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Gauge geometrodynamics. (English) Zbl 0695.58028

This is an exposition of the geometrical framework for gauge theories, from the bundle viewpoint. In order to handle properly full covariance for physical fields under spacetime transformations the superbundle of geometric objects is introduced. This leads to an interpretation of a spinor field as a superfield of geometric objects, precisely ensuring full covariance for spinor fields.
Contents: 1. Introduction. 2. Functors and fibre bundles. 3. Derivative spaces and differential equations. 4. Derivative spaces and variational calculus. 5. Connections and derivative spaces. 6. Geometrodynamics of gauge continuum systems and symmetry properties. 7. Classification of gauge continuum sytems. 8. Spinor superbundles of geometric objects and dynamics. 9. Conclusions.
Appendix A: Categories, final and initial objects and related structures. Appendix B: Hamiltonian formulation of Noether theorem. Appendix C: 0- sequences, exact sequences and splittings. Appendix D: Homotopy and covering. Appendix E: Euclidean spaces and related isometry groups. Appendix F: Gauge geometrodynamics vs. physical language.

MSC:

58J90 Applications of PDEs on manifolds
58C50 Analysis on supermanifolds or graded manifolds
53C27 Spin and Spin\({}^c\) geometry
81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
83E05 Geometrodynamics and the holographic principle
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