Geometry of Spinors*
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Topology and Physics 2019 - Lecture 2
Topology and Physics 2019 - lecture 2 Marcel Vonk February 12, 2019 2.1 Maxwell theory in differential form notation Maxwell's theory of electrodynamics is a great example of the usefulness of differential forms. A nice reference on this topic, though somewhat outdated when it comes to notation, is [1]. For notational simplicity, we will work in units where the speed of light, the vacuum permittivity and the vacuum permeability are all equal to 1: c = 0 = µ0 = 1. 2.1.1 The dual field strength In three dimensional space, Maxwell's electrodynamics describes the physics of the electric and magnetic fields E~ and B~ . These are three-dimensional vector fields, but the beauty of the theory becomes much more obvious if we (a) use a four-dimensional relativistic formulation, and (b) write it in terms of differential forms. For example, let us look at Maxwells two source-free, homogeneous equations: r · B = 0;@tB + r × E = 0: (2.1) That these equations have a relativistic flavor becomes clear if we write them out in com- ponents and organize the terms somewhat suggestively: x y z 0 + @xB + @yB + @zB = 0 x z y −@tB + 0 − @yE + @zE = 0 (2.2) y z x −@tB + @xE + 0 − @zE = 0 z y x −@tB − @xE + @yE + 0 = 0 Note that we also multiplied the last three equations by −1 to clarify the structure. All in all, we see that we have four equations (one for each space-time coordinate) which each contain terms in which the four coordinate derivatives act. Therefore, we may be tempted to write our set of equations in more \relativistic" notation as ^µν @µF = 0 (2.3) 1 with F^µν the coordinates of an antisymmetric two-tensor (i. -
21. Orthonormal Bases
21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else. -
Abstract Tensor Systems and Diagrammatic Representations
Abstract tensor systems and diagrammatic representations J¯anisLazovskis September 28, 2012 Abstract The diagrammatic tensor calculus used by Roger Penrose (most notably in [7]) is introduced without a solid mathematical grounding. We will attempt to derive the tools of such a system, but in a broader setting. We show that Penrose's work comes from the diagrammisation of the symmetric algebra. Lie algebra representations and their extensions to knot theory are also discussed. Contents 1 Abstract tensors and derived structures 2 1.1 Abstract tensor notation . 2 1.2 Some basic operations . 3 1.3 Tensor diagrams . 3 2 A diagrammised abstract tensor system 6 2.1 Generation . 6 2.2 Tensor concepts . 9 3 Representations of algebras 11 3.1 The symmetric algebra . 12 3.2 Lie algebras . 13 3.3 The tensor algebra T(g)....................................... 16 3.4 The symmetric Lie algebra S(g)................................... 17 3.5 The universal enveloping algebra U(g) ............................... 18 3.6 The metrized Lie algebra . 20 3.6.1 Diagrammisation with a bilinear form . 20 3.6.2 Diagrammisation with a symmetric bilinear form . 24 3.6.3 Diagrammisation with a symmetric bilinear form and an orthonormal basis . 24 3.6.4 Diagrammisation under ad-invariance . 29 3.7 The universal enveloping algebra U(g) for a metrized Lie algebra g . 30 4 Ensuing connections 32 A Appendix 35 Note: This work relies heavily upon the text of Chapter 12 of a draft of \An Introduction to Quantum and Vassiliev Invariants of Knots," by David M.R. Jackson and Iain Moffatt, a yet-unpublished book at the time of writing. -
Killing Spinor-Valued Forms and the Cone Construction
ARCHIVUM MATHEMATICUM (BRNO) Tomus 52 (2016), 341–355 KILLING SPINOR-VALUED FORMS AND THE CONE CONSTRUCTION Petr Somberg and Petr Zima Abstract. On a pseudo-Riemannian manifold M we introduce a system of partial differential Killing type equations for spinor-valued differential forms, and study their basic properties. We discuss the relationship between solutions of Killing equations on M and parallel fields on the metric cone over M for spinor-valued forms. 1. Introduction The subject of the present article are the systems of over-determined partial differential equations for spinor-valued differential forms, classified as atypeof Killing equations. The solution spaces of these systems of PDE’s are termed Killing spinor-valued differential forms. A central question in geometry asks for pseudo-Riemannian manifolds admitting non-trivial solutions of Killing type equa- tions, namely how the properties of Killing spinor-valued forms relate to the underlying geometric structure for which they can occur. Killing spinor-valued forms are closely related to Killing spinors and Killing forms with Killing vectors as a special example. Killing spinors are both twistor spinors and eigenspinors for the Dirac operator, and real Killing spinors realize the limit case in the eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds of positive scalar curvature. There is a classification of complete simply connected Riemannian manifolds equipped with real Killing spinors, leading to the construction of manifolds with the exceptional holonomy groups G2 and Spin(7), see [8], [1]. Killing vector fields on a pseudo-Riemannian manifold are the infinitesimal generators of isometries, hence they influence its geometrical properties. -
Arxiv:Gr-Qc/0611154 V1 30 Nov 2006 Otx Fcra Geometry
MacDowell–Mansouri Gravity and Cartan Geometry Derek K. Wise Department of Mathematics University of California Riverside, CA 92521, USA email: [email protected] November 29, 2006 Abstract The geometric content of the MacDowell–Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geomet- ric meaning to the MacDowell–Mansouri trick of combining the Levi–Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous ‘model spacetime’, including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti de Sitter, or other models. A ‘Cartan connection’ gives a prescription for parallel transport from one ‘tangent model spacetime’ to another, along any path, giving a natural interpretation of the MacDowell–Mansouri connection as ‘rolling’ the model spacetime along physical spacetime. I explain Cartan geometry, and ‘Cartan gauge theory’, in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell–Mansouri gravity, as well as its recent reformulation in terms of BF theory, in the arXiv:gr-qc/0611154 v1 30 Nov 2006 context of Cartan geometry. 1 Contents 1 Introduction 3 2 Homogeneous spacetimes and Klein geometry 8 2.1 Kleingeometry ................................... 8 2.2 MetricKleingeometry ............................. 10 2.3 Homogeneousmodelspacetimes. ..... 11 3 Cartan geometry 13 3.1 Ehresmannconnections . .. .. .. .. .. .. .. .. 13 3.2 Definition of Cartan geometry . ..... 14 3.3 Geometric interpretation: rolling Klein geometries . .............. 15 3.4 ReductiveCartangeometry . 17 4 Cartan-type gauge theory 20 4.1 Asequenceofbundles ............................. -
Tensor Manipulation in GPL Maxima
Tensor Manipulation in GPL Maxima Viktor Toth http://www.vttoth.com/ February 1, 2008 Abstract GPL Maxima is an open-source computer algebra system based on DOE-MACSYMA. GPL Maxima included two tensor manipulation packages from DOE-MACSYMA, but these were in various states of disrepair. One of the two packages, CTENSOR, implemented component-based tensor manipulation; the other, ITENSOR, treated tensor symbols as opaque, manipulating them based on their index properties. The present paper describes the state in which these packages were found, the steps that were needed to make the packages fully functional again, and the new functionality that was implemented to make them more versatile. A third package, ATENSOR, was also implemented; fully compatible with the identically named package in the commercial version of MACSYMA, ATENSOR implements abstract tensor algebras. 1 Introduction GPL Maxima (GPL stands for the GNU Public License, the most widely used open source license construct) is the descendant of one of the world’s first comprehensive computer algebra systems (CAS), DOE-MACSYMA, developed by the United States Department of Energy in the 1960s and the 1970s. It is currently maintained by 18 volunteer developers, and can be obtained in source or object code form from http://maxima.sourceforge.net/. Like other computer algebra systems, Maxima has tensor manipulation capability. This capability was developed in the late 1970s. Documentation is scarce regarding these packages’ origins, but a select collection of e-mail messages by various authors survives, dating back to 1979-1982, when these packages were actively maintained at M.I.T. When this author first came across GPL Maxima, the tensor packages were effectively non-functional. -
Two Exotic Holonomies in Dimension Four, Path Geometries, and Twistor Theory
Two Exotic Holonomies in Dimension Four, Path Geometries, and Twistor Theory by Robert L. Bryant* Table of Contents §0. Introduction §1. The Holonomy of a Torsion-Free Connection §2. The Structure Equations of H3-andG3-structures §3. The Existence Theorems §4. Path Geometries and Exotic Holonomy §5. Twistor Theory and Exotic Holonomy §6. Epilogue §0. Introduction Since its introduction by Elie´ Cartan, the holonomy of a connection has played an important role in differential geometry. One of the best known results concerning hol- onomy is Berger’s classification of the possible holonomies of Levi-Civita connections of Riemannian metrics. Since the appearance of Berger [1955], much work has been done to refine his listofpossible Riemannian holonomies. See the recent works by Besse [1987]or Salamon [1989]foruseful historical surveys and applications of holonomy to Riemannian and algebraic geometry. Less well known is that, at the same time, Berger also classified the possible pseudo- Riemannian holonomies which act irreducibly on each tangent space. The intervening years have not completely resolved the question of whether all of the subgroups of the general linear group on his pseudo-Riemannian list actually occur as holonomy groups of pseudo-Riemannian metrics, but, as of this writing, only one class of examples on his list remains in doubt, namely SO∗(2p) ⊂ GL(4p)forp ≥ 3. Perhaps the least-known aspect of Berger’s work on holonomy concerns his list of the possible irreducibly-acting holonomy groups of torsion-free affine connections. (Torsion- free connections are the natural generalization of Levi-Civita connections in the metric case.) Part of the reason for this relative obscurity is that affine geometry is not as well known or widely usedasRiemannian geometry and global results are generally lacking. -
Multilinear Algebra and Applications July 15, 2014
Multilinear Algebra and Applications July 15, 2014. Contents Chapter 1. Introduction 1 Chapter 2. Review of Linear Algebra 5 2.1. Vector Spaces and Subspaces 5 2.2. Bases 7 2.3. The Einstein convention 10 2.3.1. Change of bases, revisited 12 2.3.2. The Kronecker delta symbol 13 2.4. Linear Transformations 14 2.4.1. Similar matrices 18 2.5. Eigenbases 19 Chapter 3. Multilinear Forms 23 3.1. Linear Forms 23 3.1.1. Definition, Examples, Dual and Dual Basis 23 3.1.2. Transformation of Linear Forms under a Change of Basis 26 3.2. Bilinear Forms 30 3.2.1. Definition, Examples and Basis 30 3.2.2. Tensor product of two linear forms on V 32 3.2.3. Transformation of Bilinear Forms under a Change of Basis 33 3.3. Multilinear forms 34 3.4. Examples 35 3.4.1. A Bilinear Form 35 3.4.2. A Trilinear Form 36 3.5. Basic Operation on Multilinear Forms 37 Chapter 4. Inner Products 39 4.1. Definitions and First Properties 39 4.1.1. Correspondence Between Inner Products and Symmetric Positive Definite Matrices 40 4.1.1.1. From Inner Products to Symmetric Positive Definite Matrices 42 4.1.1.2. From Symmetric Positive Definite Matrices to Inner Products 42 4.1.2. Orthonormal Basis 42 4.2. Reciprocal Basis 46 4.2.1. Properties of Reciprocal Bases 48 4.2.2. Change of basis from a basis to its reciprocal basis g 50 B B III IV CONTENTS 4.2.3. -
Flat Connections on Oriented 2-Manifolds
e Bull. London Math. Soc. 37 (2005) 1–14 C 2005 London Mathematical Society DOI: 10.1112/S002460930400373X FLAT CONNECTIONS ON ORIENTED 2-MANIFOLDS LISA C. JEFFREY Abstract This paper aims to provide a survey on the subject of representations of fundamental groups of 2-manifolds, or in other guises flat connections on orientable 2-manifolds or moduli spaces parametrizing holomorphic vector bundles on Riemann surfaces. It emphasizes the relationships between the different descriptions of these spaces. The final two sections of the paper outline results of the author and Kirwan on the cohomology rings of certain of the spaces described earlier (formulas for intersection numbers that were discovered by Witten (Commun. Math. Phys. 141 (1991) 153–209 and J. Geom. Phys. 9 (1992) 303–368) and given a mathematical proof by the author and Kirwan (Ann. of Math. 148 (1998) 109–196)). 1. Introduction This paper aims to provide a survey on the subject of representations of fundamental groups of 2-manifolds, or in other guises flat connections on orientable 2-manifolds or moduli spaces parametrizing holomorphic vector bundles on Riemann surfaces. Sections 1 and 2 are intended to be accessible to anyone with a background corresponding to an introductory course on differentiable manifolds (at the advanced undergraduate or beginning research student level in UK universities). Sections 3 and 4 describe recent work on the topology of the spaces treated in Sections 1 and 2, and may require more background in algebraic topology (for instance, familiarity with homology theory). The layout of the paper is as follows. Section 1.2 treats the Jacobian (which is the simplest prototype for the class of objects treated throughout the paper, corresponding to the group U(1)). -
Geometric-Algebra Adaptive Filters Wilder B
1 Geometric-Algebra Adaptive Filters Wilder B. Lopes∗, Member, IEEE, Cassio G. Lopesy, Senior Member, IEEE Abstract—This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are Faces generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well- Edges suited for the description of geometric transformations. Also, (directed lines) differently from standard adaptive-filtering theory, Geometric Calculus (the extension of GA to differential calculus) allows Fig. 1. A polyhedron (3-dimensional polytope) can be completely described for applying the same derivation techniques regardless of the by the geometric multiplication of its edges (oriented lines, vectors), which type (subalgebra) of the data, i.e., real, complex numbers, generate the faces and hypersurfaces (in the case of a general n-dimensional quaternions, etc. Relying on those characteristics (among others), polytope). a deterministic quadratic cost function is posed, from which the GAAFs are devised, providing a generalization of regular adaptive filters to subalgebras of GA. From the obtained update rule, it is shown how to recover the following least-mean squares perform calculus with hypercomplex quantities, i.e., elements (LMS) adaptive filter variants: real-entries LMS, complex LMS, that generalize complex numbers for higher dimensions [2]– and quaternions LMS. Mean-square analysis and simulations in [10]. a system identification scenario are provided, showing very good agreement for different levels of measurement noise. GA-based AFs were first introduced in [11], [12], where they were successfully employed to estimate the geometric Index Terms—Adaptive filtering, geometric algebra, quater- transformation (rotation and translation) that aligns a pair of nions. -
Matrices and Tensors
APPENDIX MATRICES AND TENSORS A.1. INTRODUCTION AND RATIONALE The purpose of this appendix is to present the notation and most of the mathematical tech- niques that are used in the body of the text. The audience is assumed to have been through sev- eral years of college-level mathematics, which included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly, and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in under- graduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles, and the vector cross and triple scalar prod- ucts. The notation, as far as possible, will be a matrix notation that is easily entered into exist- ing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad. The desire to represent the components of three-dimensional fourth-order tensors that appear in anisotropic elasticity as the components of six-dimensional second-order tensors and thus rep- resent these components in matrices of tensor components in six dimensions leads to the non- traditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in §A.11, along with the rationale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES An r-by-c matrix, M, is a rectangular array of numbers consisting of r rows and c columns: ¯MM.. -
SPINORS and SPACE–TIME ANISOTROPY
Sergiu Vacaru and Panayiotis Stavrinos SPINORS and SPACE{TIME ANISOTROPY University of Athens ————————————————— c Sergiu Vacaru and Panyiotis Stavrinos ii - i ABOUT THE BOOK This is the first monograph on the geometry of anisotropic spinor spaces and its applications in modern physics. The main subjects are the theory of grav- ity and matter fields in spaces provided with off–diagonal metrics and asso- ciated anholonomic frames and nonlinear connection structures, the algebra and geometry of distinguished anisotropic Clifford and spinor spaces, their extension to spaces of higher order anisotropy and the geometry of gravity and gauge theories with anisotropic spinor variables. The book summarizes the authors’ results and can be also considered as a pedagogical survey on the mentioned subjects. ii - iii ABOUT THE AUTHORS Sergiu Ion Vacaru was born in 1958 in the Republic of Moldova. He was educated at the Universities of the former URSS (in Tomsk, Moscow, Dubna and Kiev) and reveived his PhD in theoretical physics in 1994 at ”Al. I. Cuza” University, Ia¸si, Romania. He was employed as principal senior researcher, as- sociate and full professor and obtained a number of NATO/UNESCO grants and fellowships at various academic institutions in R. Moldova, Romania, Germany, United Kingdom, Italy, Portugal and USA. He has published in English two scientific monographs, a university text–book and more than hundred scientific works (in English, Russian and Romanian) on (super) gravity and string theories, extra–dimension and brane gravity, black hole physics and cosmolgy, exact solutions of Einstein equations, spinors and twistors, anistoropic stochastic and kinetic processes and thermodynamics in curved spaces, generalized Finsler (super) geometry and gauge gravity, quantum field and geometric methods in condensed matter physics.