Irreducible Tensor Operators and the Wigner-Eckart Theorem †
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Tensor-Tensor Products with Invertible Linear Transforms
Tensor-Tensor Products with Invertible Linear Transforms Eric Kernfelda, Misha Kilmera, Shuchin Aeronb aDepartment of Mathematics, Tufts University, Medford, MA bDepartment of Electrical and Computer Engineering, Tufts University, Medford, MA Abstract Research in tensor representation and analysis has been rising in popularity in direct response to a) the increased ability of data collection systems to store huge volumes of multidimensional data and b) the recognition of potential modeling accuracy that can be provided by leaving the data and/or the operator in its natural, multidi- mensional form. In comparison to matrix theory, the study of tensors is still in its early stages. In recent work [1], the authors introduced the notion of the t-product, a generalization of matrix multiplication for tensors of order three, which can be extended to multiply tensors of arbitrary order [2]. The multiplication is based on a convolution-like operation, which can be implemented efficiently using the Fast Fourier Transform (FFT). The corresponding linear algebraic framework from the original work was further developed in [3], and it allows one to elegantly generalize all classical algorithms from linear algebra. In the first half of this paper, we develop a new tensor-tensor product for third-order tensors that can be implemented by us- ing a discrete cosine transform, showing that a similar linear algebraic framework applies to this new tensor operation as well. The second half of the paper is devoted to extending this development in such a way that tensor-tensor products can be de- fined in a so-called transform domain for any invertible linear transform. -
Squeezed and Entangled States of a Single Spin
SQUEEZED AND ENTANGLED STATES OF A SINGLE SPIN Barı¸s Oztop,¨ Alexander A. Klyachko and Alexander S. Shumovsky Faculty of Science, Bilkent University Bilkent, Ankara, 06800 Turkey e-mail: [email protected] (Received 23 December 2007; accepted 1 March 2007) Abstract We show correspondence between the notions of spin squeez- ing and spin entanglement. We propose a new measure of spin squeezing. We consider a number of physical examples. Concepts of Physics, Vol. IV, No. 3 (2007) 441 DOI: 10.2478/v10005-007-0020-0 Barı¸s Oztop,¨ Alexander A. Klyachko and Alexander S. Shumovsky It is well known that the concept of squeezed states [1] was orig- inated from the famous work by N.N. Bogoliubov [2] on the super- fluidity of liquid He4 and canonical transformations. Initially, it was developed for the Bose-fields. Later on, it has been extended on spin systems as well. The two main objectives of the present paper are on the one hand to show that the single spin s 1 can be prepared in a squeezed state and on the other hand to demonstrate≥ one-to-one correspondence between the notions of spin squeezing and entanglement. The results are illustrated by physical examples. Spin-coherent states | Historically, the notion of spin-coherent states had been introduced [3] before the notion of spin-squeezed states. In a sense, it just reflected the idea of Glauber [4] about creation of Bose-field coherent states from the vacuum by means of the displacement operator + α field = D(α) vac ;D(α) = exp(αa α∗a); (1) j i j i − where α C is an arbitrary complex parameter and a+; a are the Boson creation2 and annihilation operators. -
The Pattern Calculus for Tensor Operators in Quantum Groups Mark Gould and L
The pattern calculus for tensor operators in quantum groups Mark Gould and L. C. Biedenharn Citation: Journal of Mathematical Physics 33, 3613 (1992); doi: 10.1063/1.529909 View online: http://dx.doi.org/10.1063/1.529909 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/33/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Irreducible tensor operators in the regular coaction formalisms of compact quantum group algebras J. Math. Phys. 37, 4590 (1996); 10.1063/1.531643 Tensor operators for quantum groups and applications J. Math. Phys. 33, 436 (1992); 10.1063/1.529833 Tensor operators of finite groups J. Math. Phys. 14, 1423 (1973); 10.1063/1.1666196 On the structure of the canonical tensor operators in the unitary groups. I. An extension of the pattern calculus rules and the canonical splitting in U(3) J. Math. Phys. 13, 1957 (1972); 10.1063/1.1665940 Irreducible tensor operators for finite groups J. Math. Phys. 13, 1892 (1972); 10.1063/1.1665928 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.42.98 On: Thu, 29 Sep 2016 02:42:08 The pattern calculus for tensor operators in quantum groups Mark Gould Department of Applied Mathematics, University of Queensland,St. Lucia, Queensland,Australia 4072 L. C. Biedenharn Department of Physics, Duke University, Durham, North Carolina 27706 (Received 28 February 1992; accepted for publication 1 June 1992) An explicit algebraic evaluation is given for all q-tensor operators (with unit norm) belonging to the quantum group U&(n)) and having extremal operator shift patterns acting on arbitrary U&u(n)) irreps. -
Arxiv:2012.13347V1 [Physics.Class-Ph] 15 Dec 2020
KPOP E-2020-04 Bra-Ket Representation of the Inertia Tensor U-Rae Kim, Dohyun Kim, and Jungil Lee∗ KPOPE Collaboration, Department of Physics, Korea University, Seoul 02841, Korea (Dated: June 18, 2021) Abstract We employ Dirac's bra-ket notation to define the inertia tensor operator that is independent of the choice of bases or coordinate system. The principal axes and the corresponding principal values for the elliptic plate are determined only based on the geometry. By making use of a general symmetric tensor operator, we develop a method of diagonalization that is convenient and intuitive in determining the eigenvector. We demonstrate that the bra-ket approach greatly simplifies the computation of the inertia tensor with an example of an N-dimensional ellipsoid. The exploitation of the bra-ket notation to compute the inertia tensor in classical mechanics should provide undergraduate students with a strong background necessary to deal with abstract quantum mechanical problems. PACS numbers: 01.40.Fk, 01.55.+b, 45.20.dc, 45.40.Bb Keywords: Classical mechanics, Inertia tensor, Bra-ket notation, Diagonalization, Hyperellipsoid arXiv:2012.13347v1 [physics.class-ph] 15 Dec 2020 ∗Electronic address: [email protected]; Director of the Korea Pragmatist Organization for Physics Educa- tion (KPOP E) 1 I. INTRODUCTION The inertia tensor is one of the essential ingredients in classical mechanics with which one can investigate the rotational properties of rigid-body motion [1]. The symmetric nature of the rank-2 Cartesian tensor guarantees that it is described by three fundamental parameters called the principal moments of inertia Ii, each of which is the moment of inertia along a principal axis. -
Chapter 5 ANGULAR MOMENTUM and ROTATIONS
Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another. -
Some Applications of Eigenvalue Problems for Tensor and Tensor–Block Matrices for Mathematical Modeling of Micropolar Thin Bodies
Mathematical and Computational Applications Article Some Applications of Eigenvalue Problems for Tensor and Tensor–Block Matrices for Mathematical Modeling of Micropolar Thin Bodies Mikhail Nikabadze 1,2,∗ and Armine Ulukhanyan 2 1 Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia 2 Department of Computational Mathematics and Mathematical Physics, Bauman Moscow State Technical University, 105005 Moscow, Russia; [email protected] * Correspondence: [email protected] Received: 26 January 2019; Accepted: 21 March 2019; Published: 22 March 2019 Abstract: The statement of the eigenvalue problem for a tensor–block matrix (TBM) of any order and of any even rank is formulated, and also some of its special cases are considered. In particular, using the canonical presentation of the TBM of the tensor of elastic modules of the micropolar theory, in the canonical form the specific deformation energy and the constitutive relations are written. With the help of the introduced TBM operator, the equations of motion of a micropolar arbitrarily anisotropic medium are written, and also the boundary conditions are written down by means of the introduced TBM operator of the stress and the couple stress vectors. The formulations of initial-boundary value problems in these terms for an arbitrary anisotropic medium are given. The questions on the decomposition of initial-boundary value problems of elasticity and thin body theory for some anisotropic media are considered. In particular, the initial-boundary problems of the micropolar (classical) theory of elasticity are presented with the help of the introduced TBM operators (tensors–operators). In the case of an isotropic micropolar elastic medium (isotropic and transversely isotropic classical media), the TBM operator (tensors–operators) of cofactors to TBM operators (tensors–tensors) of the initial-boundary value problems are constructed that allow decomposing initial-boundary value problems. -
Introduction to Vector and Tensor Analysis
Introduction to vector and tensor analysis Jesper Ferkinghoff-Borg September 6, 2007 Contents 1 Physical space 3 1.1 Coordinate systems . 3 1.2 Distances . 3 1.3 Symmetries . 4 2 Scalars and vectors 5 2.1 Definitions . 5 2.2 Basic vector algebra . 5 2.2.1 Scalar product . 6 2.2.2 Cross product . 7 2.3 Coordinate systems and bases . 7 2.3.1 Components and notation . 9 2.3.2 Triplet algebra . 10 2.4 Orthonormal bases . 11 2.4.1 Scalar product . 11 2.4.2 Cross product . 12 2.5 Ordinary derivatives and integrals of vectors . 13 2.5.1 Derivatives . 14 2.5.2 Integrals . 14 2.6 Fields . 15 2.6.1 Definition . 15 2.6.2 Partial derivatives . 15 2.6.3 Differentials . 16 2.6.4 Gradient, divergence and curl . 17 2.6.5 Line, surface and volume integrals . 20 2.6.6 Integral theorems . 24 2.7 Curvilinear coordinates . 26 2.7.1 Cylindrical coordinates . 27 2.7.2 Spherical coordinates . 28 3 Tensors 30 3.1 Definition . 30 3.2 Outer product . 31 3.3 Basic tensor algebra . 31 1 3.3.1 Transposed tensors . 32 3.3.2 Contraction . 33 3.3.3 Special tensors . 33 3.4 Tensor components in orthonormal bases . 34 3.4.1 Matrix algebra . 35 3.4.2 Two-point components . 38 3.5 Tensor fields . 38 3.5.1 Gradient, divergence and curl . 39 3.5.2 Integral theorems . 40 4 Tensor calculus 42 4.1 Tensor notation and Einsteins summation rule . -
Operators and States
Operators and States University Press Scholarship Online Oxford Scholarship Online Methods in Theoretical Quantum Optics Stephen Barnett and Paul Radmore Print publication date: 2002 Print ISBN-13: 9780198563617 Published to Oxford Scholarship Online: January 2010 DOI: 10.1093/acprof:oso/9780198563617.001.0001 Operators and States STEPHEN M. BARNETT PAUL M. RADMORE DOI:10.1093/acprof:oso/9780198563617.003.0003 Abstract and Keywords This chapter is concerned with providing the necessary rules governing the manipulation of the operators and the properties of the states to enable us top model and describe some physical systems. Atom and field operators for spin, angular momentum, the harmonic oscillator or single mode field and for continuous fields are introduced. Techniques are derived for treating functions of operators and ordering theorems for manipulating these. Important states of the electromagnetic field and their properties are presented. These include the number states, thermal states, coherent states and squeezed states. The coherent and squeezed states are generated by the actions of the Glauber displacement operator and the squeezing operator. Angular momentum coherent states are described and applied to the coherent evolution of a two-state atom and to the action of a beam- splitter. Page 1 of 63 PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter -
Beyond Scalar, Vector and Tensor Harmonics in Maximally Symmetric Three-Dimensional Spaces
Beyond scalar, vector and tensor harmonics in maximally symmetric three-dimensional spaces Cyril Pitrou1, ∗ and Thiago S. Pereira2, y 1Institut d'Astrophysique de Paris, CNRS UMR 7095, 98 bis Bd Arago, 75014 Paris, France. 2Departamento de F´ısica, Universidade Estadual de Londrina, Rod. Celso Garcia Cid, Km 380, 86057-970, Londrina, Paran´a,Brazil. (Dated: October 1, 2019) We present a comprehensive construction of scalar, vector and tensor harmonics on max- imally symmetric three-dimensional spaces. Our formalism relies on the introduction of spin-weighted spherical harmonics and a generalized helicity basis which, together, are ideal tools to decompose harmonics into their radial and angular dependencies. We provide a thorough and self-contained set of expressions and relations for these harmonics. Being gen- eral, our formalism also allows to build harmonics of higher tensor type by recursion among radial functions, and we collect the complete set of recursive relations which can be used. While the formalism is readily adapted to computation of CMB transfer functions, we also collect explicit forms of the radial harmonics which are needed for other cosmological ob- servables. Finally, we show that in curved spaces, normal modes cannot be factorized into a local angular dependence and a unit norm function encoding the orbital dependence of the harmonics, contrary to previous statements in the literature. 1. INTRODUCTION ables. Indeed, on cosmological scales, where lin- ear perturbation theory successfully accounts for the formation of structures, perturbation modes, Tensor harmonics are ubiquitous tools in that is the components in an expansion on tensor gravitational theories. Their applicability reach harmonics, evolve independently from one an- a wide spectrum of topics including black-hole other. -
The Grassmannian Sigma Model in SU(2) Yang-Mills Theory
The Grassmannian σ-model in SU(2) Yang-Mills Theory David Marsh Undergraduate Thesis Supervisor: Antti Niemi Department of Theoretical Physics Uppsala University Contents 1 Introduction 4 1.1 Introduction ............................ 4 2 Some Quantum Field Theory 7 2.1 Gauge Theories .......................... 7 2.1.1 Basic Yang-Mills Theory ................. 8 2.1.2 Faddeev, Popov and Ghosts ............... 10 2.1.3 Geometric Digression ................... 14 2.2 The nonlinear sigma models .................. 15 2.2.1 The Linear Sigma Model ................. 15 2.2.2 The Historical Example ................. 16 2.2.3 Geometric Treatment ................... 17 3 Basic properties of the Grassmannian 19 3.1 Standard Local Coordinates ................... 19 3.2 Plücker coordinates ........................ 21 3.3 The Grassmannian as a homogeneous space .......... 23 3.4 The complexied four-vector ................... 25 4 The Grassmannian σ Model in SU(2) Yang-Mills Theory 30 4.1 The SU(2) Yang-Mills Theory in spin-charge separated variables 30 4.1.1 Variables .......................... 30 4.1.2 The Physics of Spin-Charge Separation ......... 34 4.2 Appearance of the ostensible model ............... 36 4.3 The Grassmannian Sigma Model ................ 38 4.3.1 Gauge Formalism ..................... 38 4.3.2 Projector Formalism ................... 39 4.3.3 Induced Metric Formalism ................ 40 4.4 Embedding Equations ...................... 43 4.5 Hodge Star Decomposition .................... 48 2 4.6 Some Complex Dierential Geometry .............. 50 4.7 Quaternionic Decomposition ................... 54 4.7.1 Verication ........................ 56 5 The Interaction 58 5.1 The Interaction Terms ...................... 58 5.1.1 Limits ........................... 59 5.2 Holomorphic forms and Dolbeault Cohomology ........ 60 5.3 The Form of the interaction .................. -
Mach's Principle: Exact Frame-Dragging Via
Mach’s principle: Exact frame-dragging via gravitomagnetism in perturbed Friedmann-Robertson-Walker universes with K =( 1, 0) ± Christoph Schmid∗ ETH Zurich, Institute for Theoretical Physics, 8093 Zurich, Switzerland (Dated: October 30, 2018) We show that there is exact dragging of the axis directions of local inertial frames by a weighted average of the cosmological energy currents via gravitomagnetism for all linear perturbations of all Friedmann-Robertson-Walker (FRW) universes and of Einstein’s static closed universe, and for all energy-momentum-stress tensors and in the presence of a cosmolgical constant. This includes FRW universes arbitrarily close to the Milne Universe and the de Sitter universe. Hence the postulate formulated by Ernst Mach about the physical cause for the time-evolution of inertial axes is shown to hold in General Relativity for linear perturbations of FRW universes. — The time-evolution of local inertial axes (relative to given local fiducial axes) is given experimentally by the precession angular velocity Ω~ gyro of local gyroscopes, which in turn gives the operational definition of the gravitomagnetic field: B~ g ≡−2 Ω~ gyro. The gravitomagnetic field is caused by energy currents J~ε via 0ˆ − 2 ~ − ~ ~ ~ the momentum constraint, Einstein’s G ˆi equation, ( ∆+ µ )Ag = 16πGNJε with Bg = curl Ag. This equation is analogous to Amp`ere’s law, but it holds for all time-dependent situations. ∆ is the de Rham-Hodge Laplacian, and ∆ = −curl curl for the vorticity sector in Riemannian 3- space. — In the solution for an open universe the 1/r2-force of Amp`ere is replaced by a Yukawa force Yµ(r)=(−d/dr)[(1/R) exp(−µr)], form-identical for FRW backgrounds with K = (−1, 0). -
Chapter 9 Symmetries and Transformation Groups
Chapter 9 Symmetries and transformation groups When contemporaries of Galilei argued against the heliocentric world view by pointing out that we do not feel like rotating with high velocity around the sun he argued that a uniform motion cannot be recognized because the laws of nature that govern our environment are invariant under what we now call a Galilei transformation between inertial systems. Invariance arguments have since played an increasing role in physics both for conceptual and practical reasons. In the early 20th century the mathematician Emmy Noether discovered that energy con- servation, which played a central role in 19th century physics, is just a special case of a more general relation between symmetries and conservation laws. In particular, energy and momen- tum conservation are equivalent to invariance under translations of time and space, respectively. At about the same time Einstein discovered that gravity curves space-time so that space-time is in general not translation invariant. As a consequence, energy is not conserved in cosmology. In the present chapter we discuss the symmetries of non-relativistic and relativistic kine- matics, which derive from the geometrical symmetries of Euclidean space and Minkowski space, respectively. We decompose transformation groups into discrete and continuous parts and study the infinitesimal form of the latter. We then discuss the transition from classical to quantum mechanics and use rotations to prove the Wigner-Eckhart theorem for matrix elements of tensor operators. After discussing the discrete symmetries parity, time reversal and charge conjugation we conclude with the implications of gauge invariance in the Aharonov–Bohm effect.