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Gravity, Spinors and Gauge-Natural Bundles University of Southampton Gravity, spinors and gauge-natural bundles Paolo Matteucci, M.Sc. Faculty of Mathematical Studies A thesis submitted for the degree of Doctor of Philosophy February UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF MATHEMATICAL STUDIES MATHEMATICS Doctor of Philosophy GRAVITY, SPINORS AND GAUGE-NATURAL BUNDLES by Paolo Matteucci, M.Sc. 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 Ap- pendix 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. Ai miei genitori Contents Acknowledgements xi Introduction 1 1 Background differential geometry 3 1.1 Preliminaries and notation . 3 1.2 Fibre bundles . 4 1.3 More on principal bundles . 7 1.3.1 Bundle of linear frames and G-structures . 12 1.4 Associated bundles . 13 1.5 Principal connections . 14 1.5.1 Linear connections . 17 1.6 Natural bundles . 19 1.7 Jets . 22 1.8 Horizontal and vertical differential . 23 1.8.1 Examples . 25 1.9 First order jet bundle . 26 1.10 Gauge-natural bundles . 27 1.10.1 Examples . 31 2 General theory of Lie derivatives 35 2.1 Generalized notion of a Lie derivative . 36 2.2 Lie derivatives and Lie algebras . 40 2.3 Reductive G-structures and their prolongations . 42 2.4 Split structures on principal bundles . 45 2.5 Lie derivatives on reductive G-structures: the Lie derivative of spinor fields 51 2.6 G-tetrads and G-tensors . 57 2.6.1 Holonomic gauge . 62 2.6.2 Lie derivative of a G-connection . 64 2.7 Critical review of some Lie derivatives . 66 2.7.1 Penrose’s Lie derivative of “spinor fields” . 66 2.7.2 Gauge-covariant Lie derivative of G-tensor valued q-forms . 69 2.7.3 Hehl et al.’s () “ordinary” Lie derivative . 69 vii Contents 3 Gauge-natural field theories 71 3.1 Variational principle . 71 3.1.1 Classical approach . 71 3.1.2 Geometric formulation on gauge-natural bundles . 73 3.2 Symmetries and conserved quantities . 77 3.2.1 Energy-momentum tensors . 83 3.3 Examples . 88 3.3.1 Classical mechanics . 88 3.3.2 Scalar fields . 89 3.3.3 Proca fields . 90 3.3.4 Yang-Mills and Maxwell fields . 91 3.3.5 General relativity . 94 4 Gauge-natural gravitation theory 99 4.1 Motivation . 99 4.2 Spin-tetrads, spin-connections and spinors . 101 4.3 Riemann-Cartan geometry on spin manifolds . 103 4.4 Einstein-Cartan-Dirac theory . 104 4.4.1 Natural approach . 108 4.5 Einstein-Dirac theory . 109 4.5.1 First order Einstein-Hilbert gravity . 110 4.6 The indeterminacy . 111 4.7 Comparison with Giachetta et al.’s () approach . 112 5 Multisymplectic derivation of bi-instantaneous dynamics 113 5.1 Motivation . 113 5.2 Multisymplectic formulation of a field theory . 114 5.3 Transition from the multisymplectic to the instantaneous formalism . 117 5.4 Bi-instantaneous dynamics . 122 5.5 Future work . 127 Conclusions and perspectives 129 A Categories and functors 131 A.1 Categories . 131 A.2 Functors . 133 B Vector fields and flows 135 B.1 General definitions . 135 B.2 A simple proposition . 137 C Lie groups and Lie algebras 139 C.1 Lie groups . 139 C.2 Lie algebras . 140 C.3 Lie group actions on manifolds . 143 viii Contents D Clifford algebras and spinors 145 D.1 Clifford algebra, γ matrices and spin group . 145 D.2 Spin structures and spinors . 146 D.3 2-spinors . 148 Bibliography 149 ix Acknowledgements It is a pleasure to acknowledge the supervision and constant encouragement of Prof. James A. Vickers throughout all the stages of this research project. The author would also like to express his deep gratitude to Dr. Marco Go- dina, University of Turin, Italy, for guidance and useful discussions on the more strictly geometrical aspects of the project. Moreover, some of the re- sults presented in this thesis are, in many respects, the climax of a long-term research project carried out over several years by the mathematical physics group of the Department of Mathematics of the University of Turin, whose indirect contribution to this work is hereby gratefully acknowledged. In partic- ular, the recent doctoral theses of Dr. Lorenzo Fatibene () and Dr. Marco Raiteri () have proved to be invaluable sources of background material. Finally, the author gladly acknowledges an EPSRC research studentship and a Faculty Research Studentship from the University of Southampton for fi- nancial support throughout his doctorate. xi Introduction Vos igitur, doctrinae et sapientiae filii, perquirite in hoc libro colli- gendo nostram dispersam intentionem quam in diversis locis propo- suimus et quod occultatum est a nobis in uno loco, manifestus fecimus illud in alio, ut sapientibus vobis patefiat. H. C. Agrippa von Nettesheim, De occulta philosophia, III, lxv It is commonly accepted nowadays that the appropriate mathematical arena for classical field theory is that of fibre bundles or, more precisely, of their jet prolongations (cf., e.g., Atiyah ; Trautman ; Saunders ; Giachetta et al. ). What is less often realized or stressed is that, in physics, fibre bundles are always considered together with some special class of morphisms, i.e. as elements of a particular category. In other words, fields are always considered together with a particular class of transformations. The category of natural bundles was introduced about thirty years ago and proved to be an extremely fruitful concept in differential geometry. But it was not until recently, when a suitable generalization was introduced, that of gauge-natural bundles, that the relevance of this functorial approach to physical applications began to be clearly per- ceived. The notion of naturality traditionally relates to the idea of coordinate invariance, or “covariance”. The more recent introduction of gauge invariance into physics gives rise to the very idea of gauge-natural bundles. Indeed, every classical field theory can be regarded as taking place on some jet pro- longation of some gauge-natural (vector or affine) bundle associated with some principal bundle over a given base manifold (Eck ; Kol´aˇr et al. ; Fatibene ). On the other hand, it is well known that one of the most powerful tools of Lagrangian field theory is the so-called “Noether theorem”. It turns out that, when phrased in modern geometrical terms, this theorem crucially involves the concept of a Lie derivative, and here is where the aforementioned functorial approach is not only useful, but also intrinsically unavoidable. By using the general theory of Lie derivatives, one can see that the concept of Lie differentiation is, crucially, a category-dependent one, and it makes a real difference in taking the Lie derivative of, say, a vector field if one regards the tangent bundle as a purely natural bundle or, alternatively, as a more general gauge-natural bundle associated with some suitable principal bundle (cf. Chapter 2). In Chapter 4 we shall show that this functorial approach is essential for a correct geometrical formulation of the Einstein (-Cartan) -Dirac theory and, at the same time, yields an unexpected freedom in the concept of conserved quantities. As we shall see, this freedom originates from the very fact that, when coupled with Dirac fields, Einstein’s general relativity can no longer be regarded as a purely natural theory, because, in order to incorporate spinors, one must enlarge the class of morphisms of the theory. 1 Introduction This is the general idea which underlies the present work. An interesting by-product of this analysis is the successful systematization of the long-debated concept of a Lie derivative of spinor fields. A synopsis of the thesis follows. 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. 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, i.e.
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