Gravity, spinors and gauge-natural bundles
The purpose of this thesis is to give a fully gauge-natural formulation of gravitation theory, which turns out to be essential for a correct geometrical formulation of the coupling between gravity and spinor fields. In Chapter 1 we recall the necessary background material from differential geometry and introduce the fundamental notion of a gauge-natural bundle. Chapter 2 is devoted to expounding the general theory of Lie derivatives, its specialization to the gauge-natural context and, in particular, to spinor structures. In Chapter 3 we describe the geometric approach to the calculus of variations and the theory of conserved quantities. Then, in Chapter 4 we give our gauge-natural formulation of the Einstein (-Cartan) -Dirac theory and, on applying the formalism developed in the previous chapter, derive a new gravitational superpotential, which exhibits an unexpected freedom of a functorial origin. Finally, in Chapter 5 we complete the picture by presenting the Hamiltonian counterpart of the Lagrangian formalism developed in Chapter 3, and proposing a multisymplectic derivation of bi-instantaneous dynamics. Appendices supplement the core of the thesis by providing the reader with useful background information, which would nevertheless disrupt the main development of the work. Appendix A is devoted to a concise account of categories and functors. In Appendix B we review some fundamental notions on vector fields and flows, and prove a simple, but useful, proposition. In Appendix C we collect the basic results that we need on Lie groups, Lie algebras and Lie group actions on manifolds. Finally, Appendix D consists of a short introduction to Clifford algebras and spinors.