DOCSLIB.ORG

Tensors Notation • Contravariant Denoted by Superscript Ai Took a Vector and Gave for Vector Calculus Us a Vector

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tensors Notation • Contravariant Denoted by Superscript Ai Took a Vector and Gave for Vector Calculus Us a Vector Tensors Notation • contravariant denoted by superscript Ai took a vector and gave For vector calculus us a vector • covariant denoted by subscript Ai took a scaler and gave us a vector Review To avoid confusion in cartesian coordinates both types are the same so • Vectors we just opt for the subscript. Thus a vector x would be x1,x2,x3 R3 • Summation representation of an n by n array As it turns out in an cartesian space and other rectilinear coordi- nate systems there is no difference between contravariant and covariant • Gradient, Divergence and Curl vectors. This will not be the case for other coordinate systems such a • Spherical Harmonics (maybe) curvilinear coordinate systems or in 4 dimensions. These definitions are closely related to the Jacobian. Motivation If you tape a book shut and try to spin it in the air on each indepen- Definitions for Tensors of Rank 2 dent axis you will notice that it spins fine on two axes but not on the Rank 2 tensors can be written as a square array. They have con- third. That’s the inertia tensor in your hands. Similar are the polar- travariant, mixed, and covariant forms. As we might expect in cartesian izations tensor, index of refraction tensor and stress tensor. But tensors coordinates these are the same. also show up in all sorts of places that don’t connect to an anisotropic material property, in fact even spherical harmonics are tensors. What are the similarities and differences between such a plethora of tensors? Vector Calculus and Identifers The mathematics of tensors is particularly useful for de- Tensor analysis extends deep into coordinate transformations of all scribing properties of substances which vary in direction– kinds of spaces and coordinate systems. At a rather shallow level it can although that’s only one example of their use. -Feynman II be used to significantly simplify the operations and identifers of vector 31-1 calculus. Definitions Scaler Product • q = Scaler Take for example the Scaler or Dot Product. a.k.a. Rank 0 tensor scaler One real number, invariant under rotation A B • Ai = Vector · = AiBi = AiBi i a.k.a Rank 1 tensor where i=1,2,3... X z }| { Components will transform under rotations. Simply put, an ar- Clearly i is just a general way of specifying the components to multiply. row does not need to start at the origin or point in any specific Here we sum the product of the two components with i is the same. The orientation, but it’s still the same arrow with the same magnitude. equation is also written without the summation. They are equivalent • Aij = Rank 2 tensor because of the Einstein summation convention. a.k.a. tensor Can be writhen out as a square array. Not all square arrays are Scaler Product with an invariant tensor (Kronecker tensors, there are some specific requirements. In cartesian space they must be an orthogonal norm preserving matrix. delta) • In N-dimensional space a tensor of rank n has N n components. In a more thorough treatment we can also take the Scaler product k using a mixed tensor of rank 2, δl more commonly recognized as the Rank 1 Tensors (Vectors) Kronecker delta δij 1 i=j The definitions for contravariant and covariant tensors are inevitably δij = (0 i =6 j defined at the beginning of all discussion on tensors. Their definitions or are inviably without explanation. In that spirit we begin our discussion of rank 1 tensors. The main difference between contravaariant and co- 1 0 δk = variant tensors is in how they are transformed. For example, a rotation l 0 1 of a vector. µ ¶ Clearly, the Kronecker delta is a rank two tensor that can be expressed as a matrix, but rarely takes such an inherently dangerous form. The Contravariant Kronecker delta is a so called invariant tensor. Now we can write our In general for we can define a contravariant vector as one that can be dot product as transformed as Aj is in the following: 0 A · B = Aiδij Bj = Aiδij Bj = AiBi 0 ∂xi A = Aj j i ∂x X j j The sum is, of course, neglected because of the summation convention. X 0 Clearly the Kronecker delta facilitates the scaler product. Furthermore, here might xi is the transformed x-axis, and xj is the old x-axis. the Kronecker delta can be applied to any situation where the product of orthogonal complements must be zero. Covariant The partial derivative above may have you thinking of a gradient. Summation Convention and Contraction The gradient is the prototype for a covariant vector which is defined as 2 0 Summation Convention When a subscript (letter, not number) ∂ϕ ∂ϕ ∂xj = appears twice on one side of an equation, summation with respect to ∂x0 ∂x ∂x0 i j j i that subscript is implied. X Covariant vectors are actually a linear form an not a vector. The linear Contraction Consists of setting two unlike indices (subscripts) equal form is a mapping of vectors into scalars which is additive and homoge- to each other and then summing as implied by the summation conven- neous under multiplication by scalars. 1 tion. 1http://xxx.lanl.gov/PS cache/gr-qc/pdf/9807/9807044.pdf 2From Arfkin p138 Author: W. Schlotter Last Modified: May 20, 2003 One-pager [email protected] 1.02 Cross Product and the invariant Levi-Civita Symbol Symmetry The rank three analog of the Kronecker delta is known as the Levi- symetric antisymetric Civita Symbol, εijk. 1 1 Aij = (Aij + Aji)+ (Aij − Aji) 2 2 1 if(ijk)= even permutation:ε123 = ε231 = ε312 z }| { z }| { εijk = −1 if(ijk) = odd permutation: ε132 = ε213 = ε321 As used in P220 0 if any of the two indexes are the same The functions of position are given by 0 It’s a three dimensional 27-element array with only six non-zero com- f = x ponents. Levi-Civita is a completely antisymmetric unit tensor. We can i 0 use it to preform some amazing operations. g = xj 0 h = xk(not given in class) Cross product See Griffiths Appendix A on Vector Calculus and Curvilinear Coor- dinates. A = B × C Vector identity examples Ai = εijkBj Ck Let’s take a closer look εijkεlmk = δjm(δilδjm − δimδjl) 1 −1 = δilδji − δimδml Ax = εxjkBj Ck = εxyz ByCz + εxzy BzCy = 3δ − δ = 2δ −1 1 il il il z}|{ z}|{ Ay = εyjkBj Ck = εyxz BxCz + εyzx BzCx Okay now lets derive identities 1 −1 z}|{ z}|{ d = a × (b × c) Az = εzjkBj Ck = εzxy BxCy + εzyx ByCx e z}|{ z}|{ di = εijka|j e{zk =}εijkaj εklmblcm Now you can see why we just write Ai = εijkBj Ck. = (δilδjm − δimδjl)aj blcm = bi(aj cj ) − ci(aj bj ) Partial derivative, Grad, Div and Curl d = b(a · c) − c(a · b) We can write a partial derivative in general form as follows And now for another identity . ∂ ∇ × (∇φ) = 0 ∂i = ∂xi εijk∂j (∂kφ) = 0 Grad And another A Thus ∇ · (∇ × ) = 0 . ∇φ = ∂iφ ∂iεijk∂j Ax = 0 We know this will generate a vector since there is nothing to sum And now a really complicated one over. c = ∇ × (a × b) Divergence ci = εijk∂j (εkmlalbm) ci = (δilδjm − δimδlj ) [(∂j al)bm + al(∂j bm)] Thus . c = (∂ a )b + a (∂ b ) − (∂ a )b − a (∂ b ) ∇ · A = ∇ · Ai = ∂iAi = ∂iδij Aj i j i i i j j j j i j j j c b a a b b a a b Since two indexes are the same we will sum to generate at scaler. = ( · ∇) + (∇ · ) − (∇ · ) − ( · ∇) Curl References and Further Reading th Thus Arfkin and Weber, Mathematical Methods for Physicists 5 ed. (2001), Section 2.6 Tensor Analysis ∇ × A = ∇ × Ai = εijk∂j Ak = ci Feynman, Lectures on Physics, Lecture II 31 Now we have three indexes so we sum twice to get a vector. This is a prime example of contraction. Related topics Rank 2 Tensors • Spherical Harmonics The Kronoker delta was previously mentioned to be rank 2 mixed • Group Theory tensor, but without further explanation. There are three types of rank Spherical Harmonics 2 tensors Group Theory 0 0 0 ∂x ∂xj A ij = i Akl contravariant ∂x ∂x kl k l X 0 0i ∂xi ∂xl k Bj = 0 Bl mixed ∂xk ∂xj Xkl 0 ∂xk ∂xl Cij = 0 0 Ckl covariant ∂xi ∂xj Xkl Author: W. Schlotter Last Modified: May 20, 2003 One-pager [email protected] 1.02.
Load more

Recommended publications
  • Tensors and Differential Forms on Vector Spaces
    APPENDIX A TENSORS AND DIFFERENTIAL FORMS ON VECTOR SPACES Since only so much of the vast and growing field of differential forms and differentiable manifolds will be actually used in this survey, we shall attempt to briefly review how the calculus of exterior differential forms on vector spaces can serve as a replacement for the more conventional vector calculus and then introduce only the most elementary notions regarding more topologically general differentiable manifolds, which will mostly be used as the basis for the discussion of Lie groups, in the following appendix. Since exterior differential forms are special kinds of tensor fields – namely, completely-antisymmetric covariant ones – and tensors are important to physics, in their own right, we shall first review the basic notions concerning tensors and multilinear algebra. Presumably, the reader is familiar with linear algebra as it is usually taught to physicists, but for the “basis-free” approach to linear and multilinear algebra (which we shall not always adhere to fanatically), it would also help to have some familiarity with the more “abstract-algebraic” approach to linear algebra, such as one might learn from Hoffman and Kunze [ 1], for instance. 1. Tensor algebra. – A tensor algebra is a type of algebra in which multiplication takes the form of the tensor product. a. Tensor product. – Although the tensor product of vector spaces can be given a rigorous definition in a more abstract-algebraic context (See Greub [ 2], for instance), for the purposes of actual calculations with tensors and tensor fields, it is usually sufficient to say that if V and W are vector spaces of dimensions n and m, respectively, then the tensor product V ⊗ W will be a vector space of dimension nm whose elements are finite linear combinations of elements of the form v ⊗ w, where v is a vector in V and w is a vector in W.
    [Show full text]
  • Multilinear Algebra
    Appendix A Multilinear Algebra This chapter presents concepts from multilinear algebra based on the basic properties of finite dimensional vector spaces and linear maps. The primary aim of the chapter is to give a concise introduction to alternating tensors which are necessary to define differential forms on manifolds. Many of the stated definitions and propositions can be found in Lee [1], Chaps. 11, 12 and 14. Some definitions and propositions are complemented by short and simple examples. First, in Sect. A.1 dual and bidual vector spaces are discussed. Subsequently, in Sects. A.2–A.4, tensors and alternating tensors together with operations such as the tensor and wedge product are introduced. Lastly, in Sect. A.5, the concepts which are necessary to introduce the wedge product are summarized in eight steps. A.1 The Dual Space Let V be a real vector space of finite dimension dim V = n.Let(e1,...,en) be a basis of V . Then every v ∈ V can be uniquely represented as a linear combination i v = v ei , (A.1) where summation convention over repeated indices is applied. The coefficients vi ∈ R arereferredtoascomponents of the vector v. Throughout the whole chapter, only finite dimensional real vector spaces, typically denoted by V , are treated. When not stated differently, summation convention is applied. Definition A.1 (Dual Space)Thedual space of V is the set of real-valued linear functionals ∗ V := {ω : V → R : ω linear} . (A.2) The elements of the dual space V ∗ are called linear forms on V . © Springer International Publishing Switzerland 2015 123 S.R.
    [Show full text]
  • Tensor Calculus and Differential Geometry
    Course Notes Tensor Calculus and Differential Geometry 2WAH0 Luc Florack March 10, 2021 Cover illustration: papyrus fragment from Euclid’s Elements of Geometry, Book II [8]. Contents Preface iii Notation 1 1 Prerequisites from Linear Algebra 3 2 Tensor Calculus 7 2.1 Vector Spaces and Bases . .7 2.2 Dual Vector Spaces and Dual Bases . .8 2.3 The Kronecker Tensor . 10 2.4 Inner Products . 11 2.5 Reciprocal Bases . 14 2.6 Bases, Dual Bases, Reciprocal Bases: Mutual Relations . 16 2.7 Examples of Vectors and Covectors . 17 2.8 Tensors . 18 2.8.1 Tensors in all Generality . 18 2.8.2 Tensors Subject to Symmetries . 22 2.8.3 Symmetry and Antisymmetry Preserving Product Operators . 24 2.8.4 Vector Spaces with an Oriented Volume . 31 2.8.5 Tensors on an Inner Product Space . 34 2.8.6 Tensor Transformations . 36 2.8.6.1 “Absolute Tensors” . 37 CONTENTS i 2.8.6.2 “Relative Tensors” . 38 2.8.6.3 “Pseudo Tensors” . 41 2.8.7 Contractions . 43 2.9 The Hodge Star Operator . 43 3 Differential Geometry 47 3.1 Euclidean Space: Cartesian and Curvilinear Coordinates . 47 3.2 Differentiable Manifolds . 48 3.3 Tangent Vectors . 49 3.4 Tangent and Cotangent Bundle . 50 3.5 Exterior Derivative . 51 3.6 Affine Connection . 52 3.7 Lie Derivative . 55 3.8 Torsion . 55 3.9 Levi-Civita Connection . 56 3.10 Geodesics . 57 3.11 Curvature . 58 3.12 Push-Forward and Pull-Back . 59 3.13 Examples . 60 3.13.1 Polar Coordinates in the Euclidean Plane .
    [Show full text]
  • C:\Book\Booktex\Notind.Dvi 0
    362 INDEX A Cauchy stress law 216 Absolute differentiation 120 Cauchy-Riemann equations 293,321 Absolute scalar field 43 Charge density 323 Absolute tensor 45,46,47,48 Christoffel symbols 108,110,111 Acceleration 121, 190, 192 Circulation 293 Action integral 198 Codazzi equations 139 Addition of systems 6, 51 Coefficient of viscosity 285 Addition of tensors 6, 51 Cofactors 25, 26, 32 Adherence boundary condition 294 Compatibility equations 259, 260, 262 Aelotropic material 245 Completely skew symmetric system 31 Affine transformation 86, 107 Compound pendulum 195,209 Airy stress function 264 Compressible material 231 Almansi strain tensor 229 Conic sections 151 Alternating tensor 6,7 Conical coordinates 74 Ampere’s law 176,301,337,341 Conjugate dyad 49 Angle between vectors 80, 82 Conjugate metric tensor 36, 77 Angular momentum 218, 287 Conservation of angular momentum 218, 295 Angular velocity 86,87,201,203 Conservation of energy 295 Arc length 60, 67, 133 Conservation of linear momentum 217, 295 Associated tensors 79 Conservation of mass 233, 295 Auxiliary Magnetic field 338 Conservative system 191, 298 Axis of symmetry 247 Conservative electric field 323 B Constitutive equations 242, 251,281, 287 Continuity equation 106,234, 287, 335 Basic equations elasticity 236, 253, 270 Contraction 6, 52 Basic equations for a continuum 236 Contravariant components 36, 44 Basic equations of fluids 281, 287 Contravariant tensor 45 Basis vectors 1,2,37,48 Coordinate curves 37, 67 Beltrami 262 Coordinate surfaces 37, 67 Bernoulli’s Theorem 292 Coordinate transformations
    [Show full text]
  • A Mathematica Package for Doing Tensor Calculations in Differential Geometry User's Manual
    Ricci A Mathematica package for doing tensor calculations in differential geometry User’s Manual Version 1.32 By John M. Lee assisted by Dale Lear, John Roth, Jay Coskey, and Lee Nave 2 Ricci A Mathematica package for doing tensor calculations in differential geometry User’s Manual Version 1.32 By John M. Lee assisted by Dale Lear, John Roth, Jay Coskey, and Lee Nave Copyright c 1992–1998 John M. Lee All rights reserved Development of this software was supported in part by NSF grants DMS-9101832, DMS-9404107 Mathematica is a registered trademark of Wolfram Research, Inc. This software package and its accompanying documentation are provided as is, without guarantee of support or maintenance. The copyright holder makes no express or implied warranty of any kind with respect to this software, including implied warranties of merchantability or fitness for a particular purpose, and is not liable for any damages resulting in any way from its use. Everyone is granted permission to copy, modify and redistribute this software package and its accompanying documentation, provided that: 1. All copies contain this notice in the main program file and in the supporting documentation. 2. All modified copies carry a prominent notice stating who made the last modifi- cation and the date of such modification. 3. No charge is made for this software or works derived from it, with the exception of a distribution fee to cover the cost of materials and/or transmission. John M. Lee Department of Mathematics Box 354350 University of Washington Seattle, WA 98195-4350 E-mail: [email protected] Web: http://www.math.washington.edu/~lee/ CONTENTS 3 Contents 1 Introduction 6 1.1Overview..............................
    [Show full text]
  • Appendix a Relations Between Covariant and Contravariant Bases
    Appendix A Relations Between Covariant and Contravariant Bases The contravariant basis vector gk of the curvilinear coordinate of uk at the point P is perpendicular to the covariant bases gi and gj, as shown in Fig. A.1.This contravariant basis gk can be defined as or or a gk g  g ¼  ðA:1Þ i j oui ou j where a is the scalar factor; gk is the contravariant basis of the curvilinear coordinate of uk. Multiplying Eq. (A.1) by the covariant basis gk, the scalar factor a results in k k ðgi  gjÞ: gk ¼ aðg : gkÞ¼ad ¼ a ÂÃk ðA:2Þ ) a ¼ðgi  gjÞ : gk gi; gj; gk The scalar triple product of the covariant bases can be written as pffiffiffi a ¼ ½¼ðg1; g2; g3 g1  g2Þ : g3 ¼ g ¼ J ðA:3Þ where Jacobian J is the determinant of the covariant basis tensor G. The direction of the cross product vector in Eq. (A.1) is opposite if the dummy indices are interchanged with each other in Einstein summation convention. Therefore, the Levi-Civita permutation symbols (pseudo-tensor components) can be used in expression of the contravariant basis. ffiffiffi p k k g g ¼ J g ¼ðgi  gjÞ¼Àðgj  giÞ eijkðgi  gjÞ eijkðgi  gjÞ ðA:4Þ ) gk ¼ pffiffiffi ¼ g J where the Levi-Civita permutation symbols are defined by 8 <> þ1ifði; j; kÞ is an even permutation; eijk ¼ > À1ifði; j; kÞ is an odd permutation; : A:5 0ifi ¼ j; or i ¼ k; or j ¼ k ð Þ 1 , e ¼ ði À jÞÁðj À kÞÁðk À iÞ for i; j; k ¼ 1; 2; 3 ijk 2 H.
    [Show full text]
  • Mechanics of Continuous Media in $(\Bar {L} N, G) $-Spaces. I
    Mechanics of Continuous Media in (Ln, g)-spaces. I. Introduction and mathematical tools S. Manoff Bulgarian Academy of Sciences, Institute for Nuclear Research and Nuclear Energy, Department of Theoretical Physics, Blvd. Tzarigradsko Chaussee 72 1784 Sofia - Bulgaria e-mail address: [email protected] Abstract Basic notions and mathematical tools in continuum media mechanics are recalled. The notion of exponent of a covariant differential operator is intro- duced and on its basis the geometrical interpretation of the curvature and the torsion in (Ln,g)-spaces is considered. The Hodge (star) operator is general- ized for (Ln,g)-spaces. The kinematic characteristics of a flow are outline in brief. PACS numbers: 11.10.-z; 11.10.Ef; 7.10.+g; 47.75.+f; 47.90.+a; 83.10.Bb 1 Introduction 1.1 Differential geometry and space-time geometry arXiv:gr-qc/0203016v1 5 Mar 2002 In the last years, the evolution of the relations between differential geometry and space-time geometry has made some important steps toward applications of more comprehensive differential-geometric structures in the models of space-time than these used in (pseudo) Riemannian spaces without torsion (Vn-spaces) [1], [2]. 1. Recently, it has been proved that every differentiable manifold with one affine connection and metrics [(Ln,g)-space] [3], [4] could be used as a model for a space- time. In it, the equivalence principle (related to the vanishing of the components of an affine connection at a point or on a curve in a manifold) holds [5] ÷ [11], [12]. Even if the manifold has two different (not only by sign) connections for tangent and co-tangent vector fields [(Ln,g)-space] [13], [14] the principle of equivalence is fulfilled at least for one of the two types of vector fields [15].
    [Show full text]
  • Arxiv:1412.2393V4 [Gr-Qc] 27 Feb 2019 2.6 Geodesics and Normal Coordinates
    Riemannian Geometry: Definitions, Pictures, and Results Adam Marsh February 27, 2019 Abstract A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations. As much material as possible is developed for manifolds with connection (omitting a metric) to make clear which aspects can be readily generalized to gauge theories. The presentation in most cases does not assume a coordinate frame or zero torsion, and the coordinate-free, tensor, and Cartan formalisms are developed in parallel. Contents 1 Introduction 2 2 Parallel transport 3 2.1 The parallel transporter . 3 2.2 The covariant derivative . 4 2.3 The connection . 5 2.4 The covariant derivative in terms of the connection . 6 2.5 The parallel transporter in terms of the connection . 9 arXiv:1412.2393v4 [gr-qc] 27 Feb 2019 2.6 Geodesics and normal coordinates . 9 2.7 Summary . 10 3 Manifolds with connection 11 3.1 The covariant derivative on the tensor algebra . 12 3.2 The exterior covariant derivative of vector-valued forms . 13 3.3 The exterior covariant derivative of algebra-valued forms . 15 3.4 Torsion . 16 3.5 Curvature .
    [Show full text]
  • Differential Geometric Algebra with Leibniz and Grassmann Michael Reed (Crucial Flow Research)
    Proceedings of JuliaCon Differential geometric algebra with Leibniz and Grassmann Michael Reed (Crucial Flow Research) The nature of the geometric algebra code generation enables one to easily extend the abstract product operations to any specific number field type (including differential operators with Leibniz.jl or symbolic coefficients with Reduce.jl), by making use of Julia’s type system. Mixed tensor products with their coefficients are constructed from these operations to work with bivector elements of Lie groups [7][10]. 1. Direct sum parametric type polymorphism The DirectSum.jl package is a work in progress providing the necessary tools to work with an arbitrary Manifold specified github.com/chakravala/Grassmann.jl by an encoding. Due to the parametric type system for the generating VectorBundle, the Julia compiler can fully pre- allocate and often cache values efficiently ahead of run-time. Although intended for use with the Grassmann.jl package, ABSTRACT DirectSum can be used independently. The Grassmann.jl package provides tools for computations based on multi-linear algebra and spin groups using the Definition 1 Vector bundle of submanifolds. Let 푀 = 푇 휇푉 ∈ TensorBundle<:Manifold 푛 extended geometric algebra known as Leibniz-Grassmann- Vect핂 be a of rank , Clifford-Hestenes algebra. Combinatorial products include 휇 푇 푉 = (푛, ℙ, 푔, 휈, 휇), ℙ ⊆ ⟨푣∞, 푣∅⟩ , 푔 ∶ 푉 × 푉 → 핂 exterior, regressive, inner, and geometric; along with the Hodge star, adjoint, reversal, and boundary operators. The The type TensorBundle{n,ℙ,g,휈, 휇} uses byte-encoded data kernelized operations are built up from composite sparse available at pre-compilation, where ℙ specifies the basis for tensor products and Hodge duality, with high dimensional up and down projection, 푔 is a bilinear form that specifies the support for up to 62 indices using staged caching and pre- metric of the space, and 휇 is an integer specifying the order of compilation.
    [Show full text]
  • Induced Representations of Tensors and Spinors of Any Rank in the SHP Theory
    Induced Representations of Tensors and Spinors of any Rank in the SHP Theory Lawrence P. Horwitz Raymond and Beverly Sackler Faculty of Exact Science, School of Physic, Tel Aviv University, Ramat Aviv 69978, Israel. Physics Department, Bar Ilan University, Ramat Gan 52900, Israel. Israel Physics Department, Ariel University, Ariel 40700, Israel Email: [email protected] Meir Zeilig-Hess Raymond and Beverly Sackler Faculty of Exact Science, School of Physic, Tel Aviv University, Ramat Aviv 69978, Israel. Email: [email protected] Abstract We show that a modification of Wigner's induced representation for the description of a relativistic particle with spin can be used to construct spinors and tensors of arbitrary rank, with invariant decomposition over angular momentum. In particular, scalar and vector fields, as well as the representations of their transformations, are constructed. The method that is developed here admits the construction of wave packets and states of a many body relativistic system with definite total angular momentum. Furthermore, a Pauli-Lubanski operator is constructed on the orbit of the induced representation which provides a Casimir operator for the Poincaré group and which contains the physical intrinsic angular momentum of the particle covariantly. 1 Introduction 1.1 Foundations of the SHP theory The theory of Stueckelberg-Horwitz-Piron (called SHP henceforth) was initiated by Stueckelberg's suggestion, according to which pairs of particles can be created and/or annihilated throughout the motion of a single particle, in a way which can be described by classical equations of motions in space-time[1]. The evolution parameter, , of such particle trajectories is an invariant under Lorentz transformations.
    [Show full text]
  • On Dynamic Equations for Interaction of the Affinor Field with Affine Connection
    SUffOWf объединенный ИНСТИТУТ ядерных исследований дубна Е2-8 7-826 A.B.Peetov ON DYNAMIC EQUATIONS FOR INTERACTION OF THE AFFINOR FIELD WITH AFFINE CONNECTION Submitted to XXI International Symposium on Elementary Particle Theory, GDR, 1987. 1987 INTRODUCTION Of fundamental importance for the theory of strong, elect­ romagnetic, and gravitational interactions are such tensor fields as a scalar field, covariant vector field, covariant skew-symmetric tensor field of the second rank, covariant sym­ metric tensor field of the second rank; this list of the sim­ plest tensor fields of rank not higher than the second is in­ complete since it does not contain a mixed tensor field of the second rank that is called the affinor field. If one re­ calls that general relativity is essentially the theory of a symmetric tensor field of the second rank, it is quite natu­ ral to assume that the theory of the affinor field is not on­ ly of a pure mathematical interest. In this report devoted to the theory of affinor field I consider a model of two interacting fields, one of which is an affinor field and the other is a non-tensor field called the affine connection. I. LAGRANGIAN OF THE AFFINOR FIELD Out of two affinors So and Тд an affinor may be construc­ ted called their product P0 = Sa T/3- (I) If Is/8 I Ф 0, there exists an affinor Sg inverse to 8я , S"S^1<7=S_10S^= 8j§. With the operation of multiplication thus defined, the set of affinors with a nonzero determinant forms a group, denoted by Gi , which plays a fundamental role in the theory of affinor field as the group of internal sym­ metry.
    [Show full text]
  • From Schouten to Mackenzie: Notes on Brackets
    From Schouten to Mackenzie: notes on brackets∗ Yvette Kosmann-Schwarzbach In Memory of Kirill Mackenzie (1951–2020) Abstract In this paper, dedicated to the memory of Kirill Mackenzie, I relate the origins and early development of the theory of graded Lie brackets, first in the pu- blications on differential geometry of Schouten, Nijenhuis, and Fr¨olicher–Nijenhuis, then in the work of Gerstenhaber and Nijenhuis–Richardson in cohomology theory. Keywords: graded Lie algebra, Schouten bracket, Nijenhuis torsion, Fr¨olicher– Nijenhuis bracket, Nijenhuis–Richardson bracket, Gerstenhaber algebra, Macken- zie theory. Introduction Many aspects of the contributions of Kirill Mackenzie to the theory of Lie groupoids and Lie algebroids, in particular to that of the double and multiple structures, are already classical and often cited. As a tribute to his memory, arXiv:2105.14828v1 [math.HO] 31 May 2021 I shall survey the history of the emergence of some theories that were tools – among many others – in the development and description of his mathematics. In his General Theory of Lie Groupoids and Lie Algebroids, in 2005, Mackenzie inserted a section of “Notes” at the end of each chapter. Yet he ad- vised his readers that his notes were “not comprehensive historical surveys”. After the chapter “Cohomology and Schouten calculus for Lie algebroids”, he wrote: “The Schouten bracket of multivector fields on a manifold goes back to Schouten [1954] and Nijenhuis [1955]. The same alge- braic structures on the cohomology of associative algebras were introduced by Gerstenhaber [1963].” ∗To appear in the Journal of Geometric Mechanics, 2021. 1 I shall try to expand on these elements of a story, and survey some of the early developments in the theory of the various graded brackets that Mackenzie cited and used1.
    [Show full text]