Group theory - QMII 2017
|
|
- Ross Simpson
- 5 years ago
- Views:
Transcription
1 Group theory - QMII 7 Daniel Aloni References. Lecture notes - Gilad Perez. Lie algebra in particle physics - H. Georgi. Google... Motivation As a warm up let us motivate the need for Group theory in physics. It comes out that Group theory and symmetries share very similar properties. This motivates us to learn more sophisticated tools of Group theory and how to implement them in modern physics. We will take rotations as an example but the following holds for any symmetry:. Arotationofarotationisalsoarotation-R = R R.. Multiplication of rotations is associative - R (R R )=(R R ) R.. The identity is a unique rotation which leaves the system unchanged and commutes with all other rotations. 4. For each rotation we can rotate the system backwards. This inverse rotation is unique and satisfies - R R = R R =.
2 This list is exactly the list of axioms that defines a group. A Group is a pair (G, ) ofasetg and a product s.t.. Closure - 8g,g G, g g G.. Associativity - 8g,g,g G, (g g ) g = g (g g ).. There is an identity e G, s.t.8g G, e g = g e = g. 4. Every element g G has an inverse element g G, s.t. g g = g g = e. What we would like to learn? Here are few examples, considering the rotations again. A rotation of a three-vector v, isgivenbya matrixr. How it acts on other vector spaces, for instance the Hilbert space in quantum mechanics? What are the conserved quantities? How to make it infinitesimal? (and what is it good for...) How a rotation of one space is induced to another space? As we will see, Group theory and in particular Lie groups and Lie algebra will teach us how to answer these questions in a systematic way, and how to do that for any symmetry. Classification Adictionaryformostimportantgroups(inphysics...).. Discrete: Adiscretegroupmightcontainafiniteorinfinitenumberofelements. If the number of elements is finite, thenthenumber of elements is called the order of the group.
3 Cyclic group Z n -ThesetofintegernumbersG =(,,,..n ) with addition mod n. Itisequivalenttocyclicpermutationsofn objects. Symmetric group S n - All possible permutations of n objects.
4 Example - Z, S and the triangle Z is also the rotation group of the triangle. The elements of the rotation group are: e =donothing, a =(,, ), a =(,, ) where a (a )areunderstoodascyclic(anti-cyclic)interchangingofcorners in positions (,,). Graphically We can write the multiplication table for Z,andcomparetorotationsof triangle: mod n, e a a e e a a a a a e a a e a The symmetric group includes three additional elements - mirroring around the altitudes. The elements of S are: a =(, ), a 4 =(, ), a 5 =(, ) where a for instance is understood as interchanging the corners in positions (,). Graphically Exercise - write the multiplication table for S. 4
5 Additive group of integers - The elements are all the integer numbers n Z. Theproductofthegroupisadditionofnumbers.Thisis the most trivial example of infinite discrete group.. Lie groups: By Lie groups we will always mean Matrix Lie Groups. A Lie group have an infinite number of elements which are given by a smooth function of a finite number of parameters f(x,x..., x n ) G, x i A. For us A = R, C. The group multiplication law is just matrix multiplication. General Linear group GL(n, V )={A M n (V ) det(a) 6= }. The set of all invertible n n matrices on a field V,withmatrixmultiplication. Orthogonal group O(n) ={A GL(n, R) A T A =}. Equivalently, the column vectors of A are an orthonormal set. Also equivalently, A preserve the canonical inner product in R n. Note that det A = ±. Special Orthogonal Group. SO(n) = {A O(n) det A = }. SO(n) isthegroupofrotationsindimensionn. O(n) containsrotations and reflections. Question: does taking det A = give a group? Unitary Group. U(n) ={A GL(n, C) A A =}. Equivalently, the column vectors are an orthonormal set in C n. Also equivalently, A preserves the canonical inner product. Note that det A =. Special Unitary Group. SU(n) ={A U(n) det A =}. Generalized Orthogonal Group. O(n, k) ={A GL(n+k, R) A T ga = z } { g}, whereg =diag(,...,, n times k times z } {,..., ). Lets check that this is a group. If A, B O(n, k) then(a B) T A B = z } { B T A T A B = B T B = ) A B O(n, k). The Lorentz group is O(, ). 5
6 Subgroups For a given group G, ifasubsetofelementsh G, formagroupwiththe same product of G, wesaythath is a subgroup of G. Examples: We already saw that Z is a subgroup of S. Consider the two dimensional unitary group U(). This is a set of unitary matrices which acts on two dimensional complex vectors.. The group U() which changes the overall phase of those vectors is a subgroup of U().. The group SU() is also a subgroup of U() since for all matrices A B = A B. Note that the two subgroups commutes. We will make use of these fact to show that U() can be decomposed completely to this two subgroups, denoted by U() = SU() U(). SO() is a subgroup of SO(4). For instance we can choose a subset of matrices of the following form O = SO() Note that unlike the previous case, in this case the remaining is not a group, namely there is no group G, s.t. SO(4) = SO() G. C A 4 Representation For physicists, the theory of representation is the link between group theory and applications in physics. It tells us how an element g in an abstract group G, actsonaphysicalsystem.moreformally, 6
7 A Representation ( homomorphism) is a mapping, D of elements of G onto a set of linear operators D : G GL(V ) s.t. D(g )D(g )=D(g g ) We use rep as a short hand notation. 4. Discrete The action of Z on complex numbers D : Z C: D(e) =,D(a )=e i/,d(a )=e 4 i/. This representation is Isomorphism. andonto. Suchrepresentationsarecalled Regular representation of a discrete group of order n, is constructed as follows:. For each element g i G associate a vector g i i s.t. they form an orthonormal basis, namely hg i g j i = ij.. Define the regular representation on this vector space as D(g i ) g j i = g i g j i This is indeed a representation since D(g i )D(g j )=D(g i g j ).. The components of the matrices are given by [D(g)] ij = hg i D(g) g j i A homomorphism is a mapping from a group G to a group H which is compatible with the group product, namely : G H satisfies 8g,g G, (g G g )= (g ) H (g ). A representation is a homomorphism to the set of linear operators. 7
8 Let us find it explicitly for Z : B C B C B C ei A, a i A, a i A Clearly D(e) g i i = eg i i = g i i)d(e) =. By using the multiplication table we find D(a ) ei = a i,d(a ) ei = a i D(a ) a i = a i,d(a ) a i = ei D(a ) a i = ei,d(a ) a i = a i as you already saw in class... 9 >= >; D(a B C B C A,D(a )=@ A ArepresentationofS on a two dimensional vector space is given by: D(e) =,D(a )= D(a )=,D(a )= Exercises: p p p p Check that this is indeed a representation. Find the regular representation of S.,D(a )=,D(a )= p p p p 4. Lie groups The trivial representation - D(g) =, 8g G. Fundamental representation -ForaLiegroup,whichisdefinedas asetoflinearoperatorsthatactsonavectorspace,thefundamental representation is the representation of the group on its vector space: D(A) =A, 8A G. 8
9 Example - Consider a group element U SU(N) and an N-dimensional complex vector v V. Then the action of the fundamental representation on the vector space V is D fund. (U)v = Uv, U i jv j Anti-fundamental D anti. is the complex conjugation (not ) ofthe fundamental representation D, namelyd anti. (A) =D(A) = A, 8A G. Example - Consider again a group element U SU(N) and an N- dimensional complex vector w V. Then the action of the antifundamental representation on the vector space V is D anti. (U)w = U w, U i j w j = w j (U ) j i, w T U Then if v transforms under the fundamental v transforms under the anti fundamental denoted by v Uv ) v v U comment: Note that physicists always make an abuse of notation in this context. Although D fund and D anti. tells us how to act with an element g GL(V )onthevectorspacev,itisalsosaidthatthevector v lives in the fundamental and v lives in the anti fundamental. This convention will be convenient once we will start to learn field theories. Tensor representation -Byusingpreviousdefinitionswecanconstruct representations which act on tensors with any number of fundamental and anti-fundamental indices. Example - Consider a group element U U(N) andatensort ijk with three fundamental indices. Then the tensor representation is D tens. (U)T ijk = U i iu j ju k jt ijk 9
10 4.. Building invariants We will deal with physical systems which are defined by their action S[ ]. The whole purpose of learning group theory is to learn how to deal with symmetries of this action. In the language of representations if S[ ]livesin the trivial representation of some symmetry group G we are saying that G is asymmetryofthesystem. InotherwordsS[ ]isinvariant(orscalar)under the action of G. How do we build terms from general representations such that the whole object is invariant? There is a simple rule of thumb that all indices should be contracted by using Kronecker s delta. This is not a general statement but will work with most of the groups that we will deal with. Examples: Consider a vector v which transforms uncder the fundamental of SU() and a tensor T with three anti-fundamental indices. Then the most trivial invariant that we can think of is v i v i v i v i = v k(u ) k i U i mv m = v i and a more complicated one will be v i v j v k T ijk Ui i v i U j j vj Uk k v k Ui l U m j Uk m T lmn = l i i mv m = v i v i, j m n k v i v j v k T lmn = v i v j v k T ijk. For SU(N) there is an additional way to contract indices by using the N- dimensional anti symmetric tensor. fundamental of SU(). Then Consider three vectors v, w, z in the ijk v i w j z k i j i kui U j j U k k v i w j z k where =. We see that if ijk = i j i kui U j j U k k then this object is invariant. Let us choose ijk =. Then we observe that i j i ku U j U k = Det(U) =, and for each anti-cyclic permutations of ijk we get a minus SU(N) sign. Therefore indeed ijk = i j i ku and this construction is an SU(N) invariant. i U j j U k k Exercise - Show that if i = j the right hand side also vanishes.
Lecture 10: A (Brief) Introduction to Group Theory (See Chapter 3.13 in Boas, 3rd Edition)
Lecture 0: A (Brief) Introduction to Group heory (See Chapter 3.3 in Boas, 3rd Edition) Having gained some new experience with matrices, which provide us with representations of groups, and because symmetries
More informationBasic Concepts of Group Theory
Chapter 1 Basic Concepts of Group Theory The theory of groups and vector spaces has many important applications in a number of branches of modern theoretical physics. These include the formal theory of
More informationSymmetries, Groups, and Conservation Laws
Chapter Symmetries, Groups, and Conservation Laws The dynamical properties and interactions of a system of particles and fields are derived from the principle of least action, where the action is a 4-dimensional
More informationQuantum Field Theory III
Quantum Field Theory III Prof. Erick Weinberg January 19, 2011 1 Lecture 1 1.1 Structure We will start with a bit of group theory, and we will talk about spontaneous symmetry broken. Then we will talk
More informationTHE EULER CHARACTERISTIC OF A LIE GROUP
THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth
More informationGroup Theory - QMII 2017
Group Theory - QMII 017 Reminder Last time we said that a group element of a matrix lie group can be written as an exponent: U = e iαaxa, a = 1,..., N. We called X a the generators, we have N of them,
More informationIntroduction to Group Theory
Chapter 10 Introduction to Group Theory Since symmetries described by groups play such an important role in modern physics, we will take a little time to introduce the basic structure (as seen by a physicist)
More informationClifford Algebras and Spin Groups
Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of
More informationPhysics 557 Lecture 5
Physics 557 Lecture 5 Group heory: Since symmetries and the use of group theory is so much a part of recent progress in particle physics we will take a small detour to introduce the basic structure (as
More informationGROUP THEORY PRIMER. D(g 1 g 2 ) = D(g 1 )D(g 2 ), g 1, g 2 G. and, as a consequence, (2) (3)
GROUP THEORY PRIMER New terms: representation, irreducible representation, completely reducible representation, unitary representation, Mashke s theorem, character, Schur s lemma, orthogonality theorem,
More informationGroup Theory in Particle Physics
Group Theory in Particle Physics Joshua Albert Phy 205 http://en.wikipedia.org/wiki/image:e8_graph.svg Where Did it Come From? Group Theory has it's origins in: Algebraic Equations Number Theory Geometry
More informationCLASSICAL GROUPS DAVID VOGAN
CLASSICAL GROUPS DAVID VOGAN 1. Orthogonal groups These notes are about classical groups. That term is used in various ways by various people; I ll try to say a little about that as I go along. Basically
More informationSymmetries, Fields and Particles. Examples 1.
Symmetries, Fields and Particles. Examples 1. 1. O(n) consists of n n real matrices M satisfying M T M = I. Check that O(n) is a group. U(n) consists of n n complex matrices U satisfying U U = I. Check
More informationPart III Symmetries, Fields and Particles
Part III Symmetries, Fields and Particles Theorems Based on lectures by N. Dorey Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often
More informationGROUP THEORY PRIMER. New terms: tensor, rank k tensor, Young tableau, Young diagram, hook, hook length, factors over hooks rule
GROUP THEORY PRIMER New terms: tensor, rank k tensor, Young tableau, Young diagram, hook, hook length, factors over hooks rule 1. Tensor methods for su(n) To study some aspects of representations of a
More informationProblems in Linear Algebra and Representation Theory
Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific
More informationGroup Theory. PHYS Southern Illinois University. November 15, PHYS Southern Illinois University Group Theory November 15, / 7
Group Theory PHYS 500 - Southern Illinois University November 15, 2016 PHYS 500 - Southern Illinois University Group Theory November 15, 2016 1 / 7 of a Mathematical Group A group G is a set of elements
More informationarxiv:math-ph/ v1 31 May 2000
An Elementary Introduction to Groups and Representations arxiv:math-ph/0005032v1 31 May 2000 Author address: Brian C. Hall University of Notre Dame, Department of Mathematics, Notre Dame IN 46556 USA E-mail
More informationIntroduction to Lie Groups
Introduction to Lie Groups MAT 4144/5158 Winter 2015 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike
More informationQUATERNIONS AND ROTATIONS
QUATERNIONS AND ROTATIONS SVANTE JANSON 1. Introduction The purpose of this note is to show some well-known relations between quaternions and the Lie groups SO(3) and SO(4) (rotations in R 3 and R 4 )
More informationTopics in Representation Theory: Lie Groups, Lie Algebras and the Exponential Map
Topics in Representation Theory: Lie Groups, Lie Algebras and the Exponential Map Most of the groups we will be considering this semester will be matrix groups, i.e. subgroups of G = Aut(V ), the group
More informationPhysics 129b Lecture 1 Caltech, 01/08/19
Physics 129b Lecture 1 Caltech, 01/08/19 Introduction What is a group? From Wikipedia: A group is an algebraic structure consisting of a set of elements together with an operation that combines any two
More informationRepresentation theory and quantum mechanics tutorial Spin and the hydrogen atom
Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Justin Campbell August 3, 2017 1 Representations of SU 2 and SO 3 (R) 1.1 The following observation is long overdue. Proposition
More informationA group G is a set of discrete elements a, b, x alongwith a group operator 1, which we will denote by, with the following properties:
1 Why Should We Study Group Theory? Group theory can be developed, and was developed, as an abstract mathematical topic. However, we are not mathematicians. We plan to use group theory only as much as
More informationMAT 445/ INTRODUCTION TO REPRESENTATION THEORY
MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations
More informationNotation. For any Lie group G, we set G 0 to be the connected component of the identity.
Notation. For any Lie group G, we set G 0 to be the connected component of the identity. Problem 1 Prove that GL(n, R) is homotopic to O(n, R). (Hint: Gram-Schmidt Orthogonalization.) Here is a sequence
More informationGroup Theory Crash Course
Group Theory Crash Course M. Bauer WS 10/11 1 What is a group? In physics, symmetries play a central role. If it were not for those you would not find an advantage in polar coordinates for the Kepler problem
More informationUnitary rotations. October 28, 2014
Unitary rotations October 8, 04 The special unitary group in dimensions It turns out that all orthogonal groups SO n), rotations in n real dimensions) may be written as special cases of rotations in a
More informationSymmetries, Fields and Particles 2013 Solutions
Symmetries, Fields and Particles 013 Solutions Yichen Shi Easter 014 1. (a) Define the groups SU() and SO(3), and find their Lie algebras. Show that these Lie algebras, including their bracket structure,
More informationThe Geometry of Root Systems. Brian C. Hall
The Geometry of Root Systems A E Z S Brian C. Hall T G R S T G R S 1 1. I Root systems arise in the theory of Lie groups and Lie algebras, but can also be studied as mathematical objects in their own right.
More information1 Classifying Unitary Representations: A 1
Lie Theory Through Examples John Baez Lecture 4 1 Classifying Unitary Representations: A 1 Last time we saw how to classify unitary representations of a torus T using its weight lattice L : the dual of
More informationLECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are
More informationThe groups SO(3) and SU(2) and their representations
CHAPTER VI The groups SO(3) and SU() and their representations Two continuous groups of transformations that play an important role in physics are the special orthogonal group of order 3, SO(3), and the
More informationPhysics 129B, Winter 2010 Problem Set 4 Solution
Physics 9B, Winter Problem Set 4 Solution H-J Chung March 8, Problem a Show that the SUN Lie algebra has an SUN subalgebra b The SUN Lie group consists of N N unitary matrices with unit determinant Thus,
More information232A Lecture Notes Representation Theory of Lorentz Group
232A Lecture Notes Representation Theory of Lorentz Group 1 Symmetries in Physics Symmetries play crucial roles in physics. Noether s theorem relates symmetries of the system to conservation laws. In quantum
More informationTopics in Representation Theory: Fourier Analysis and the Peter Weyl Theorem
Topics in Representation Theory: Fourier Analysis and the Peter Weyl Theorem 1 Fourier Analysis, a review We ll begin with a short review of simple facts about Fourier analysis, before going on to interpret
More informationWeyl Group Representations and Unitarity of Spherical Representations.
Weyl Group Representations and Unitarity of Spherical Representations. Alessandra Pantano, University of California, Irvine Windsor, October 23, 2008 β ν 1 = ν 2 S α S β ν S β ν S α ν S α S β S α S β ν
More information1 Hermitian symmetric spaces: examples and basic properties
Contents 1 Hermitian symmetric spaces: examples and basic properties 1 1.1 Almost complex manifolds............................................ 1 1.2 Hermitian manifolds................................................
More informationContinuous symmetries and conserved currents
Continuous symmetries and conserved currents based on S-22 Consider a set of scalar fields, and a lagrangian density let s make an infinitesimal change: variation of the action: setting we would get equations
More informationGRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.
GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,
More informationTOPICS IN HARMONIC ANALYSIS WITH APPLICATIONS TO RADAR AND SONAR. Willard Miller
TOPICS IN HARMONIC ANALYSIS WITH APPLICATIONS TO RADAR AND SONAR Willard Miller October 23 2002 These notes are an introduction to basic concepts and tools in group representation theory both commutative
More informationAn homomorphism to a Lie algebra of matrices is called a represetation. A representation is faithful if it is an isomorphism.
Lecture 3 1. LIE ALGEBRAS 1.1. A Lie algebra is a vector space along with a map [.,.] : L L L such that, [αa + βb,c] = α[a,c] + β[b,c] bi linear [a,b] = [b,a] Anti symmetry [[a,b],c] + [[b,c],a][[c,a],b]
More informationBRST and Dirac Cohomology
BRST and Dirac Cohomology Peter Woit Columbia University Dartmouth Math Dept., October 23, 2008 Peter Woit (Columbia University) BRST and Dirac Cohomology October 2008 1 / 23 Outline 1 Introduction 2 Representation
More informationModern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur
Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Lecture 02 Groups: Subgroups and homomorphism (Refer Slide Time: 00:13) We looked
More informationHodge Structures. October 8, A few examples of symmetric spaces
Hodge Structures October 8, 2013 1 A few examples of symmetric spaces The upper half-plane H is the quotient of SL 2 (R) by its maximal compact subgroup SO(2). More generally, Siegel upper-half space H
More informationLie Algebra and Representation of SU(4)
EJTP, No. 8 9 6 Electronic Journal of Theoretical Physics Lie Algebra and Representation of SU() Mahmoud A. A. Sbaih, Moeen KH. Srour, M. S. Hamada and H. M. Fayad Department of Physics, Al Aqsa University,
More informationRepresentation Theory. Ricky Roy Math 434 University of Puget Sound
Representation Theory Ricky Roy Math 434 University of Puget Sound May 2, 2010 Introduction In our study of group theory, we set out to classify all distinct groups of a given order up to isomorphism.
More informationRepresentations of Lorentz Group
Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: How do we find the smallest (irreducible) representations of the
More informationChapter 2 The Group U(1) and its Representations
Chapter 2 The Group U(1) and its Representations The simplest example of a Lie group is the group of rotations of the plane, with elements parametrized by a single number, the angle of rotation θ. It is
More informationGROUP THEORY PRIMER. New terms: so(2n), so(2n+1), symplectic algebra sp(2n)
GROUP THEORY PRIMER New terms: so(2n), so(2n+1), symplectic algebra sp(2n) 1. Some examples of semi-simple Lie algebras In the previous chapter, we developed the idea of understanding semi-simple Lie algebras
More informationQuantum Theory and Group Representations
Quantum Theory and Group Representations Peter Woit Columbia University LaGuardia Community College, November 1, 2017 Queensborough Community College, November 15, 2017 Peter Woit (Columbia University)
More informationThe Spinor Representation
The Spinor Representation Math G4344, Spring 2012 As we have seen, the groups Spin(n) have a representation on R n given by identifying v R n as an element of the Clifford algebra C(n) and having g Spin(n)
More informationCategories and Quantum Informatics: Hilbert spaces
Categories and Quantum Informatics: Hilbert spaces Chris Heunen Spring 2018 We introduce our main example category Hilb by recalling in some detail the mathematical formalism that underlies quantum theory:
More informationOctober 25, 2013 INNER PRODUCT SPACES
October 25, 2013 INNER PRODUCT SPACES RODICA D. COSTIN Contents 1. Inner product 2 1.1. Inner product 2 1.2. Inner product spaces 4 2. Orthogonal bases 5 2.1. Existence of an orthogonal basis 7 2.2. Orthogonal
More informationIntroduction to relativistic quantum mechanics
Introduction to relativistic quantum mechanics. Tensor notation In this book, we will most often use so-called natural units, which means that we have set c = and =. Furthermore, a general 4-vector will
More informationChapter 2. Symmetries of a system
Chapter 2 Symmetries of a system 1 CHAPTER 2. SYMMETRIES OF A SYSTEM 2 2.1 Classical symmetry Symmetries play a central role in the study of physical systems. They dictate the choice of the dynamical variables
More informationChapter 3. Introducing Groups
Chapter 3 Introducing Groups We need a super-mathematics in which the operations are as unknown as the quantities they operate on, and a super-mathematician who does not know what he is doing when he performs
More informationREPRESENTATION THEORY NOTES FOR MATH 4108 SPRING 2012
REPRESENTATION THEORY NOTES FOR MATH 4108 SPRING 2012 JOSEPHINE YU This note will cover introductory material on representation theory, mostly of finite groups. The main references are the books of Serre
More informationREPRESENTATION THEORY OF S n
REPRESENTATION THEORY OF S n EVAN JENKINS Abstract. These are notes from three lectures given in MATH 26700, Introduction to Representation Theory of Finite Groups, at the University of Chicago in November
More informationQuantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras
Quantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 6 Postulates of Quantum Mechanics II (Refer Slide Time: 00:07) In my last lecture,
More informationQuantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras
Quantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 4 Postulates of Quantum Mechanics I In today s lecture I will essentially be talking
More informationQuantum Physics and the Representation Theory of SU(2)
Quantum Physics and the Representation Theory of SU(2 David Urbanik University of Waterloo Abstract. Over the past several decades, developments in Quantum Physics have provided motivation for research
More informationTHE QUANTUM DOUBLE AS A HOPF ALGEBRA
THE QUANTUM DOUBLE AS A HOPF ALGEBRA In this text we discuss the generalized quantum double construction. treatment of the results described without proofs in [2, Chpt. 3, 3]. We give several exercises
More informationGroup Theory and the Quark Model
Version 1 Group Theory and the Quark Model Milind V Purohit (U of South Carolina) Abstract Contents 1 Introduction Symmetries and Conservation Laws Introduction Finite Groups 4 1 Subgroups, Cosets, Classes
More informationINTRODUCTION TO REPRESENTATION THEORY AND CHARACTERS
INTRODUCTION TO REPRESENTATION THEORY AND CHARACTERS HANMING ZHANG Abstract. In this paper, we will first build up a background for representation theory. We will then discuss some interesting topics in
More informationFunctional determinants
Functional determinants based on S-53 We are going to discuss situations where a functional determinant depends on some other field and so it cannot be absorbed into the overall normalization of the path
More informationNote that a unit is unique: 1 = 11 = 1. Examples: Nonnegative integers under addition; all integers under multiplication.
Algebra fact sheet An algebraic structure (such as group, ring, field, etc.) is a set with some operations and distinguished elements (such as 0, 1) satisfying some axioms. This is a fact sheet with definitions
More informationPlan for the rest of the semester. ψ a
Plan for the rest of the semester ϕ ψ a ϕ(x) e iα(x) ϕ(x) 167 Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: and
More informationSimple Lie algebras. Classification and representations. Roots and weights
Chapter 3 Simple Lie algebras. Classification and representations. Roots and weights 3.1 Cartan subalgebra. Roots. Canonical form of the algebra We consider a semi-simple (i.e. with no abelian ideal) Lie
More information5 Irreducible representations
Physics 129b Lecture 8 Caltech, 01/1/19 5 Irreducible representations 5.5 Regular representation and its decomposition into irreps To see that the inequality is saturated, we need to consider the so-called
More informationTopics in Representation Theory: Roots and Weights
Topics in Representation Theory: Roots and Weights 1 The Representation Ring Last time we defined the maximal torus T and Weyl group W (G, T ) for a compact, connected Lie group G and explained that our
More informationM3/4/5P12 PROBLEM SHEET 1
M3/4/5P12 PROBLEM SHEET 1 Please send any corrections or queries to jnewton@imperialacuk Exercise 1 (1) Let G C 4 C 2 s, t : s 4 t 2 e, st ts Let V C 2 with the stard basis Consider the linear transformations
More informationLecture 21 Relevant sections in text: 3.1
Lecture 21 Relevant sections in text: 3.1 Angular momentum - introductory remarks The theory of angular momentum in quantum mechanics is important in many ways. The myriad of results of this theory, which
More informationPhysics 251 Solution Set 1 Spring 2017
Physics 5 Solution Set Spring 07. Consider the set R consisting of pairs of real numbers. For (x,y) R, define scalar multiplication by: c(x,y) (cx,cy) for any real number c, and define vector addition
More information2. The center of G, denoted by Z(G), is the abelian subgroup which commutes with every elements of G. The center always contains the unit element e.
Chapter 2 Group Structure To be able to use groups in physics, or mathematics, we need to know what are the important features distinguishing one group from another. This is under the heading of group
More informationGroup representations
Group representations A representation of a group is specified by a set of hermitian matrices that obey: (the original set of NxN dimensional matrices for SU(N) or SO(N) corresponds to the fundamental
More informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More information8. COMPACT LIE GROUPS AND REPRESENTATIONS
8. COMPACT LIE GROUPS AND REPRESENTATIONS. Abelian Lie groups.. Theorem. Assume G is a Lie group and g its Lie algebra. Then G 0 is abelian iff g is abelian... Proof.. Let 0 U g and e V G small (symmetric)
More informationRepresentations of Matrix Lie Algebras
Representations of Matrix Lie Algebras Alex Turzillo REU Apprentice Program, University of Chicago aturzillo@uchicago.edu August 00 Abstract Building upon the concepts of the matrix Lie group and the matrix
More informationL(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that
ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive
More informationLECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori
LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.
More informationPAPER 43 SYMMETRY AND PARTICLE PHYSICS
MATHEMATICAL TRIPOS Part III Monday, 31 May, 2010 1:30 pm to 4:30 pm PAPER 43 SYMMETRY AND PARTICLE PHYSICS Attempt no more than THREE questions. There are FOUR questions in total. The questions carry
More informationThese notes are incomplete they will be updated regularly.
These notes are incomplete they will be updated regularly. LIE GROUPS, LIE ALGEBRAS, AND REPRESENTATIONS SPRING SEMESTER 2008 RICHARD A. WENTWORTH Contents 1. Lie groups and Lie algebras 2 1.1. Definition
More informationRepresentation Theory
Representation Theory Representations Let G be a group and V a vector space over a field k. A representation of G on V is a group homomorphism ρ : G Aut(V ). The degree (or dimension) of ρ is just dim
More informationSymmetries, Fields and Particles 2013 Solutions
Symmetries, Fields and Particles 03 Solutions Yichen Shi July 9, 04. a Define the groups SU and SO3, and find their Lie algebras. Show that these Lie algebras, including their bracket structure, are isomorphic.
More informationTutorial 5 Clifford Algebra and so(n)
Tutorial 5 Clifford Algebra and so(n) 1 Definition of Clifford Algebra A set of N Hermitian matrices γ 1, γ,..., γ N obeying the anti-commutator γ i, γ j } = δ ij I (1) is the basis for an algebra called
More informationLie Theory in Particle Physics
Lie Theory in Particle Physics Tim Roethlisberger May 5, 8 Abstract In this report we look at the representation theory of the Lie algebra of SU(). We construct the general finite dimensional irreducible
More informationLECTURE 2: SYMPLECTIC VECTOR BUNDLES
LECTURE 2: SYMPLECTIC VECTOR BUNDLES WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic Vector Spaces Definition 1.1. A symplectic vector space is a pair (V, ω) where V is a finite dimensional vector space (over
More informationAssignment 3. A tutorial on the applications of discrete groups.
Assignment 3 Given January 16, Due January 3, 015. A tutorial on the applications of discrete groups. Consider the group C 3v which is the cyclic group with three elements, C 3, augmented by a reflection
More informationRisi Kondor, The University of Chicago
Risi Kondor, The University of Chicago Data: {(x 1, y 1 ),, (x m, y m )} algorithm Hypothesis: f : x y 2 2/53 {(x 1, y 1 ),, (x m, y m )} {(ϕ(x 1 ), y 1 ),, (ϕ(x m ), y m )} algorithm Hypothesis: f : ϕ(x)
More informationLecture notes on Quantum Computing. Chapter 1 Mathematical Background
Lecture notes on Quantum Computing Chapter 1 Mathematical Background Vector states of a quantum system with n physical states are represented by unique vectors in C n, the set of n 1 column vectors 1 For
More informationParticles I, Tutorial notes Sessions I-III: Roots & Weights
Particles I, Tutorial notes Sessions I-III: Roots & Weights Kfir Blum June, 008 Comments/corrections regarding these notes will be appreciated. My Email address is: kf ir.blum@weizmann.ac.il Contents 1
More informationRepresentations Are Everywhere
Representations Are Everywhere Nanghua Xi Member of Chinese Academy of Sciences 1 What is Representation theory Representation is reappearance of some properties or structures of one object on another.
More informationne varieties (continued)
Chapter 2 A ne varieties (continued) 2.1 Products For some problems its not very natural to restrict to irreducible varieties. So we broaden the previous story. Given an a ne algebraic set X A n k, we
More informationCLASSIFICATION OF COMPLETELY POSITIVE MAPS 1. INTRODUCTION
CLASSIFICATION OF COMPLETELY POSITIVE MAPS STEPHAN HOYER ABSTRACT. We define a completely positive map and classify all completely positive linear maps. We further classify all such maps that are trace-preserving
More information1 Mathematical preliminaries
1 Mathematical preliminaries The mathematical language of quantum mechanics is that of vector spaces and linear algebra. In this preliminary section, we will collect the various definitions and mathematical
More informationLecture 5: Sept. 19, 2013 First Applications of Noether s Theorem. 1 Translation Invariance. Last Latexed: September 18, 2013 at 14:24 1
Last Latexed: September 18, 2013 at 14:24 1 Lecture 5: Sept. 19, 2013 First Applications of Noether s Theorem Copyright c 2005 by Joel A. Shapiro Now it is time to use the very powerful though abstract
More informationCS 468: Computational Topology Group Theory Fall b c b a b a c b a c b c c b a
Q: What s purple and commutes? A: An abelian grape! Anonymous Group Theory Last lecture, we learned about a combinatorial method for characterizing spaces: using simplicial complexes as triangulations
More informationOn Vector Product Algebras
On Vector Product Algebras By Markus Rost. This text contains some remarks on vector product algebras and the graphical techniques. It is partially contained in the diploma thesis of D. Boos and S. Maurer.
More informationChapter 1 LORENTZ/POINCARE INVARIANCE. 1.1 The Lorentz Algebra
Chapter 1 LORENTZ/POINCARE INVARIANCE 1.1 The Lorentz Algebra The requirement of relativistic invariance on any fundamental physical system amounts to invariance under Lorentz Transformations. These transformations
More information