Gravity, spinors and gauge-natural bundles

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.

Matteucci, Paolo

aa8b1ee7-e701-42c4-abae-37831e72d20f

February 2003

Matteucci, Paolo

aa8b1ee7-e701-42c4-abae-37831e72d20f

Vickers, James

719cd73f-c462-417d-a341-0b042db88634

Matteucci, Paolo
(2003)
Gravity, spinors and gauge-natural bundles.
*University of Southampton, Department of Mathematics, Doctoral Thesis*, 221pp.

Record type:
Thesis
(Doctoral)

## Abstract

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.

Text

** 00247964.pdf
- Other**
## More information

Published date: February 2003

Organisations:
University of Southampton

## Identifiers

Local EPrints ID: 50610

URI: http://eprints.soton.ac.uk/id/eprint/50610

PURE UUID: 4a96811e-5819-493c-b232-b941eb98a292

## Catalogue record

Date deposited: 27 Mar 2008

Last modified: 14 Mar 2019 01:56

## Export record

## Contributors

Author:
Paolo Matteucci

## University divisions

## Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics