the evolution operator. The paper concludes with an extensive appendix in which requirements from the theory of bundles are described.
scientific article; zbMATH DE number 3923624
scientific article