Lorentz Group Projector Technique for Decomposing Reducible Representations and Applications to High Spins
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Representations of Certain Semidirect Product Groups* H
JOURNAL OF FUNCTIONAL ANALYSIS 19, 339-372 (1975) Representations of Certain Semidirect Product Groups* JOSEPH A. WOLF Department of Mathematics, University of California, Berkeley, California 94720 Communicated by the Editovs Received April I, 1974 We study a class of semidirect product groups G = N . U where N is a generalized Heisenberg group and U is a generalized indefinite unitary group. This class contains the Poincare group and the parabolic subgroups of the simple Lie groups of real rank 1. The unitary representations of G and (in the unimodular cases) the Plancherel formula for G are written out. The problem of computing Mackey obstructions is completely avoided by realizing the Fock representations of N on certain U-invariant holomorphic cohomology spaces. 1. INTRODUCTION AND SUMMARY In this paper we write out the irreducible unitary representations for a type of semidirect product group that includes the Poincare group and the parabolic subgroups of simple Lie groups of real rank 1. If F is one of the four finite dimensional real division algebras, the corresponding groups in question are the Np,Q,F . {U(p, 4; F) xF+}, where FP~*: right vector space F”+Q with hermitian form h(x, y) = i xiyi - y xipi ) 1 lJ+1 N p,p,F: Im F + F”q* with (w. , ~o)(w, 4 = (w, + 7.0+ Im h(z, ,4, z. + 4, U(p, Q; F): unitary group of Fp,q, F+: subgroup of the group generated by F-scalar multiplication on Fp>*. * Research partially supported by N.S.F. Grant GP-16651. 339 Copyright 0 1975 by Academic Press, Inc. -
On Manifolds of Tensors of Fixed Tt-Rank
ON MANIFOLDS OF TENSORS OF FIXED TT-RANK SEBASTIAN HOLTZ, THORSTEN ROHWEDDER, AND REINHOLD SCHNEIDER Abstract. Recently, the format of TT tensors [19, 38, 34, 39] has turned out to be a promising new format for the approximation of solutions of high dimensional problems. In this paper, we prove some new results for the TT representation of a tensor U ∈ Rn1×...×nd and for the manifold of tensors of TT-rank r. As a first result, we prove that the TT (or compression) ranks ri of a tensor U are unique and equal to the respective seperation ranks of U if the components of the TT decomposition are required to fulfil a certain maximal rank condition. We then show that d the set T of TT tensors of fixed rank r forms an embedded manifold in Rn , therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices [7], we introduce certain gauge conditions to obtain a unique representation of the tangent space TU T of T and deduce a local parametrization of the TT manifold. The parametrisation of TU T is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems [33]. We conclude with remarks on those applications and present some numerical examples. 1. Introduction The treatment of high-dimensional problems, typically of problems involving quantities from Rd for larger dimensions d, is still a challenging task for numerical approxima- tion. This is owed to the principal problem that classical approaches for their treatment normally scale exponentially in the dimension d in both needed storage and computa- tional time and thus quickly become computationally infeasable for sensible discretiza- tions of problems of interest. -
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.. -
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. -
1.3 Cartesian Tensors a Second-Order Cartesian Tensor Is Defined As A
1.3 Cartesian tensors A second-order Cartesian tensor is defined as a linear combination of dyadic products as, T Tijee i j . (1.3.1) The coefficients Tij are the components of T . A tensor exists independent of any coordinate system. The tensor will have different components in different coordinate systems. The tensor T has components Tij with respect to basis {ei} and components Tij with respect to basis {e i}, i.e., T T e e T e e . (1.3.2) pq p q ij i j From (1.3.2) and (1.2.4.6), Tpq ep eq TpqQipQjqei e j Tij e i e j . (1.3.3) Tij QipQjqTpq . (1.3.4) Similarly from (1.3.2) and (1.2.4.6) Tij e i e j Tij QipQjqep eq Tpqe p eq , (1.3.5) Tpq QipQjqTij . (1.3.6) Equations (1.3.4) and (1.3.6) are the transformation rules for changing second order tensor components under change of basis. In general Cartesian tensors of higher order can be expressed as T T e e ... e , (1.3.7) ij ...n i j n and the components transform according to Tijk ... QipQjqQkr ...Tpqr... , Tpqr ... QipQjqQkr ...Tijk ... (1.3.8) The tensor product S T of a CT(m) S and a CT(n) T is a CT(m+n) such that S T S T e e e e e e . i1i2 im j1j 2 jn i1 i2 im j1 j2 j n 1.3.1 Contraction T Consider the components i1i2 ip iq in of a CT(n). -
Tensor-Spinor Theory of Gravitation in General Even Space-Time Dimensions
Physics Letters B 817 (2021) 136288 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Tensor-spinor theory of gravitation in general even space-time dimensions ∗ Hitoshi Nishino a, ,1, Subhash Rajpoot b a Department of Physics, College of Natural Sciences and Mathematics, California State University, 2345 E. San Ramon Avenue, M/S ST90, Fresno, CA 93740, United States of America b Department of Physics & Astronomy, California State University, 1250 Bellflower Boulevard, Long Beach, CA 90840, United States of America a r t i c l e i n f o a b s t r a c t Article history: We present a purely tensor-spinor theory of gravity in arbitrary even D = 2n space-time dimensions. Received 18 March 2021 This is a generalization of the purely vector-spinor theory of gravitation by Bars and MacDowell (BM) in Accepted 9 April 2021 4D to general even dimensions with the signature (2n − 1, 1). In the original BM-theory in D = (3, 1), Available online 21 April 2021 the conventional Einstein equation emerges from a theory based on the vector-spinor field ψμ from a Editor: N. Lambert m lagrangian free of both the fundamental metric gμν and the vierbein eμ . We first improve the original Keywords: BM-formulation by introducing a compensator χ, so that the resulting theory has manifest invariance = =− = Bars-MacDowell theory under the nilpotent local fermionic symmetry: δψ Dμ and δ χ . We next generalize it to D Vector-spinor (2n − 1, 1), following the same principle based on a lagrangian free of fundamental metric or vielbein Tensors-spinors rs − now with the field content (ψμ1···μn−1 , ωμ , χμ1···μn−2 ), where ψμ1···μn−1 (or χμ1···μn−2 ) is a (n 1) (or Metric-less formulation (n − 2)) rank tensor-spinor. -
SUPERSYMMETRY 1. Introduction the Purpose
SUPERSYMMETRY JOSH KANTOR 1. Introduction The purpose of these notes is to give a short and (overly?)simple description of supersymmetry for Mathematicians. Our description is far from complete and should be thought of as a first pass at the ideas that arise from supersymmetry. Fundamental to supersymmetry is the mathematics of Clifford algebras and spin groups. We will describe the mathematical results we are using but we refer the reader to the references for proofs. In particular [4], [1], and [5] all cover spinors nicely. 2. Spin and Clifford Algebras We will first review the definition of spin, spinors, and Clifford algebras. Let V be a vector space over R or C with some nondegenerate quadratic form. The clifford algebra of V , l(V ), is the algebra generated by V and 1, subject to the relations v v = v, vC 1, or equivalently v w + w v = 2 v, w . Note that elements of· l(V ) !can"b·e written as polynomials· in V · and this! giv"es a splitting l(V ) = l(VC )0 l(V )1. Here l(V )0 is the set of elements of l(V ) which can bCe writtenC as a linear⊕ C combinationC of products of even numbers ofCvectors from V , and l(V )1 is the set of elements which can be written as a linear combination of productsC of odd numbers of vectors from V . Note that more succinctly l(V ) is just the quotient of the tensor algebra of V by the ideal generated by v vC v, v 1. -
Low-Level Image Processing with the Structure Multivector
Low-Level Image Processing with the Structure Multivector Michael Felsberg Bericht Nr. 0202 Institut f¨ur Informatik und Praktische Mathematik der Christian-Albrechts-Universitat¨ zu Kiel Olshausenstr. 40 D – 24098 Kiel e-mail: [email protected] 12. Marz¨ 2002 Dieser Bericht enthalt¨ die Dissertation des Verfassers 1. Gutachter Prof. G. Sommer (Kiel) 2. Gutachter Prof. U. Heute (Kiel) 3. Gutachter Prof. J. J. Koenderink (Utrecht) Datum der mundlichen¨ Prufung:¨ 12.2.2002 To Regina ABSTRACT The present thesis deals with two-dimensional signal processing for computer vi- sion. The main topic is the development of a sophisticated generalization of the one-dimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance – equivariance constraint, and by its relation to complex analysis and potential theory, a two-dimensional approach is derived. This method is called the monogenic signal and it is based on the Riesz transform instead of the Hilbert transform. By means of this linear approach it is possible to estimate the local orientation and the local phase of signals which are projections of one-dimensional functions to two dimensions. For general two-dimensional signals, however, the monogenic signal has to be further extended, yielding the structure multivector. The latter approach combines the ideas of the structure tensor and the quaternionic analytic signal. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitudes, the local anisotropy, the local orientation, and two local phases. Both, the monogenic signal and the struc- ture multivector are combined with an appropriate scale-space approach, resulting in generalized quadrature filters. -
The Unitary Representations of the Poincaré Group in Any Spacetime Dimension Abstract Contents
SciPost Physics Lecture Notes Submission The unitary representations of the Poincar´egroup in any spacetime dimension X. Bekaert1, N. Boulanger2 1 Institut Denis Poisson, Unit´emixte de Recherche 7013, Universit´ede Tours, Universit´e d'Orl´eans,CNRS, Parc de Grandmont, 37200 Tours (France) [email protected] 2 Service de Physique de l'Univers, Champs et Gravitation, Universit´ede Mons, UMONS Research Institute for Complex Systems, Place du Parc 20, 7000 Mons (Belgium) [email protected] December 31, 2020 1 Abstract 2 An extensive group-theoretical treatment of linear relativistic field equations 3 on Minkowski spacetime of arbitrary dimension D > 3 is presented. An exhaus- 4 tive treatment is performed of the two most important classes of unitary irre- 5 ducible representations of the Poincar´egroup, corresponding to massive and 6 massless fundamental particles. Covariant field equations are given for each 7 unitary irreducible representation of the Poincar´egroup with non-negative 8 mass-squared. 9 10 Contents 11 1 Group-theoretical preliminaries 2 12 1.1 Universal covering of the Lorentz group 2 13 1.2 The Poincar´egroup and algebra 3 14 1.3 ABC of unitary representations 4 15 2 Elementary particles as unitary irreducible representations of the isom- 16 etry group 5 17 3 Classification of the unitary representations 7 18 3.1 Induced representations 7 19 3.2 Orbits and stability subgroups 8 20 3.3 Classification 10 21 4 Tensorial representations and Young diagrams 12 22 4.1 Symmetric group 12 23 4.2 General linear -
The Riemann Curvature Tensor
The Riemann Curvature Tensor Jennifer Cox May 6, 2019 Project Advisor: Dr. Jonathan Walters Abstract A tensor is a mathematical object that has applications in areas including physics, psychology, and artificial intelligence. The Riemann curvature tensor is a tool used to describe the curvature of n-dimensional spaces such as Riemannian manifolds in the field of differential geometry. The Riemann tensor plays an important role in the theories of general relativity and gravity as well as the curvature of spacetime. This paper will provide an overview of tensors and tensor operations. In particular, properties of the Riemann tensor will be examined. Calculations of the Riemann tensor for several two and three dimensional surfaces such as that of the sphere and torus will be demonstrated. The relationship between the Riemann tensor for the 2-sphere and 3-sphere will be studied, and it will be shown that these tensors satisfy the general equation of the Riemann tensor for an n-dimensional sphere. The connection between the Gaussian curvature and the Riemann curvature tensor will also be shown using Gauss's Theorem Egregium. Keywords: tensor, tensors, Riemann tensor, Riemann curvature tensor, curvature 1 Introduction Coordinate systems are the basis of analytic geometry and are necessary to solve geomet- ric problems using algebraic methods. The introduction of coordinate systems allowed for the blending of algebraic and geometric methods that eventually led to the development of calculus. Reliance on coordinate systems, however, can result in a loss of geometric insight and an unnecessary increase in the complexity of relevant expressions. Tensor calculus is an effective framework that will avoid the cons of relying on coordinate systems. -
Tensor, Exterior and Symmetric Algebras
Tensor, Exterior and Symmetric Algebras Daniel Murfet May 16, 2006 Throughout this note R is a commutative ring, all modules are left R-modules. If we say a ring is noncommutative, we mean it is not necessarily commutative. Unless otherwise specified, all rings are noncommutative (except for R). If A is a ring then the center of A is the set of all x ∈ A with xy = yx for all y ∈ A. Contents 1 Definitions 1 2 The Tensor Algebra 5 3 The Exterior Algebra 6 3.1 Dimension of the Exterior Powers ............................ 11 3.2 Bilinear Forms ...................................... 14 3.3 Other Properties ..................................... 18 3.3.1 The determinant formula ............................ 18 4 The Symmetric Algebra 19 1 Definitions Definition 1. A R-algebra is a ring morphism φ : R −→ A where A is a ring and the image of φ is contained in the center of A. This is equivalent to A being an R-module and a ring, with r · (ab) = (r · a)b = a(r · b), via the identification of r · 1 and φ(r). A morphism of R-algebras is a ring morphism making the appropriate diagram commute, or equivalently a ring morphism which is also an R-module morphism. In this section RnAlg will denote the category of these R-algebras. We use RAlg to denote the category of commutative R-algebras. A graded ring is a ring A together with a set of subgroups Ad, d ≥ 0 such that A = ⊕d≥0Ad as an abelian group, and st ∈ Ad+e for all s ∈ Ad, t ∈ Ae. -
Lecture 12. Tensors
Lecture 12. Tensors In this lecture we define tensors on a manifold, and the associated bundles, and operations on tensors. 12.1 Basic definitions We have already seen several examples of the idea we are about to introduce, namely linear (or multilinear) operators acting on vectors on M. For example, the metric is a bilinear operator which takes two vectors to give a real number, i.e. gx : TxM × TxM → R for each x is defined by u, v → gx(u, v). The difference between two connections ∇(1) and ∇(2) is a bilinear op- erator which takes two vectors and gives a vector, i.e. a bilinear operator Sx : TxM × TxM → TxM for each x ∈ M. Similarly, the torsion of a connec- tion has this form. Definition 12.1.1 A covector ω at x ∈ M is a linear map from TxM to R. The set of covectors at x forms an n-dimensional vector space, which we ∗ denote Tx M.Atensor of type (k, l)atx is a multilinear map which takes k vectors and l covectors and gives a real number × × × ∗ × × ∗ → R Tx : .TxM .../0 TxM1 .Tx M .../0 Tx M1 . k times l times Note that a covector is just a tensor of type (1, 0), and a vector is a tensor of type (0, 1), since a vector v acts linearly on a covector ω by v(ω):=ω(v). Multilinearity means that i1 ik j1 jl T c vi1 ,..., c vik , aj1 ω ,..., ajl ω i i j j 1 k 1 l . i1 ik j1 jl = c ...c aj1 ...ajl T (vi1 ,...,vik ,ω ,...,ω ) i1,...,ik,j1,...,jl 110 Lecture 12.