DOCSLIB.ORG

Scalars, Vectors, Tensors and All That C Alex R

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Scalars, Vectors, Tensors and All That C Alex R Lecture 26 Friday - March 31, 2006 Written or last updated: March 31, 2006 P442 – Analytical Mechanics - II Scalars, Vectors, Tensors and All That c Alex R. Dzierba Some Notation We start by laying out notation some of which many of you may already be familiar with and others may not. We are also going to make the assumption for now that we are dealing with three dimensional space. Implied summation by repeated index The components of a vector a are aj where j = 1, 3. When an index is repeated in expressions that usually implies a summation. Here are some examples: aibi ⇒ a1b1 + a2b2 + a3b3 ∂a ∂a ∂a ∂a i ⇒ 1 + 2 + 3 ∂xi ∂x1 ∂x2 ∂x3 Assuming that cij and dij are elements of a matrix: cijdjk ⇒ ci1d1k + ci2d2k + ci3d3k and the result would yield the ik-th element of the product matrix. Here is product of a matrix times a matrix times a vector and the result is a vector: 3 3 X X cijdjkak ⇒ cijdjkak j=1 k=1 Kronecker Delta This is defined as 1 if i = j δ = (1) ij 0 otherwise As will be noted below, the δij are components of a second-order tensor. 1 Levi-Civita Tensor 0 if any two labels are equal ijk = 1 if the labels are an even permutation of 1,2,3 (2) −1 if the labels are an odd permutation of 1,2,3 The Levi-Civita tensor has 3 × 3 × 3 = 27 components of which 21 equal zero, three equal +1 and three equal −1. The determinant of a matrix A can be written as: a11 a12 a13 |A| = a21 a22 a23 = ijka1ia2ja3k (3) a31 a32 a33 where a repeated index implies a summation. Vectors and Tensors Suppose that the 3 × 3 matrix Lij is the transformation matrix that specifies how the components (ai) of a vector are changed when the Cartesian axes are rotated. Then: 0 ai = Lijaj A vector is a first-order tensor. If a and b are any two vectors, the nine numbers aibj (products of their components) form a tensor and in this case it will be a second-order tensor whose elements we will call Aij. Under a rotation the new elements of our tensor are: 0 Aij = LikLjmAkm The δij are components of a second-order tensor. The ijk are components of a third-order tensor. Examples of Tensors in Physics The moment of inertia is a tensor. Recall how it was defined: Z Ijk = ρ(x)(x · xδjk − xjxk)dV V In the above δjk and xjxk are components of a tensor. We saw the the angular momentum L and angular velocity ω are related through the moment of inertia tensor: 2 Li = Iikωk In electrodynamics we are used to Ohm’s law which states that I = V/R where I is current, V is voltage and R is resistance. This is often recast at J = σE where J is current per area, σ is conductivity (the inverse of resistivity) and E is the electric field. But what we wrote down here only applies to materials that are isotropic. But for anisotropic crystals the conductivity is different along different crystal axes and the material is more correctly described by: Ji = σikEk where σjk is a second-order tensor. There is a similar equation relating the magnetic moment per volume M to the magnetic field B through the magnetic susceptibility χ which is also a tensor: Mi = χikBk In the above examples we have relations between vectors involving second-order tensors. We have just started talking about the theory of elasticity that relates two second-order tensors through a fourth-order tensor. One of the second order tensors is the strain tensor which describes the local deformation of an elastic body at any interior point as: 1 ∂ui ∂uj Tij = + 2 ∂uj ∂ui where the displacement vector u describes the strain of a small volume element located by the vector x. We also have a second-order tensor Sij called the stress tensor where the i-th component of stress (pressure or shear force) acting normal to or along the plane whose normal is specified by the index j. The stress and strain tensor are related by: Tij = γijkmSkm The fourth-order tensor γijkm completely describes the elastic properties of the material. Although this beast requires 34 = 81 numbers in principle, symmetry arguments reduce this number to 21. This number is even smaller for crystals with higher symmetry and an isotropic material only requires two numbers to describe its elastic properties. Cross Product The components of the cross product of two vectors a and b can be written as: (a × b)i = ijkajbk = ajbk − akbj (4) 3 we see that we have here a mathematical object written as products of components of two vectors – so what we have here is a second-order tensor. So let’s write this as: cjk = ajbk − akbj (5) Forget for the moment that we are talking about a vector defined as the cross product of two vectors but rather look at this as a tensor. This particular tensor is clearly anti-symmetric since cij = −cji. That means the diagonal elements are zero since cii = −cii ⇒ cii = 0. We start with 3 × 3 = 9 elements of which three are zero so that leaves us with six but only three are needed by the antisymmetry property. So we only need three numbers. The three numbers, it turns out, match the dimensionality of our vectors and furthermore they transform, under rotations, just like a vector. Transforming a Cross Product Vector In doing this example, let me use the x, y, z for indices rather than 1, 2, 3. So we have: cxy = axby − aybx cyz = aybz − azby czx = azbx − axbz Suppose we do a rotation of coordinates rotating through angle θ about the z-axis. Then look at how our vectors transform starting with a: ax0 = ax cos θ + ay sin θ ay0 = ay cos θ − ax sin θ az0 = az There will be similar equations for b in the primed frame. Here is what you find when you evaluate: cx0y0 = ax0 by0 − ay0 bx0 = cxy and you also find: cy0z0 = cyz cos θ + czx sin θ cz0x0 = czx cos θ − cyz sin θ So cyz and czx and cxy behave like the x, y, z components, respectively, of a vector. So this tensor transforms like a vector. But this fortuitous accident only happens in three dimensions. The four-dimension analogy 4 would be an anti-symmetric tensor with 4 × 4 = 16 elements which requires six numbers (the four diagonal elements are zero and leaving 12 but of these we only need six). That means this tensor in four dimension cannot transform like a four-dimensional vector. Polar and Axial Vectors So our in three dimensional world we see that vectors, like displacement and momentum, transform like vectors. Cross products, like torque and angular momentum transform like vectors as well. But there is a way that cross products are different. If we invert coordinates, so that x → −x and y → −y and z → −z then vectors like displacement x and velocity v and momentum p change sign. This is also true for the electric field E. Coordinate inversions are sometimes called parity operations or space reflections. For these vectors a → −a. Such vectors, that change sign under a coordinate inversion, are called polar vectors. Cross product vectors, on the other hand, are called axial vectors or pseudovectors – they don’t change sign under a parity operation. This is all because cross-products require a handedness in their definition. This is true of torque τ , angular momentum L, angular velocity ω and the magnetic field B, for example. For polar vectors all you need is a length and direction. Axial vectors also require a handedness. The cross product of two vectors is given by the right-hand rule. Handedness remains unchanged under space reflections. Also note that you cannot get to a space reflection through rotations. Scalars and Pseudoscalars A scalar (like mass or energy or the dot product of two polar vectors) does not change sign under a space inversion. But it is possible to define a scalar that does change sign and such scalars are called pseudoscalars. An example is the product of three polar vectors: a · (b × c). 5.
Load more

Recommended publications
  • Generic Properties of Symmetric Tensors
    2006 – 1/48 – P.Comon Generic properties of Symmetric Tensors Pierre COMON I3S - CNRS other contributors: Bernard MOURRAIN INRIA institute Lek-Heng LIM, Stanford University I3S 2006 – 2/48 – P.Comon Tensors & Arrays Definitions Table T = {Tij..k} Order d of T def= # of its ways = # of its indices def Dimension n` = range of the `th index T is Square when all dimensions n` = n are equal T is Symmetric when it is square and when its entries do not change by any permutation of indices I3S 2006 – 3/48 – P.Comon Tensors & Arrays Properties Outer (tensor) product C = A ◦ B: Cij..` ab..c = Aij..` Bab..c Example 1 outer product between 2 vectors: u ◦ v = u vT Multilinearity. An order-3 tensor T is transformed by the multi-linear map {A, B, C} into a tensor T 0: 0 X Tijk = AiaBjbCkcTabc abc Similarly: at any order d. I3S 2006 – 4/48 – P.Comon Tensors & Arrays Example Example 2 Take 1 v = −1 Then 1 −1 −1 1 v◦3 = −1 1 1 −1 This is a “rank-1” symmetric tensor I3S 2006 – 5/48 – P.Comon Usefulness of symmetric arrays CanD/PARAFAC vs ICA .. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... CanD/PARAFAC: = + ... + . ... ... ... ... ... ... ... ... I3S 2006 – 6/48 – P.Comon Usefulness of symmetric arrays CanD/PARAFAC vs ICA .. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... CanD/PARAFAC: = + ... + . ... ... ... ... ... ... ... ... PARAFAC cannot be used when: • Lack of diversity I3S 2006 – 7/48 – P.Comon Usefulness of symmetric arrays CanD/PARAFAC vs ICA .. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... CanD/PARAFAC: = + ... + . ... ... ... ... ... ... ... ... PARAFAC cannot be used when: • Lack of diversity • Proportional slices I3S 2006 – 8/48 – P.Comon Usefulness of symmetric arrays CanD/PARAFAC vs ICA .
    [Show full text]
  • Parallel Spinors and Connections with Skew-Symmetric Torsion in String Theory*
    ASIAN J. MATH. © 2002 International Press Vol. 6, No. 2, pp. 303-336, June 2002 005 PARALLEL SPINORS AND CONNECTIONS WITH SKEW-SYMMETRIC TORSION IN STRING THEORY* THOMAS FRIEDRICHt AND STEFAN IVANOV* Abstract. We describe all almost contact metric, almost hermitian and G2-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the V- parallel spinors. In particular, we obtain partial solutions of the type // string equations in dimension n = 5, 6 and 7. 1. Introduction. Linear connections preserving a Riemannian metric with totally skew-symmetric torsion recently became a subject of interest in theoretical and mathematical physics. For example, the target space of supersymmetric sigma models with Wess-Zumino term carries a geometry of a metric connection with skew-symmetric torsion [23, 34, 35] (see also [42] and references therein). In supergravity theories, the geometry of the moduli space of a class of black holes is carried out by a metric connection with skew-symmetric torsion [27]. The geometry of NS-5 brane solutions of type II supergravity theories is generated by a metric connection with skew-symmetric torsion [44, 45, 43]. The existence of parallel spinors with respect to a metric connection with skew-symmetric torsion on a Riemannian spin manifold is of importance in string theory, since they are associated with some string solitons (BPS solitons) [43]. Supergravity solutions that preserve some of the supersymmetry of the underlying theory have found many applications in the exploration of perturbative and non-perturbative properties of string theory.
    [Show full text]
  • 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..
    [Show full text]
  • A Some Basic Rules of Tensor Calculus
    A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a b + ... +c d. The scalars, vectors and tensors are handled as invariant (independent⊗ from the choice⊗ of the coordinate system) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write a = aig , A = aibj + ... + cidj g g i i ⊗ j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k k ik ik a gk ∑ a gk, A bk ∑ A bk ≡ k=1 ≡ k=1 In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e.
    [Show full text]
  • Symmetric Tensors and Symmetric Tensor Rank Pierre Comon, Gene Golub, Lek-Heng Lim, Bernard Mourrain
    Symmetric tensors and symmetric tensor rank Pierre Comon, Gene Golub, Lek-Heng Lim, Bernard Mourrain To cite this version: Pierre Comon, Gene Golub, Lek-Heng Lim, Bernard Mourrain. Symmetric tensors and symmetric tensor rank. SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2008, 30 (3), pp.1254-1279. hal-00327599 HAL Id: hal-00327599 https://hal.archives-ouvertes.fr/hal-00327599 Submitted on 8 Oct 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. SYMMETRIC TENSORS AND SYMMETRIC TENSOR RANK PIERRE COMON∗, GENE GOLUB†, LEK-HENG LIM†, AND BERNARD MOURRAIN‡ Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank-1 order-k tensor is the outer product of k non-zero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it.
    [Show full text]
  • Chapter 04 Rotational Motion
    Chapter 04 Rotational Motion P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 04: Rotational Motion Angular Momentum Angular momentum of particle with respect to origin, O, is given by l⃗ = ⃗r × p⃗ Rate of change of angular momentum is given z by cross product of ⃗r with applied force. p m dl⃗ dp⃗ = ⃗r × = ⃗r × F⃗ = ⃗휏 r dt dt O y Cross product is defined as applied torque, ⃗휏. x Unlike linear momentum, angular momentum depends on origin choice. P. J. Grandinetti Chapter 04: Rotational Motion Conservation of Angular Momentum Consider system of N Particles z m5 m 2 Rate of change of angular momentum is m3 ⃗ ∑N l⃗ ∑N ⃗ m1 dL d 훼 dp훼 = = ⃗r훼 × dt dt dt 훼=1 훼=1 y which becomes m4 x ⃗ ∑N dL ⃗ net = ⃗r훼 × F dt 훼 Total angular momentum is 훼=1 ∑N ∑N ⃗ ⃗ L = l훼 = ⃗r훼 × p⃗훼 훼=1 훼=1 P. J. Grandinetti Chapter 04: Rotational Motion Conservation of Angular Momentum ⃗ ∑N dL ⃗ net = ⃗r훼 × F dt 훼 훼=1 Taking an earlier expression for a system of particles from chapter 1 ∑N ⃗ net ⃗ ext ⃗ F훼 = F훼 + f훼훽 훽=1 훽≠훼 we obtain ⃗ ∑N ∑N ∑N dL ⃗ ext ⃗ = ⃗r훼 × F + ⃗r훼 × f훼훽 dt 훼 훼=1 훼=1 훽=1 훽≠훼 and then obtain 0 > ⃗ ∑N ∑N ∑N dL ⃗ ext ⃗ rd ⃗ ⃗ = ⃗r훼 × F + ⃗r훼 × f훼훽 double sum disappears from Newton’s 3 law (f = *f ) dt 훼 12 21 훼=1 훼=1 훽=1 훽≠훼 P.
    [Show full text]
  • 8.09(F14) Chapter 2: Rigid Body Dynamics
    Chapter 2 Rigid Body Dynamics 2.1 Coordinates of a Rigid Body A set of N particles forms a rigid body if the distance between any 2 particles is fixed: rij ≡ jri − rjj = cij = constant: (2.1) Given these constraints, how many generalized coordinates are there? If we know 3 non-collinear points in the body, the remaing points are fully determined by triangulation. The first point has 3 coordinates for translation in 3 dimensions. The second point has 2 coordinates for spherical rotation about the first point, as r12 is fixed. The third point has one coordinate for circular rotation about the axis of r12, as r13 and r23 are fixed. Hence, there are 6 independent coordinates, as represented in Fig. 2.1. This result is independent of N, so this also applies to a continuous body (in the limit of N ! 1). Figure 2.1: 3 non-collinear points can be fully determined by using only 6 coordinates. Since the distances between any two other points are fixed in the rigid body, any other point of the body is fully determined by the distance to these 3 points. 29 CHAPTER 2. RIGID BODY DYNAMICS The translations of the body require three spatial coordinates. These translations can be taken from any fixed point in the body. Typically the fixed point is the center of mass (CM), defined as: 1 X R = m r ; (2.2) M i i i where mi is the mass of the i-th particle and ri the position of that particle with respect to a fixed origin and set of axes (which will notationally be unprimed) as in Fig.
    [Show full text]
  • Connections with Skew-Symmetric Ricci Tensor on Surfaces
    Connections with skew-symmetric Ricci tensor on surfaces Andrzej Derdzinski Abstract. Some known results on torsionfree connections with skew-symmet- ric Ricci tensor on surfaces are extended to connections with torsion, and Wong’s canonical coordinate form of such connections is simplified. Mathematics Subject Classification (2000). 53B05; 53C05, 53B30. Keywords. Skew-symmetric Ricci tensor, locally homogeneous connection, left-invariant connection on a Lie group, curvature-homogeneity, neutral Ein- stein metric, Jordan-Osserman metric, Petrov type III. 1. Introduction This paper generalizes some results concerning the situation where ∇ is a connection on a surface Σ, and the Ricci (1.1) tensor ρ of ∇ is skew symmetric at every point. What is known about condition (1.1) can be summarized as follows. Norden [18], [19, §89] showed that, for a torsionfree connection ∇ on a surface, skew-sym- metry of the Ricci tensor is equivalent to flatness of the connection obtained by projectivizing ∇, and implies the existence of a fractional-linear first integral for the geodesic equation. Wong [25, Theorem 4.2] found three coordinate expressions which, locally, represent all torsionfree connections ∇ with (1.1) such that ρ 6= 0 everywhere in Σ. Kowalski, Opozda and Vl´aˇsek [13] used an approach different from Wong’s to classify, locally, all torsionfree connections ∇ satisfying (1.1) that are also locally homogeneous, while in [12] they proved that, for real-analytic tor- sionfree connections ∇ with (1.1), third-order curvature-homogeneity implies local homogeneity (but one cannot replace the word ‘third’ with ‘second’). Blaˇzi´cand Bokan [2] showed that the torus T 2 is the only closed surface Σ admitting both a torsionfree connection ∇ with (1.1) and a ∇-parallel almost complex structure.
    [Show full text]
  • Symmetric Tensor Decomposition and Algorithms
    Symmetric Tensor Decomposition and Algorithms Author: María López Quijorna Directors: M.E.Alonso and A.Díaz-Cano Universidad Complutense de Madrid Academic Year: 2010-2011 Final Project of Master in Mathematic Research 2 Contents Contents 3 1 Introduction 7 2 Preliminaires 9 2.1 Applications....................................... 11 2.2 From symmetric tensor to homogeneous polynomials................ 12 3 The binary case 13 4 Problem Formulations 15 4.1 Polynomial Decomposition............................... 15 4.2 Geometric point of view................................ 16 4.2.1 Big Waring Problem.............................. 16 4.2.2 Veronese and secant varieties......................... 17 4.3 Decomposition using duality.............................. 18 5 Inverse systems and duality 23 5.1 Duality and formal series................................ 23 5.2 Inverse systems..................................... 25 5.3 Inverse system of a single point............................ 27 6 Gorenstein Algebras 31 7 Hankel operators and quotient algebra 33 8 Truncated Hankel Operators 39 9 Symmetric tensor decomposition algorithm 51 9.1 Symmetric tensor decomposition algorithm..................... 54 9.2 Future work....................................... 56 Bibliography 57 3 4 CONTENTS Abstract Castellano: El objetivo de este trabajo es el estudio de la descomposición de tensores simétricos de dimensión "n" y orden "d". Equivalentemente el estudio de la descomposición de polinomios homogéneos de grado "d" en "n" variables como suma de "r" potencias d-ésimas de formas lineales. Este problema tiene una interpretación geométrica en términos de incidencia de variedades se- cantes de variedades de Veronese: Problema de Waring [12],[6]. Clásicamente, en el caso de formas binarias el resultado completo se debe a Sylvester. El principal objeto de estudio del trabajo es el algoritmo de descomposición de tensores simétricos, que es una generalización del teorema de Sylvester y ha sido tomado de [1].
    [Show full text]
  • Exploiting Symmetry in Tensors for High Performance: Multiplication with Symmetric Tensors∗
    SIAM J. SCI. COMPUT. c 2014 Society for Industrial and Applied Mathematics Vol. 36, No. 5, pp. C453–C479 EXPLOITING SYMMETRY IN TENSORS FOR HIGH PERFORMANCE: MULTIPLICATION WITH SYMMETRIC TENSORS∗ † † † MARTIN D. SCHATZ ,TZEMENGLOW,ROBERTA.VANDEGEIJN, AND ‡ TAMARA G. KOLDA Abstract. Symmetric tensor operations arise in a wide variety of computations. However, the benefits of exploiting symmetry in order to reduce storage and computation is in conflict with a desire to simplify memory access patterns. In this paper, we propose a blocked data structure (blocked compact symmetric storage) wherein we consider the tensor by blocks and store only the unique blocks of a symmetric tensor. We propose an algorithm by blocks, already shown of benefit for matrix computations, that exploits this storage format by utilizing a series of temporary tensors to avoid redundant computation. Further, partial symmetry within temporaries is exploited to further avoid redundant storage and redundant computation. A detailed analysis shows that, relative to storing and computing with tensors without taking advantage of symmetry and partial symmetry, storage requirements are reduced by a factor of O(m!) and computational requirements by a factor of O((m +1)!/2m), where m is the order of the tensor. However, as the analysis shows, care must be taken in choosing the correct block size to ensure these storage and computational benefits are achieved (particularly for low-order tensors). An implementation demonstrates that storage is greatly reduced and the complexity introduced by storing and computing with tensors by blocks is manageable. Preliminary results demonstrate that computational time is also reduced. The paper concludes with a discussion of how insights in this paper point to opportunities for generalizing recent advances in the domain of linear algebra libraries to the field of multilinear computation.
    [Show full text]
  • Two-Spinors and Symmetry Operators
    Two-spinors and Symmetry Operators An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools Master’s thesis in Physics Simon Stefanus Jacobsson DEPARTMENT OF MATHEMATICAL SCIENCES CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2021 www.chalmers.se Master’s thesis 2021 Two-spinors and Symmetry Operators ¦ An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools SIMON STEFANUS JACOBSSON Department of Mathematical Sciences Division of Analysis and Probability Theory Mathematical Physics Chalmers University of Technology Gothenburg, Sweden 2021 Two-spinors and Symmetry Operators An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools SIMON STEFANUS JACOBSSON © SIMON STEFANUS JACOBSSON, 2021. Supervisor: Thomas Bäckdahl, Mathematical Sciences Examiner: Simone Calogero, Mathematical Sciences Master’s Thesis 2021 Department of Mathematical Sciences Division of Analysis and Probability Theory Mathematical Physics Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Typeset in LATEX Printed by Chalmers Reproservice Gothenburg, Sweden 2021 iii Two-spinors and Symmetry Operators An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools SIMON STEFANUS JACOBSSON Department of Mathematical Sciences Chalmers University of Technology Abstract This thesis employs spinor techniques to find what conditions a curved spacetime must satisfy for there to exist a second order symmetry operator for the massive Dirac equation. Conditions are of the form of the existence of a set of Killing spinors satisfying some set of covariant differential equations.
    [Show full text]
  • Structure of Symmetric Tensors of Type (0, 2)
    TRANSACTIONSOF THE AMERICAN MATHEMATICALSOCIETY Volume 234, Number I, 1977 STRUCTUREOF SYMMETRICTENSORS OF TYPE (0, 2) AND TENSORS OF TYPE (1, 1) ON THE TANGENTBUNDLE BY KAM-PING MOK, E. M. PATTERSONAND YUNG-CHOW WONG Abstract. The concepts of M-tensor and Af-connection on the tangent bundle TM of a smooth manifold M are used in a study of symmetric tensors of type (0, 2) and tensors of type (1, 1) on TM. The constructions make use of certain local frames adapted to an A/-connection. They involve extending known results on TM using tensors on M to cases in which these tensors are replaced by A/-tensors. Particular attention is devoted to (pseudo-) Riemannian metrics on TM, notably those for which the vertical distribution on TM is null or nonnull, and to the construction of almost product and almost complex structures on TM. 1. Introduction. Let M be a smooth manifold and TM its tangent bundle. In his 1958 paper [1], S. Sasaki constructed a Riemannian metric on TM from a Riemannian metric on M, heralding the beginning of the differential geome- try of the tangent bundle. Since then, other Riemannian metrics on TM have been constructed (see Yano and Ishihara [3, Chapter IV]) but no general method of construction has emerged. Recently, two of us (Wong and Mok [1]) introduced the concepts of A/-tensor and three types of connections on TM, and used them to clarify the relationship between several related known concepts on TM. In this paper, we shall show how the concepts of A/-tensor and one of these connections (which we now call A/-connection) enable us to have a complete picture of the structures of the symmetric tensors of type (0, 2) and tensors of type (1,1) on TM.
    [Show full text]