The Exponential Map, Lie Groups, and Lie Algebras
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Field Theory 1. Introduction to Quantum Field Theory (A Primer for a Basic Education)
Field Theory 1. Introduction to Quantum Field Theory (A Primer for a Basic Education) Roberto Soldati February 23, 2019 Contents 0.1 Prologue ............................ 4 1 Basics in Group Theory 8 1.1 Groups and Group Representations . 8 1.1.1 Definitions ....................... 8 1.1.2 Theorems ........................ 12 1.1.3 Direct Products of Group Representations . 14 1.2 Continuous Groups and Lie Groups . 16 1.2.1 The Continuous Groups . 16 1.2.2 The Lie Groups .................... 18 1.2.3 An Example : the Rotations Group . 19 1.2.4 The Infinitesimal Operators . 21 1.2.5 Lie Algebra ....................... 23 1.2.6 The Exponential Representation . 24 1.2.7 The Special Unitary Groups . 27 1.3 The Non-Homogeneous Lorentz Group . 34 1.3.1 The Lorentz Group . 34 1.3.2 Semi-Simple Groups . 43 1.3.3 The Poincar´eGroup . 49 2 The Action Functional 57 2.1 The Classical Relativistic Wave Fields . 57 2.1.1 Field Variations .................... 58 2.1.2 The Scalar and Vector Fields . 62 2.1.3 The Spinor Fields ................... 65 2.2 The Action Principle ................... 76 2.3 The N¨otherTheorem ................... 79 2.3.1 Problems ........................ 87 1 3 The Scalar Field 94 3.1 General Features ...................... 94 3.2 Normal Modes Expansion . 101 3.3 Klein-Gordon Quantum Field . 106 3.4 The Fock Space .......................112 3.4.1 The Many-Particle States . 112 3.4.2 The Lorentz Covariance Properties . 116 3.5 Special Distributions . 129 3.5.1 Euclidean Formulation . 133 3.6 Problems ............................137 4 The Spinor Field 151 4.1 The Dirac Equation . -
MATH 2370, Practice Problems
MATH 2370, Practice Problems Kiumars Kaveh Problem: Prove that an n × n complex matrix A is diagonalizable if and only if there is a basis consisting of eigenvectors of A. Problem: Let A : V ! W be a one-to-one linear map between two finite dimensional vector spaces V and W . Show that the dual map A0 : W 0 ! V 0 is surjective. Problem: Determine if the curve 2 2 2 f(x; y) 2 R j x + y + xy = 10g is an ellipse or hyperbola or union of two lines. Problem: Show that if a nilpotent matrix is diagonalizable then it is the zero matrix. Problem: Let P be a permutation matrix. Show that P is diagonalizable. Show that if λ is an eigenvalue of P then for some integer m > 0 we have λm = 1 (i.e. λ is an m-th root of unity). Hint: Note that P m = I for some integer m > 0. Problem: Show that if λ is an eigenvector of an orthogonal matrix A then jλj = 1. n Problem: Take a vector v 2 R and let H be the hyperplane orthogonal n n to v. Let R : R ! R be the reflection with respect to a hyperplane H. Prove that R is a diagonalizable linear map. Problem: Prove that if λ1; λ2 are distinct eigenvalues of a complex matrix A then the intersection of the generalized eigenspaces Eλ1 and Eλ2 is zero (this is part of the Spectral Theorem). 1 Problem: Let H = (hij) be a 2 × 2 Hermitian matrix. Use the Min- imax Principle to show that if λ1 ≤ λ2 are the eigenvalues of H then λ1 ≤ h11 ≤ λ2. -
Unitary Group - Wikipedia
Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group Unitary group In mathematics, the unitary group of degree n, denoted U( n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL( n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group. The unitary group U( n) is a real Lie group of dimension n2. The Lie algebra of U( n) consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator. The general unitary group (also called the group of unitary similitudes ) consists of all matrices A such that A∗A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix. Contents Properties Topology Related groups 2-out-of-3 property Special unitary and projective unitary groups G-structure: almost Hermitian Generalizations Indefinite forms Finite fields Degree-2 separable algebras Algebraic groups Unitary group of a quadratic module Polynomial invariants Classifying space See also Notes References Properties Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group 1 of 7 2/23/2018, 10:13 AM Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group homomorphism The kernel of this homomorphism is the set of unitary matrices with determinant 1. -
Material on Algebraic and Lie Groups
2 Lie groups and algebraic groups. 2.1 Basic Definitions. In this subsection we will introduce the class of groups to be studied. We first recall that a Lie group is a group that is also a differentiable manifold 1 and multiplication (x, y xy) and inverse (x x ) are C1 maps. An algebraic group is a group7! that is also an algebraic7! variety such that multi- plication and inverse are morphisms. Before we can introduce our main characters we first consider GL(n, C) as an affi ne algebraic group. Here Mn(C) denotes the space of n n matrices and GL(n, C) = g Mn(C) det(g) =) . Now Mn(C) is given the structure nf2 2 j 6 g of affi ne space C with the coordinates xij for X = [xij] . This implies that GL(n, C) is Z-open and as a variety is isomorphic with the affi ne variety 1 Mn(C) det . This implies that (GL(n, C)) = C[xij, det ]. f g O Lemma 1 If G is an algebraic group over an algebraically closed field, F , then every point in G is smooth. Proof. Let Lg : G G be given by Lgx = gx. Then Lg is an isomorphism ! 1 1 of G as an algebraic variety (Lg = Lg ). Since isomorphisms preserve the set of smooth points we see that if x G is smooth so is every element of Gx = G. 2 Proposition 2 If G is an algebraic group over an algebraically closed field F then the Z-connected components Proof. -
Matrix Lie Groups
Maths Seminar 2007 MATRIX LIE GROUPS Claudiu C Remsing Dept of Mathematics (Pure and Applied) Rhodes University Grahamstown 6140 26 September 2007 RhodesUniv CCR 0 Maths Seminar 2007 TALK OUTLINE 1. What is a matrix Lie group ? 2. Matrices revisited. 3. Examples of matrix Lie groups. 4. Matrix Lie algebras. 5. A glimpse at elementary Lie theory. 6. Life beyond elementary Lie theory. RhodesUniv CCR 1 Maths Seminar 2007 1. What is a matrix Lie group ? Matrix Lie groups are groups of invertible • matrices that have desirable geometric features. So matrix Lie groups are simultaneously algebraic and geometric objects. Matrix Lie groups naturally arise in • – geometry (classical, algebraic, differential) – complex analyis – differential equations – Fourier analysis – algebra (group theory, ring theory) – number theory – combinatorics. RhodesUniv CCR 2 Maths Seminar 2007 Matrix Lie groups are encountered in many • applications in – physics (geometric mechanics, quantum con- trol) – engineering (motion control, robotics) – computational chemistry (molecular mo- tion) – computer science (computer animation, computer vision, quantum computation). “It turns out that matrix [Lie] groups • pop up in virtually any investigation of objects with symmetries, such as molecules in chemistry, particles in physics, and projective spaces in geometry”. (K. Tapp, 2005) RhodesUniv CCR 3 Maths Seminar 2007 EXAMPLE 1 : The Euclidean group E (2). • E (2) = F : R2 R2 F is an isometry . → | n o The vector space R2 is equipped with the standard Euclidean structure (the “dot product”) x y = x y + x y (x, y R2), • 1 1 2 2 ∈ hence with the Euclidean distance d (x, y) = (y x) (y x) (x, y R2). -
LECTURE 12: LIE GROUPS and THEIR LIE ALGEBRAS 1. Lie
LECTURE 12: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups Definition 1.1. A Lie group G is a smooth manifold equipped with a group structure so that the group multiplication µ : G × G ! G; (g1; g2) 7! g1 · g2 is a smooth map. Example. Here are some basic examples: • Rn, considered as a group under addition. • R∗ = R − f0g, considered as a group under multiplication. • S1, Considered as a group under multiplication. • Linear Lie groups GL(n; R), SL(n; R), O(n) etc. • If M and N are Lie groups, so is their product M × N. Remarks. (1) (Hilbert's 5th problem, [Gleason and Montgomery-Zippin, 1950's]) Any topological group whose underlying space is a topological manifold is a Lie group. (2) Not every smooth manifold admits a Lie group structure. For example, the only spheres that admit a Lie group structure are S0, S1 and S3; among all the compact 2 dimensional surfaces the only one that admits a Lie group structure is T 2 = S1 × S1. (3) Here are two simple topological constraints for a manifold to be a Lie group: • If G is a Lie group, then TG is a trivial bundle. n { Proof: We identify TeG = R . The vector bundle isomorphism is given by φ : G × TeG ! T G; φ(x; ξ) = (x; dLx(ξ)) • If G is a Lie group, then π1(G) is an abelian group. { Proof: Suppose α1, α2 2 π1(G). Define α : [0; 1] × [0; 1] ! G by α(t1; t2) = α1(t1) · α2(t2). Then along the bottom edge followed by the right edge we have the composition α1 ◦ α2, where ◦ is the product of loops in the fundamental group, while along the left edge followed by the top edge we get α2 ◦ α1. -
The Classical Groups and Domains 1. the Disk, Upper Half-Plane, SL 2(R
(June 8, 2018) The Classical Groups and Domains Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ The complex unit disk D = fz 2 C : jzj < 1g has four families of generalizations to bounded open subsets in Cn with groups acting transitively upon them. Such domains, defined more precisely below, are bounded symmetric domains. First, we recall some standard facts about the unit disk, the upper half-plane, the ambient complex projective line, and corresponding groups acting by linear fractional (M¨obius)transformations. Happily, many of the higher- dimensional bounded symmetric domains behave in a manner that is a simple extension of this simplest case. 1. The disk, upper half-plane, SL2(R), and U(1; 1) 2. Classical groups over C and over R 3. The four families of self-adjoint cones 4. The four families of classical domains 5. Harish-Chandra's and Borel's realization of domains 1. The disk, upper half-plane, SL2(R), and U(1; 1) The group a b GL ( ) = f : a; b; c; d 2 ; ad − bc 6= 0g 2 C c d C acts on the extended complex plane C [ 1 by linear fractional transformations a b az + b (z) = c d cz + d with the traditional natural convention about arithmetic with 1. But we can be more precise, in a form helpful for higher-dimensional cases: introduce homogeneous coordinates for the complex projective line P1, by defining P1 to be a set of cosets u 1 = f : not both u; v are 0g= × = 2 − f0g = × P v C C C where C× acts by scalar multiplication. -
Lie Group and Geometry on the Lie Group SL2(R)
INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear -
Lecture 2: Spectral Theorems
Lecture 2: Spectral Theorems This lecture introduces normal matrices. The spectral theorem will inform us that normal matrices are exactly the unitarily diagonalizable matrices. As a consequence, we will deduce the classical spectral theorem for Hermitian matrices. The case of commuting families of matrices will also be studied. All of this corresponds to section 2.5 of the textbook. 1 Normal matrices Definition 1. A matrix A 2 Mn is called a normal matrix if AA∗ = A∗A: Observation: The set of normal matrices includes all the Hermitian matrices (A∗ = A), the skew-Hermitian matrices (A∗ = −A), and the unitary matrices (AA∗ = A∗A = I). It also " # " # 1 −1 1 1 contains other matrices, e.g. , but not all matrices, e.g. 1 1 0 1 Here is an alternate characterization of normal matrices. Theorem 2. A matrix A 2 Mn is normal iff ∗ n kAxk2 = kA xk2 for all x 2 C : n Proof. If A is normal, then for any x 2 C , 2 ∗ ∗ ∗ ∗ ∗ 2 kAxk2 = hAx; Axi = hx; A Axi = hx; AA xi = hA x; A xi = kA xk2: ∗ n n Conversely, suppose that kAxk = kA xk for all x 2 C . For any x; y 2 C and for λ 2 C with jλj = 1 chosen so that <(λhx; (A∗A − AA∗)yi) = jhx; (A∗A − AA∗)yij, we expand both sides of 2 ∗ 2 kA(λx + y)k2 = kA (λx + y)k2 to obtain 2 2 ∗ 2 ∗ 2 ∗ ∗ kAxk2 + kAyk2 + 2<(λhAx; Ayi) = kA xk2 + kA yk2 + 2<(λhA x; A yi): 2 ∗ 2 2 ∗ 2 Using the facts that kAxk2 = kA xk2 and kAyk2 = kA yk2, we derive 0 = <(λhAx; Ayi − λhA∗x; A∗yi) = <(λhx; A∗Ayi − λhx; AA∗yi) = <(λhx; (A∗A − AA∗)yi) = jhx; (A∗A − AA∗)yij: n ∗ ∗ n Since this is true for any x 2 C , we deduce (A A − AA )y = 0, which holds for any y 2 C , meaning that A∗A − AA∗ = 0, as desired. -
10 Group Theory
10 Group theory 10.1 What is a group? A group G is a set of elements f, g, h, ... and an operation called multipli- cation such that for all elements f,g, and h in the group G: 1. The product fg is in the group G (closure); 2. f(gh)=(fg)h (associativity); 3. there is an identity element e in the group G such that ge = eg = g; 1 1 1 4. every g in G has an inverse g− in G such that gg− = g− g = e. Physical transformations naturally form groups. The elements of a group might be all physical transformations on a given set of objects that leave invariant a chosen property of the set of objects. For instance, the objects might be the points (x, y) in a plane. The chosen property could be their distances x2 + y2 from the origin. The physical transformations that leave unchanged these distances are the rotations about the origin p x cos ✓ sin ✓ x 0 = . (10.1) y sin ✓ cos ✓ y ✓ 0◆ ✓− ◆✓ ◆ These rotations form the special orthogonal group in 2 dimensions, SO(2). More generally, suppose the transformations T,T0,T00,... change a set of objects in ways that leave invariant a chosen property property of the objects. Suppose the product T 0 T of the transformations T and T 0 represents the action of T followed by the action of T 0 on the objects. Since both T and T 0 leave the chosen property unchanged, so will their product T 0 T . Thus the closure condition is satisfied. -
Bott Periodicity for the Unitary Group
Bott Periodicity for the Unitary Group Carlos Salinas March 7, 2018 Abstract We will present a condensed proof of the Bott Periodicity Theorem for the unitary group U following John Milnor’s classic Morse Theory. There are many documents on the internet which already purport to do this (and do so very well in my estimation), but I nevertheless will attempt to give a summary of the result. Contents 1 The Basics 2 2 Fiber Bundles 3 2.1 First fiber bundle . .4 2.2 Second Fiber Bundle . .5 2.3 Third Fiber Bundle . .5 2.4 Fourth Fiber Bundle . .5 3 Proof of the Periodicity Theorem 6 3.1 The first equivalence . .7 3.2 The second equality . .8 4 The Homotopy Groups of U 8 1 The Basics The original proof of the Periodicity Theorem relies on a deep result of Marston Morse’s calculus of variations, the (Morse) Index Theorem. The proof of this theorem, however, goes beyond the scope of this document, the reader is welcome to read the relevant section from Milnor or indeed Morse’s own paper titled The Index Theorem in the Calculus of Variations. Perhaps the first thing we should set about doing is introducing the main character of our story; this will be the unitary group. The unitary group of degree n (here denoted U(n)) is the set of all unitary matrices; that is, the set of all A ∈ GL(n, C) such that AA∗ = I where A∗ is the conjugate of the transpose of A (conjugate transpose for short). -
Extending Symmetries of the Standard Model Towards Grand Unification
Master research Extending Symmetries of the Standard Model towards Grand Unification Anno Touwen Supervisor prof. dr. Dani¨el Boer August 3, 2018 Abstract In this master thesis the spacetime, global and gauge symmetries of the Standard Model are reviewed. These symmetries are used as a basis for finding possible extensions of this successful model, such as the two Higgs doublet model and Left-Right model. Methods of finding subgroups and the slitting up of representations are discussed based on Dynkin diagrams. These methods are applied to analyse the subgroups of the exceptional group E6 as candidates for Grand Unified 3 Theory groups. In this study SU(5), SO(10) and SU(3) and SU(4)×SU(2)×SU(2) are most important. A phenomenological comparison between these models is given focussed on the different types of leptoquarks that could be responsible for the not yet observed proton decay. Contents 1 Introduction 3 2 Symmetry groups and Gauge theories 5 2.1 Group theory . .5 2.2 Unitary and Special Unitary groups . .8 2.3 Orthogonal and Special Orthogonal groups . 10 2.4 Gauge theories and fields . 11 2.5 Spacetime symmetries . 13 2.6 C, P and T transformations . 14 3 The Standard Model 16 3.1 The Electroweak interaction . 16 3.2 The Strong interaction . 19 3.3 Fermion content . 19 3.4 Eigenstates and the CKM-matrix . 22 3.5 Global symmetries . 23 4 Beyond the Standard Model 28 4.1 Two Higgs doublet model . 28 4.2 Left-Right models . 34 5 A further study of Lie Groups and Algebras 37 5.1 Root systems .