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Higher Spin Fields on Curved Spacetimes

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Higher Spin Fields on Curved Spacetimes Universit¨at Leipzig Fakult¨at fur¨ Mathematik und Informatik Mathematisches Institut Higher Spin Fields on Curved Spacetimes Diplomarbeit im Studiengang Mathematik vorgelegt von Rainer Muhlhoff¨ Betreuender Hochschullehrer: Prof. Dr. Rainer Verch Institut fur¨ Theoretische Physik Leipzig, im Dezember 2007 Contents Preface v Notation and Conventions ix I. Spinors and Representation Theory 1 1. Background in Representation Theory 3 1.1. Basic Definitions and Important Theorems . ..... 3 1.2. Forming New Representations Out of Existing Ones . ....... 5 2. Some Complex Algebra 9 2.1. Complex Conjugation of Vector Spaces and Lie Algebras . ........ 9 2.2. Complex Conjugation of Linear Maps and Representations........ 13 3. Representation Theory of SL(2, C) 17 3.1. Review: Representation Theory of SU(2) . ..... 17 3.2. Lie Algebras of and SL(2, C) ....................... 23 L 3.3. Representation Theory of SL(2, C) ..................... 25 4. The 2-Spinor Formalism 37 4.1. Foundationsof2-SpinorNotation . .... 37 4.2. Vector-Spinor-Correspondence . ..... 43 5. Clifford Algebras and Dirac Spinors 49 5.1. Clifford Algebras and Related Definitions . ...... 49 5.2. The Spinor Representation of Cl1,3 andDirac-Matrices . 52 5.3. TheDirac-SpinorFormalism . .. 58 II. Differential Operators on Spinor Fields 63 6. Spinor Fields on Curved Spacetime 65 6.1. Pseudo-Riemannian and Lorentzian Manifolds . ....... 65 6.2. Principal Bundles and Associated Bundles . ...... 69 6.3. Connections on Principal Bundles . .... 74 6.4. Spin Structures and Spinor Bundles . .... 79 iii Contents 7. Normally Hyperbolic Differential Operators 87 7.1. DifferentialOperatorsonManifolds . ..... 87 7.2. Normally Hyperbolic Differential Operators . ....... 93 7.3. ExistenceofGreen’sOperators . ... 96 III.Buchdahl’s Generalised Dirac Equations 99 8. Solving Buchdahl’s Generalised Dirac Equations 101 8.1. Statement of Buchdahl’s Generalised Dirac Equations . .......... 101 8.2. Lemmas for Buchdahl’s Equations in 2-Spinor Notation . ......... 105 8.3. Normal Hyperbolicity and Existence of Green’s Operators ........ 108 8.4. TheCauchyProblem ............................. 109 9. CAR-Algebraic Quantisation of Buchdahl Fields 115 9.1. TheGeneralIdea ............................... 115 9.2. Generalisation of Dimock’s Quantisation . ....... 117 9.3. QuantisationinIllge’sFramework . ..... 122 Table of Relevant Lie Groups 127 Bibliography 131 iv Preface The present thesis investigates Buchdahl’s equations for the description of massive par- s ticles of arbitrary spin 2 , s Æ, on curved spacetime manifolds. This is a system of two first order linear differential∈ equations for spinor fields, first studied by Buchdahl [Buc82a] and given by Wunsch¨ [Wun85]¨ in the equivalent formulation AA ...As A1...As−1 ˙ ψ 1 −1 µϕ =0 ∇XA − X˙ (A X˙ A1...As−1) AA1...As−1 | ϕ | ν ψ =0 . (∇ X˙ − Assuming the spacetime manifold (M,g) to be a 4-dimensional, globally hyperbolic, ori- ented and time-oriended Lorentzian manifold, existence and uniqueness of advanced and retarded Green’s operators for a certain quite general class of first order linear differen- tial operators (including Buchdahl’s equations and Dirac’s equation) will be proved, the Cauchy problem for Buchdahl’s equations will be solved globally for all s Æ, and two possible constructions for quantizing Buchdahl fields using CAR-algebras in∈ the fashion of [Dim82] will be given. This is a mathematics diploma thesis on a topic in theoretical physics. It is one of the goals to present all considerations in an abstract and rigerous mathematical formulation. To this end, and to make this document accessible to both readers from a mathematical and readers from a physical background, one further main aspect of this thesis (apart from the study of Buchdahl’s equations) is fully developing a comprehensive mathemat- ical framework for dealing with spinor fields on Lorentzian manifolds. This approach and all the in-depth preparatory work will pay off strikingly when finally starting the investigation of Buchdahl’s equations in chapter 8, as they will have made the concrete situation accessible to the powerful results on normally hyperbolic linear differential operators on globally hyperbolic Lorentzian manifolds by B¨ar, Ginoux and Pf¨affle [BGP]. Using these, the main results sketched above, which are generalisations of results by Dimock [Dim82] and Wunsch¨ [Wun85],¨ will be obtained in an elegant and straightforward manner. Overview and Structure of this Thesis As stated above, Buchdahl’s equations are a system of differential equations on spinor fields on a spacetime manifold (M,g). By a spinor field we1 mean a section of an associated vector bundle DM := (M) D ∆ on M, where (M) denotes a spin structure +S × S (which has structure group Spin (1, 3) ∼= SL(2, C) in our case) and (D, ∆) is a finite- dimensional vector space representation of SL(2, C). More precisely, sections of DM are called spinor fields of type D. Spinor fields are used for the description of particles in physics, and different types of SL(2, C)-representations, i. e. different types of spinor fields, generally correspond with different (classes/types of) particles. 1Throughout this document, “we” will be used in the sense of “the author and you—the reader”. v Preface This indicates that the representation theory of SL(2, C), notably the classification of finite-dimensional irreducible representations of SL(2, C), is of particular relevance for the physical theory. More precisely, as we will see in chapter 3, of relevance are the (equivalence classes of) finite-dimensional irreducible representations of SL(2, C) as real Lie group, which makes the classification procedure significantly more complicated com- C C pared to the “sl(2, C) = su(2) ”-case (this is the complex Lie algebra of SL(2, )) presented in many introductory⊗ text books on representation theory. This thesis will be subdivided into three parts and the first big goal in part I will be classifying finite-dimensional irreducible representations of the real Lie group SL(2, C). We will do this in chapter 3 by facilitating the well known classification of irreducible vector space representations of SU(2) in a way that is indicated but not carried out with satisfactory mathematical rigour in [SU01]. As a preparation for chapter 3 we collect relevant basic facts from representation theory in chapter 1. In chapter 2, we will develop a “little theory of complex algebra”, i. e. of complex conjugate vector spaces, linear maps, tensor products, representations etc. This will be a core ingredient for our mathematical approach to the representation theory of SL(2, C). In chapter 4 we will give a thorough introduction to the 2-spinor formalism in a way based on abstract indices. The 2-spinor formalism is an elementary formalism for dealing ( 1 ,0) (0, 1 ) with spinors of type D 2 , D 2 , and their tensor products. It (historically) originates from physics and usually comprises a notation in components with respect to standard bases. But by adopting abstract index notation and in virtue of our considerations in chapter 2 we will be able to set up a strict formalism which deals with abstract objects. Chapter 5 is entirely devoted to the illumination of the interrelationship of the Clifford 4 algebra Cl1,3 of Minkowski spacetime (Ê , η), its spinor representations, Dirac matrices and the Dirac-spinor formalism. We will propose a precise notion of“a collection of Dirac matrices”, learn how collections of Dirac matrices correspond with spinor representations of Cl1,3 and prove a general version of Pauli’s fundamental theorem on Dirac matrices. Then we will propose a formal notion of “the standard representation”, which is a re- alisation of the spinor representation of Cl1,3 connected with the 2-spinor formalism in a canonical way. This will then lead to the declaration of the Dirac-spinor formalism, again in the fashion of abstract index notation. Part II of this thesis starts with chapter 6, where the construction of spinor bundles on our 4-dimensional oriented and time-oriented Lorentzian spacetime manifold (M,g) is carried out in full detail. To make this thesis sufficiently self-contained, we shall begin by reviewing relevant facts from Lorentzian geometry and about principal bundles, associated vector bundles and spin structures. Eventually we will construct the spinor (j,j′) AB AX˙ bundles D M := (M) D(j,j′) ∆j,j′ and extend the (tensor-)spinors ǫ , σa and A˜ S × γa B˜ as given fiberwise in part I to (tensor-)spinor fields on M. It is a technical but very important result at the end of chapter 6 that the covariant derivatives on the spinor bundles induced by the Levi-Civita covariant derivative on T M are mutually compatible AB AX˙ A˜ (in an appropriate sense) and that ǫ , σa and γa B˜ are parallel with respect to these covariant derivatives. In chapter 7 we will introduce linear differential operators on vector bundles. We will fix the central concept of normally hyperbolic differential operators, which are second order linear differential operators with metric principal part. These operators and the vi solution theory of differential equations given by them on globally hyperbolic manifolds were studied to a large extent in [BGP]. In section 7.3 we will use results from this useful reference to prove existence and uniqueness of advanced and retarded Green’s operators for a first order linear differential operator P on a vector bundle , as soon as there is a second first order linear differential operator Q on , such thatE P Q is E normally hyperbolic. This is a generalisation of [Dim82, theorem 2.1] and turns out to s be applicable to the Buchdahl operator for all spin numbers , s Æ. 2 ∈ Part III of this thesis starts with chapter 8, where we will apply the whole theory developed so far to Buchdahl’s generalised Dirac equations for spinor fields, where M B is assumed globally hyperbolic.
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