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Coframes, spinors and torsion James Burnett A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy of University College London. Department of Mathematics University College London November 2011 I, James Burnett confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. Abstract This thesis is based on five articles, four of which have been published in the Journal of Math- ematical Physics, Physical Review D, Modern Physics Letters A and Journal of High Energy Physics. The fifth has been submitted to Mathematika. In these works we study several distinct problems within the broad subject area of Mathematical Physics. The common feature is that all these works deal with rotations of one form or another. In particular, we show an equivalence between the massless and massive Dirac equations and models based on the concept of rotating material points. We also solve an open problem in Einstein-Cartan theory, namely, we find a natural matter source for a non-trivial spin angular momentum tensor. Finally, we construct a complete class of non-standard (non-local) spinor field theories and examine their possible applications in Cosmology. Acknowledgements I would like to express profound gratitude to my main supervisor Dmitri Vassiliev, with whom I have had numerous political discussions, and my pseudo second supervisor Christian Boehmer. They have both been tremendously helpful, understanding, and above all, entertaining! I also thank my second supervisor Ofer Lahav, an inspirational academic. I have been fortunate to have a good network of friends also completing their PhDs, whose helpful discussions spurred me on when needed: Finn Gubay, Dara McCutcheon, David West- ern, Doug Lazenby, Louise Jottrand, and, of course, Hannah Fry. My non-scientific friends also helped me with relaxing: Laura Casey, Susie Hogarth, Rowan Dennison, Giuseppe Diprima, Jesse Hughes, Paul Blakey, and Luci Keeley. For those of you who can not find their name but believe it should be here, I am extending my gratitude to you: I am a forgetful person! Finally, my family have always supported me, and made this thesis possible through their kind words and brilliant dinners. I dedicate this thesis to my late grandmother, Sheelah (Peggy) Burnett. She always had something funny to say, what a great way to live one’s life! Contents 1 Introduction 9 1.1 Topicsandthemes................................. 9 1.1.1 Coframe ................................. 9 1.1.2 Spinors and torsion ........................... 10 1.1.3 New class of spinors ........................... 11 2 Notation 15 2.1 Coframes ..................................... 16 j 2.1.1 e α .................................... 17 j 2.1.2 ϑ α .................................... 17 2.2 Spinors ...................................... 18 2.3 NotationforChapters3&4 . 19 2.3.1 Torsion .................................. 22 2.4 NotationforChapters5&6 . 23 2.5 Brief introduction to teleparallelism . ......... 24 I Alternative picture of particle physics. 27 3 Weyl Lagrangian 29 3.1 Thesetup ..................................... 29 3.2 Symmetries .................................... 30 Contents 6 3.3 ProofofTheorem3.1 ............................... 33 3.4 Equivalence.................................... 33 3.5 ProofofTheorem3.2 ............................... 35 3.6 Conformalinvariance . 38 3.7 Weyl’s equation in teleparallel form . ....... 41 3.8 Discussionofresults ..... ...... ..... ...... ..... .. 45 4 Dirac Lagrangian 48 4.1 What changes have we made to our model? . 50 4.1.1 Mass and electromagnetism . 51 4.2 ThenewLagrangian ............................... 52 4.3 Switching to the language of spinors . ...... 54 4.4 Separating out the coordinate x3 ......................... 57 4.5 Mainresult .................................... 58 4.6 The sign in the inequality (4.18) . ..... 59 4.7 Planewavesolutions .............................. 60 4.8 Discussion..................................... 64 4.8.1 Problem of vanishing density . 64 4.8.2 Electron in curved spacetime . 64 4.8.3 Rigid Lorentz transformations of the coframe . ....... 64 4.8.4 Our choice of Lagrangian . 65 4.8.5 Exclusionofgravity ........................... 65 4.8.6 Density as a dynamical variable . 66 4.8.7 Electron in dimension 1+3 . 66 4.8.8 Similarity with the Ashtekar–Jacobson–Smolin construction . 67 Contents 7 II Elko spinor in cosmology. 69 5 Spinors and torsion 71 5.1 A very short introduction to Elko spinors . ........ 71 5.2 Darkenergy.................................... 72 5.2.1 Cosmological Elko spinor field equations . ..... 74 5.3 Dark spinors as dark energy . 77 5.3.1 Phantom dark energy models . 77 5.3.2 Convergingmodels............................ 78 5.3.3 Divergingmodels............................. 79 5.3.4 Oscillatingmodels ............................ 80 5.3.5 Discussion ................................ 81 5.4 Elkoasasourceoftorsion . 82 5.5 Einstein-Cartan theory with Elko spinors . ......... 83 5.6 Cosmological field equations with torsion . ........ 84 5.7 Cosmological Elko spinor dynamics . ..... 87 5.7.1 Discussion ................................ 89 6 Non Standard Spinors and Cosmology 91 6.1 Energy-momentum tensor . 93 A Nonlinear second order equations which reduce to pairs of linear first order equations 94 Bibliography 99 List of Figures 5.1 Current constraints to the w Ω parameter space. 74 − m 5.2 Equation of state for V1(ϕ)............................. 78 5.3 Equation of state for V1(ϕ) (at later time). 79 5.4 Equation of state for V1(ϕ), and different initial conditions. 79 5.5 Equation of state for V1(ϕ), and different initial conditions (at later time). 80 5.6 Equation of state for V2(ϕ)............................. 80 5.7 Oscillating equation of state for V1(ϕ)....................... 81 5.8 Left: Hubble parameter and right: torsion function h .............. 88 5.9 Torsion function h in terms of e-foldings. 89 Chapter 1 Introduction 1.1 Topics and themes This thesis is based on five articles, four of which have been published in the Journal of Math- ematical Physics, Physical Review D, Modern Physics Letters A and Journal of High Energy Physics. The fifth has been submitted to Mathematika. The reader may notice discrepancies in notation between the thesis and the articles. This is due to the nature of working on two main but distinct topics during the course of the PhD. Therefore, making such notational changes was unavoidable in preparing a coherent document. The first half of the thesis details extensive work investigating an interesting link between spinors and rotating material points. The latter half deals with spinors in cosmology, specifi- cally, with what are called non-standard spinors. 1.1.1 Coframe We suggest a new geometric interpretation of both the Dirac and Weyl (massless Dirac) equa- tions. The basic idea is to view space-time as an elastic continuum whose material points can experience no displacements, only rotations, with rotations of different material points being totally independent. The idea of rotating material points may seem exotic, however it has long been accepted in continuum mechanics within the Cosserat theory of elasticity [43]. This idea also lies at the heart of the theory of teleparallelism (= absolute parallelism = fernparallelismus), a subject promoted by A. Einstein and E.´ Cartan [37, 106, 117]. With regards to the latter it is 1.1. Topics and themes 10 interesting that Cartan acknowledged [35] that he drew inspiration from the ‘beautiful’ work of the Cosserat brothers. An elastic continuum with no displacements, only rotations, is, of course, a limit case of Cosserat elasticity. The other limit case is classical elasticity with displacements only and no (micro)rotations. Rotations of material points of the elastic continuum are described mathematically by attaching to each geometric point an orthonormal basis, which gives a field of orthonormal bases called the frame or coframe, depending on whether one prefers dealing with vectors or covectors. Our model will be built on the basis of exterior calculus so for us it will be more natural to use the coframe. 1.1.2 Spinors and torsion General relativity is a successful theory in agreement with a vast number of observations. It is based on the Einstein-Hilbert action which yields the field equations if varied with respect to the metric. If, however, the metric and the connection (more precisely the non-Riemannian part of the connection with the connection assumed to be metric compatible) are considered as a priori independent variables, two field equations are obtained. The first one relates the Einstein tensor (not necessarily symmetric) to the canonical energy-momentum tensor, while the other field equation relates the skew-symmetric part of the connection, the torsion tensor, to the spin angular momentum of matter, see e.g. [67, 68, 69, 70, 66, 115]. Spin and torsion are related by algebraic equations, and torsion vanishes in the absence of sources. The cosmological principle states that the universe is homogeneous and isotropic on very large scales. More mathematically speaking, the four dimensional spacetime (M, g) is defined by 3D space-like hypersurfaces of constant time which are orbits of a Lie group G action on M, with isometry group SO(3). We assume all fields to be invariant under the action of G which means g = 0 and T λ = 0 where denotes the Lie derivative with respect Lξ µν Lξ µν Lξ
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