DOCSLIB.ORG

Symmetry Groups in Physics Part 1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Symmetry Groups in Physics Part 1 Symmetry groups in physics Part 1 George Jorjadze Free University of Tbilisi Zielona Gora - 24.01.2017 G.J.| Symmetry groups in physics Lecture 2 1/21 Contents • Introduction • Group axioms • Examples of groups • Subgroup • Group homomorphism • Group representation • Matrix exponent • GL(n) and SL(n) groups • U(n) and SU(n) groups • SU(2) group • O(n) and SO(n) groups • SO(3) group G.J.| Symmetry groups in physics Contents 2/21 Introduction Group theory is a mathematical tool that investigates symmetries of physical systems and provides effective description of physical phenomena. The methods of group theory allow to handle the consequences of symmetries even without the dynamical content of a physical problem. Physical systems have discrete or/and continuous symmetries. Examples of discrete symmetries: • Space inversion ~x 7! −~x, • Time inversion t 7! −t, • Charge conjugation e 7! −e, • Permutations of identical particles. Examples of continuous symmetries are: • Space-time translations, • Space rotations, • Lorentz transformations, • Rescaling of space-time coordinates. G.J.| Symmetry groups in physics Introduction 3/21 Group axioms A group G is a set of elements g, h, ... with a binary operation g · h 2 G, which is called product and it satisfies the following axioms: • Asscociativity. For any three elements (g; h; k) from G one has g · (h · k) = (g · h) · k : • Unit element. There is an element e 2 G, such that for any g 2 G e · g = g · e = g : • Inverse element. Any g 2 G has an inverse element, denoted by g−1, with the property g · g−1 = g−1 · g = e : Exercise 1. Show that G contains only one unit element; and g 2 G has only one inverse element. In general, the group product is not commutative (see examples). G.J.| Symmetry groups in physics Group axioms 4/21 Examples of groups 1. Real numbers R form a group if the group product is defined as the sum of two numbers. Then, the unit element is 0 and the inverse element of x 2 R is −x. 2. Positive real numbers R+ with standard multiplication form a group. Its unit element is 1 and the inverse of a 2 R+ is 1=a. 3. Non-degenerated n × n matrices form a group under the matrix multiplications. The group unit element here is the unit matrix. 4. Unitary transformations of a Hilbert space H form a group. These transformations preserve the transition amplitudes, i.e. the scalar product in H h U j U φ i = h j φ i: The product of two transformations is defined as their composition (U1 · U2) j i = U1 (U2 j i) : G.J.| Symmetry groups in physics Examples of groups 5/21 5. Galilei transformations relate space-time coordinates of inertial systems in Newtonian mechanics t 7! t~= t + a ; xi 7! x~i = Rij xj + ai + vi t ; (i; j = 1; 2; 3): Here, a is a time-translation parameter, Rij is a rotation matrix, ai is a vector for space translations and vi is a velocity between the inertial systems. Galilei transformations form a group, where the group product is defined as a composition of two transformations, similarly to the previous example. This group is called Galilei group. µ µ µ ν µ 6. Poincare transformations, x 7! x~ = Λ ν x + a ; relate space-time coordinates of inertial systems in relativistic mechanics and they form Poincare group. We will discuss this group later in more detail. Note that the groups in the first two examples are commutative (g · h = h · g for any pair (g; h)), while others are non-commutative. In the last three examples groups are given as symmetry transformations of physical systems. Thus, groups naturally arise in physics. G.J.| Symmetry groups in physics Examples of groups 6/21 Subgroup Definition. A subset H of a group G is called subgroup, if H is a group under the product given in G. Examples: 1. Non-zero complex numbers z 6= 0 form a group under the standard multiplications. Complex numbers with unit norm jzj = 1 form its subgroup, which is denoted by U(1). 2. Space-time translations and space rotations form subgroups of Galilei group. The boosts also form its subgroup. Exercise 2. Check that the matrices cos θ − sin θ M(θ) = ; θ 2 [0; 2π) ; sin θ cos θ form a subgroup of the group of 2 × 2 non-degenerated matrices. Find the matrix M −1(θ). These group is denoted by SO(2). G.J.| Symmetry groups in physics Subgroup 7/21 Group homomorphism Definition. A map F : G 7! G~ from a group G to a group G~ is called homomorphism, if F (g · h) = F (g) · F (h) : An invertible homomorphism is called isomorphism and the isomorphism of a group to itself is called automorphism. Examples: Let us consider the group of real numbers x 2 R (see Example 1, p.5) and introduce two maps a) x 7! ex, b) x 7! eix: Exercise 3. Check that: a) provides isomorphisms of R to R+ , b) is a homomorphism of R into U(1), which is not an isomorphism. −1 Exercise 4. Check that the map Fh(g) = h · g · h , defined by a group element h 2 G, is an automorphism of G. It is called inner automorphism. Exercise 5. Check that the map θ 7! −θ is an automorphism of SO(2) (see Exercise 2), but it is not inner an automorphism. G.J.| Symmetry groups in physics Group homomorphism 8/21 Group representation Let us consider a linear space L and its one-to-one linear maps f : L 7! L. The set of all one-to-one linear maps form a group, denoted by GL. The group product in GL is defined as a composition of two maps: if f1 2 GL and f2 2 GL, then f1 · f2 (x) = f1(f2(x)), for any x 2 L. Definition. A homomorphism of a group G in some GL is a representation of G. Thus, a representation of G is a set of non-degenerated linear operators fg : L 7! L, labeled by the group elements g 2 G, which satisfy the operator relation fg · fh = fg·h . Definition. Representation of G is called irreducible if L has no subspace, apart from trivial, which is invariant under the action of all operators fg. Definition. Let fg and f~g be two representations of a group G in linear spaces L and L~, respectively. These representations are called equivalent, −1 if there exists an invertible map R : L 7! L~ such that f~g = R fg R , for all g 2 G. G.J.| Symmetry groups in physics Group representation 9/21 Definition. Representation of G is called unitary if it preserves the scalar product in L, i.e. if h fg(x) j fg(y) i = h x j y i; for all g 2 G. Examples: 2 1. R as a linear space of 2-vectors ~x = (x1; x2) with scalar product h ~x j ~y i = x1y1 + x2y2. The SO(2) matrices M(θ), given in Exercise 2, 2 define linear maps of R , which can be treated as a representation. Exercise 6. Check that it is an unitary and irreducible representation (UIR). 1 2. C is a 1-dimensional linear complex space of complex numbers z 2 C with scalar product h z j w i = z∗ w. Let us consider a linear representation 1 iθ iθ 1 of U(1) in C given by e z, where e 2 U(1) and z 2 C . Exercise 7. Check that this defines UIR of U(1). 3. The groups U(1) and SO(2) are obviously isomorphic. Hence, a representation of U(1) is a representation of SO(2) and vice versa. Exercise 6 then provides an example of UIR for U(1). 2 1 Exercise 8. Find an invertible map from R to C that makes the UIR's of U(1) given in Exercises 6 and 7 equivalent to each other. G.J.| Symmetry groups in physics Group representation 10/21 Matrix exponent Let us consider a n × n matrix A. Its exponent is defined by the series A2 An exp[A] = I + A + + ··· + + ··· 2! n! Exercise 9. Prove that this series converges for any matrix A. Exercise 10. Calculate the matrix exponents and check the identities 0 a 1 a 0 b cosh b sinh b exp = ; exp = ; 0 0 0 1 b 0 sinh b cosh b 0 −θ cos θ − sin θ exp = : θ 0 sin θ cos θ Exercise 11. Prove the Jacobi's formula for the determinant of matrix exponent det (exp[A]) = eTr[A] : G.J.| Symmetry groups in physics Matrix exponent 11/21 GL(n) group GL(n) group is given by n × n non-degenerated matrices. This group was already introduced on page 5 as an example of groups. If the matrix elements of M 2 GL(n) are only real numbers the group is denoted by GL(n; R), and it is denoted by GL(n; C), if the matrix elements are complex. The condition detM 6= 0 restricts the domain of possible values of the matrix elements of M but not their number. Therefore, GL(n; R) and 2 2 GL(n; C) are n and 2n parametric groups, respectively. For example, GL(1; R) = R+ [ R− and GL(1; C) is a complex plane without the origin. Note that the notation GL(n) is related to the fact that this group is associated with one-to-one general linear transformations of n-dimensional vector space.
Load more

Recommended publications
  • Molecular Symmetry
    Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry .
    [Show full text]
  • Arxiv:1910.10745V1 [Cond-Mat.Str-El] 23 Oct 2019 2.2 Symmetry-Protected Time Crystals
    A Brief History of Time Crystals Vedika Khemania,b,∗, Roderich Moessnerc, S. L. Sondhid aDepartment of Physics, Harvard University, Cambridge, Massachusetts 02138, USA bDepartment of Physics, Stanford University, Stanford, California 94305, USA cMax-Planck-Institut f¨urPhysik komplexer Systeme, 01187 Dresden, Germany dDepartment of Physics, Princeton University, Princeton, New Jersey 08544, USA Abstract The idea of breaking time-translation symmetry has fascinated humanity at least since ancient proposals of the per- petuum mobile. Unlike the breaking of other symmetries, such as spatial translation in a crystal or spin rotation in a magnet, time translation symmetry breaking (TTSB) has been tantalisingly elusive. We review this history up to recent developments which have shown that discrete TTSB does takes place in periodically driven (Floquet) systems in the presence of many-body localization (MBL). Such Floquet time-crystals represent a new paradigm in quantum statistical mechanics — that of an intrinsically out-of-equilibrium many-body phase of matter with no equilibrium counterpart. We include a compendium of the necessary background on the statistical mechanics of phase structure in many- body systems, before specializing to a detailed discussion of the nature, and diagnostics, of TTSB. In particular, we provide precise definitions that formalize the notion of a time-crystal as a stable, macroscopic, conservative clock — explaining both the need for a many-body system in the infinite volume limit, and for a lack of net energy absorption or dissipation. Our discussion emphasizes that TTSB in a time-crystal is accompanied by the breaking of a spatial symmetry — so that time-crystals exhibit a novel form of spatiotemporal order.
    [Show full text]
  • Chapter 1 – Symmetry of Molecules – P. 1
    Chapter 1 – Symmetry of Molecules – p. 1 - 1. Symmetry of Molecules 1.1 Symmetry Elements · Symmetry operation: Operation that transforms a molecule to an equivalent position and orientation, i.e. after the operation every point of the molecule is coincident with an equivalent point. · Symmetry element: Geometrical entity (line, plane or point) which respect to which one or more symmetry operations can be carried out. In molecules there are only four types of symmetry elements or operations: · Mirror planes: reflection with respect to plane; notation: s · Center of inversion: inversion of all atom positions with respect to inversion center, notation i · Proper axis: Rotation by 2p/n with respect to the axis, notation Cn · Improper axis: Rotation by 2p/n with respect to the axis, followed by reflection with respect to plane, perpendicular to axis, notation Sn Formally, this classification can be further simplified by expressing the inversion i as an improper rotation S2 and the reflection s as an improper rotation S1. Thus, the only symmetry elements in molecules are Cn and Sn. Important: Successive execution of two symmetry operation corresponds to another symmetry operation of the molecule. In order to make this statement a general rule, we require one more symmetry operation, the identity E. (1.1: Symmetry elements in CH4, successive execution of symmetry operations) 1.2. Systematic classification by symmetry groups According to their inherent symmetry elements, molecules can be classified systematically in so called symmetry groups. We use the so-called Schönfliess notation to name the groups, Chapter 1 – Symmetry of Molecules – p. 2 - which is the usual notation for molecules.
    [Show full text]
  • TASI 2008 Lectures: Introduction to Supersymmetry And
    TASI 2008 Lectures: Introduction to Supersymmetry and Supersymmetry Breaking Yuri Shirman Department of Physics and Astronomy University of California, Irvine, CA 92697. [email protected] Abstract These lectures, presented at TASI 08 school, provide an introduction to supersymmetry and supersymmetry breaking. We present basic formalism of supersymmetry, super- symmetric non-renormalization theorems, and summarize non-perturbative dynamics of supersymmetric QCD. We then turn to discussion of tree level, non-perturbative, and metastable supersymmetry breaking. We introduce Minimal Supersymmetric Standard Model and discuss soft parameters in the Lagrangian. Finally we discuss several mech- anisms for communicating the supersymmetry breaking between the hidden and visible sectors. arXiv:0907.0039v1 [hep-ph] 1 Jul 2009 Contents 1 Introduction 2 1.1 Motivation..................................... 2 1.2 Weylfermions................................... 4 1.3 Afirstlookatsupersymmetry . .. 5 2 Constructing supersymmetric Lagrangians 6 2.1 Wess-ZuminoModel ............................... 6 2.2 Superfieldformalism .............................. 8 2.3 VectorSuperfield ................................. 12 2.4 Supersymmetric U(1)gaugetheory ....................... 13 2.5 Non-abeliangaugetheory . .. 15 3 Non-renormalization theorems 16 3.1 R-symmetry.................................... 17 3.2 Superpotentialterms . .. .. .. 17 3.3 Gaugecouplingrenormalization . ..... 19 3.4 D-termrenormalization. ... 20 4 Non-perturbative dynamics in SUSY QCD 20 4.1 Affleck-Dine-Seiberg
    [Show full text]
  • Observability and Symmetries of Linear Control Systems
    S S symmetry Article Observability and Symmetries of Linear Control Systems Víctor Ayala 1,∗, Heriberto Román-Flores 1, María Torreblanca Todco 2 and Erika Zapana 2 1 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica 1000000, Chile; [email protected] 2 Departamento Académico de Matemáticas, Universidad, Nacional de San Agustín de Arequipa, Calle Santa Catalina, Nro. 117, Arequipa 04000, Peru; [email protected] (M.T.T.); [email protected] (E.Z.) * Correspondence: [email protected] Received: 7 April 2020; Accepted: 6 May 2020; Published: 4 June 2020 Abstract: The goal of this article is to compare the observability properties of the class of linear control systems in two different manifolds: on the Euclidean space Rn and, in a more general setup, on a connected Lie group G. For that, we establish well-known results. The symmetries involved in this theory allow characterizing the observability property on Euclidean spaces and the local observability property on Lie groups. Keywords: observability; symmetries; Euclidean spaces; Lie groups 1. Introduction For general facts about control theory, we suggest the references, [1–5], and for general mathematical issues, the references [6–8]. In the context of this article, a general control system S can be modeled by the data S = (M, D, h, N). Here, M is the manifold of the space states, D is a family of differential equations on M, or if you wish, a family of vector fields on the manifold, h : M ! N is a differentiable observation map, and N is a manifold that contains all the known information that you can see of the system through h.
    [Show full text]
  • Symmetry of Graphs. Circles
    Symmetry of graphs. Circles Symmetry of graphs. Circles 1 / 10 Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Symmetry of graphs. Circles 2 / 10 Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Symmetry of graphs. Circles 2 / 10 Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Symmetry of graphs. Circles 2 / 10 Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object.
    [Show full text]
  • The Symmetry Group of a Finite Frame
    Technical Report 3 January 2010 The symmetry group of a finite frame Richard Vale and Shayne Waldron Department of Mathematics, Cornell University, 583 Malott Hall, Ithaca, NY 14853-4201, USA e–mail: [email protected] Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand e–mail: [email protected] ABSTRACT We define the symmetry group of a finite frame as a group of permutations on its index set. This group is closely related to the symmetry group of [VW05] for tight frames: they are isomorphic when the frame is tight and has distinct vectors. The symmetry group is the same for all similar frames, in particular for a frame, its dual and canonical tight frames. It can easily be calculated from the Gramian matrix of the canonical tight frame. Further, a frame and its complementary frame have the same symmetry group. We exploit this last property to construct and classify some classes of highly symmetric tight frames. Key Words: finite frame, geometrically uniform frame, Gramian matrix, harmonic frame, maximally symmetric frames partition frames, symmetry group, tight frame, AMS (MOS) Subject Classifications: primary 42C15, 58D19, secondary 42C40, 52B15, 0 1. Introduction Over the past decade there has been a rapid development of the theory and application of finite frames to areas as diverse as signal processing, quantum information theory and multivariate orthogonal polynomials, see, e.g., [KC08], [RBSC04] and [W091]. Key to these applications is the construction of frames with desirable properties. These often include being tight, and having a high degree of symmetry. Important examples are the harmonic or geometrically uniform frames, i.e., tight frames which are the orbit of a single vector under an abelian group of unitary transformations (see [BE03] and [HW06]).
    [Show full text]
  • Introduction to Supersymmetry
    Introduction to Supersymmetry Pre-SUSY Summer School Corpus Christi, Texas May 15-18, 2019 Stephen P. Martin Northern Illinois University [email protected] 1 Topics: Why: Motivation for supersymmetry (SUSY) • What: SUSY Lagrangians, SUSY breaking and the Minimal • Supersymmetric Standard Model, superpartner decays Who: Sorry, not covered. • For some more details and a slightly better attempt at proper referencing: A supersymmetry primer, hep-ph/9709356, version 7, January 2016 • TASI 2011 lectures notes: two-component fermion notation and • supersymmetry, arXiv:1205.4076. If you find corrections, please do let me know! 2 Lecture 1: Motivation and Introduction to Supersymmetry Motivation: The Hierarchy Problem • Supermultiplets • Particle content of the Minimal Supersymmetric Standard Model • (MSSM) Need for “soft” breaking of supersymmetry • The Wess-Zumino Model • 3 People have cited many reasons why extensions of the Standard Model might involve supersymmetry (SUSY). Some of them are: A possible cold dark matter particle • A light Higgs boson, M = 125 GeV • h Unification of gauge couplings • Mathematical elegance, beauty • ⋆ “What does that even mean? No such thing!” – Some modern pundits ⋆ “We beg to differ.” – Einstein, Dirac, . However, for me, the single compelling reason is: The Hierarchy Problem • 4 An analogy: Coulomb self-energy correction to the electron’s mass A point-like electron would have an infinite classical electrostatic energy. Instead, suppose the electron is a solid sphere of uniform charge density and radius R. An undergraduate problem gives: 3e2 ∆ECoulomb = 20πǫ0R 2 Interpreting this as a correction ∆me = ∆ECoulomb/c to the electron mass: 15 0.86 10− meters m = m + (1 MeV/c2) × .
    [Show full text]
  • Group Symmetry and Covariance Regularization
    Group Symmetry and Covariance Regularization Parikshit Shah and Venkat Chandrasekaran email: [email protected]; [email protected] Abstract Statistical models that possess symmetry arise in diverse settings such as ran- dom fields associated to geophysical phenomena, exchangeable processes in Bayesian statistics, and cyclostationary processes in engineering. We formalize the notion of a symmetric model via group invariance. We propose projection onto a group fixed point subspace as a fundamental way of regularizing covariance matrices in the high- dimensional regime. In terms of parameters associated to the group we derive precise rates of convergence of the regularized covariance matrix and demonstrate that signif- icant statistical gains may be expected in terms of the sample complexity. We further explore the consequences of symmetry on related model-selection problems such as the learning of sparse covariance and inverse covariance matrices. We also verify our results with simulations. 1 Introduction An important feature of many modern data analysis problems is the small number of sam- ples available relative to the dimension of the data. Such high-dimensional settings arise in a range of applications in bioinformatics, climate studies, and economics. A fundamental problem that arises in the high-dimensional regime is the poor conditioning of sample statis- tics such as sample covariance matrices [9] ,[8]. Accordingly, a fruitful and active research agenda over the last few years has been the development of methods for high-dimensional statistical inference and modeling that take into account structure in the underlying model. Some examples of structural assumptions on statistical models include models with a few latent factors (leading to low-rank covariance matrices) [18], models specified by banded or sparse covariance matrices [9], [8], and Markov or graphical models [25, 31, 27].
    [Show full text]
  • Symmetry in Physical Law
    Week 4 - Character of Physical Law Symmetry in Physical Law Candidate Higgs decays into 4 electrons Is any momentum missing? The Character of Physical Law John Anderson – Week 4 The Academy of Lifelong Learning 1 Symmetries Rule Nature Old School1882 -1935 New Breed What key bit of knowledge would you save? • The world is made of atoms. - Richard Feynman • The laws of physics are based on symmetries. - Brian Greene Emmy Noether showed every invariance corresponds to a conservation law. (Energy, momentum, angular momentum.) 2 1 Week 4 - Character of Physical Law Symmetry Violated - Parity Chen-Ning Franklin T.D. Lee Chien-Shiung Wu Yang The Weak Nuclear Force Violates Parity • Yang and Lee scour literature. Strong force and E&M are good. • Wu crash projects measurement with supercooled Cobalt 60. Nobel Prize 1957 for Yang and Lee, but Wu gets no cigar. 3 Parity - “Mirror Symmetry” Equal numbers of electrons should be emitted parallel and antiparallel to the magnetic field if parity is conserved, but they found that more electrons were emitted in the direction opposite to the magnetic field and therefore opposite to the nuclear spin. Chien-Shiung Wu 1912-1997 Hyperphysics Georgia State Univ. 4 2 Week 4 - Character of Physical Law Parity Change - Right to Left Handed x z P y Parity Operator y x z Right-handed Coordinates Left-handed Coordinates Parity operator: (x,y,z) to (-x,-y,-z) Changes Right-hand system to Left-hand system 5 Symmetry in the Particle Zoo How to bring order out of chaos? Quarks. Pattern recognition. More latter.
    [Show full text]
  • Special Unitary Group - Wikipedia
    Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Special unitary group In mathematics, the special unitary group of degree n, denoted SU( n), is the Lie group of n×n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U( n), consisting of all n×n unitary matrices. As a compact classical group, U( n) is the group that preserves the standard inner product on Cn.[nb 1] It is itself a subgroup of the general linear group, SU( n) ⊂ U( n) ⊂ GL( n, C). The SU( n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1] The simplest case, SU(1) , is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+ I, − I}. [nb 2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. Contents Properties Lie algebra Fundamental representation Adjoint representation The group SU(2) Diffeomorphism with S 3 Isomorphism with unit quaternions Lie Algebra The group SU(3) Topology Representation theory Lie algebra Lie algebra structure Generalized special unitary group Example Important subgroups See also 1 of 10 2/22/2018, 8:54 PM Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Remarks Notes References Properties The special unitary group SU( n) is a real Lie group (though not a complex Lie group).
    [Show full text]
  • Physics of the Lorentz Group
    Physics of the Lorentz Group Physics of the Lorentz Group Sibel Ba¸skal Department of Physics, Middle East Technical University, 06800 Ankara, Turkey e-mail: [email protected] Young S. Kim Center for Fundamental Physics, University of Maryland, College Park, Maryland 20742, U.S.A. e-mail: [email protected] Marilyn E. Noz Department of Radiology, New York University, New York, NY 10016 U.S.A. e-mail: [email protected] Preface When Newton formulated his law of gravity, he wrote down his formula applicable to two point particles. It took him 20 years to prove that his formula works also for extended objects such as the sun and earth. When Einstein formulated his special relativity in 1905, he worked out the transfor- mation law for point particles. The question is what happens when those particles have space-time extensions. The hydrogen atom is a case in point. The hydrogen atom is small enough to be regarded as a particle obeying Einstein'sp law of Lorentz transformations in- cluding the energy-momentum relation E = p2 + m2. Yet, it is known to have a rich internal space-time structure, rich enough to provide the foundation of quantum mechanics. Indeed, Niels Bohr was interested in why the energy levels of the hydrogen atom are discrete. His interest led to the replacement of the orbit by a standing wave. Before and after 1927, Einstein and Bohr met occasionally to discuss physics. It is possible that they discussed how the hydrogen atom with an electron orbit or as a standing-wave looks to moving observers.
    [Show full text]