3. Continuous Groups Lectures on Symmetries in Physics –SL(N,C);SO(N); SU(N); SO(N,M) – Useful Matrix Relations in GL(N,C) – Generators and Exponential rep of Groups Apostolos Pilaftsis [ Examples: SO(2), U(1), SO(3), SU(2) ] Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom http://pilaftsi.home.cern.ch/pilaftsi/ 4. Lie Algebra and Lie Groups – Generators of a Group as Basis Vectors of a Lie Algebra 0. Literature – The Adjoint Representation – Group Theory as the Calculus of Symmetries in Physics – Normalization of Generators and Casimir Operators 1. Introduction to Group Theory 5. Tensors in SU(N) – Definition of a Group G –Preliminaries – Young Tableaux – The Discrete Groups Sn, Zn and Cn – Cosets and Coset Decomposition – Applications to Particle Physics – Normal Subgroup H and Quotient Group G/H – Morphisms between Groups 6. Lorentz and Poincar´eGroups – Lie Algebra and Generators of the Lorentz Group 2. Group Representations (Reps) – Lie Algebra and Generators of the Poincar´eGroup – Definition of a Vector Space V – Single Particle States – Definition of a Group rep. –ReducibleandIrreduciblereps(Irreps) – Direct Products and Clebsch–Gordan Series 2 7. Lagrangians in Field Theory Literature • – Variational Principle and Equation of Motion – Lagrangians for the Klein-Gordon and Maxwell equations In order of relevance and difficulty: – Lagrangian for the Dirac equation 1. H.F. Jones: Groups, Representations and Physics (IOP, 8. Gauge Groups 1998) Second Edition – Global and Local Symmetries. 2. L.H. Ryder, Quantum Field Theory (CUP, 1996) Second – Gauge Invariance of the QED Lagrangian Edition –Noether’sTheorem – Yang–Mills Theories 3. T.-P. Cheng and L.-F. Li, Gauge Theory of Elementary Particle Physics (OUP, 1984). 9. The Geometry of Gauge Transformations (Trans) 4. S. Pokorski, Gauge Field Theories (CUP, 2000) Second – Parallel Transport and Covariant Derivative Edition. – Topology of the Vacuum: the Bohm–Aharanov Effect 5. J. Wess and J. Bagger, Supersymmetry and Supergravity, 10. Supersymmetry (SUSY) (Princeton University Press, 1992) Second Edition – Graded Lie Algebra – Generators of the Super-Poincar´eGroup – The Wess–Zumino Model –Feynmanrules 3 4 A list of related problems from H.F. Jones: 1. Introduction to Group Theory 1. 2.5, 2.9, 2.12∗ – Definition of a Group G 2. 3.1, 3.3, 3.4, 3.6 A group (G, ) is a set of elements a, b, c . endowed with · { } 3. 6.1, 6.2, 6.3 a composition law · that has the following properties: 4. 9.1 (i) Closure. a, b G,theelementc = a b G. ∀ ∈ · ∈ 5. 8.1, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8∗, 8.9∗ (ii) Associativity. a, b, c G, it holds a (b c)=(a b) c 6. 10.1, 10.2, 10.3 ∀ ∈ · · · · (iii) The identity element e. e G: e a = a, a G. 7. 10.4, 10.5, 10.6, 10.7, 10.8 ∃ ∈ · ∀ ∈ 1 1 8. 11.3, 11.5, 11.7, 11.8 (iv) The inverse element a− of a. a G, a− G: 1 1 ∀ ∈ ∃ ∈ a a− = a− a = e. · · Note that more problems as exercises are included in these notes. If a b = b a, a, b G, the group G is called Abelian. · · ∀ ∈ – The Discrete Groups Sn, Zn and Cn Group G Multiplication Order Remarks Sn:permutation Successiveoperation n! Non-Abelian of n objects in general Zn:integers AdditionmodnnAbelian modulo n Cn:cyclicgroup Unspecified product nCn ∼= Zn n 1 · e, a, . a − { } with an =1 5 6 – Cosets and Coset Decomposition – Normal Subgroup H and Quotient Group G/H Coset.LetH = h ,h ,...,h be a proper (i.e. H = G Conjugate to H.IfH is a subgroup of G,thentheset { 1 2 r} ̸ and H = I = e ) subgroup of G. H = gHg 1 = gh g 1,gh g 1,...,gh g 1 , for a given ̸ { } ′ − 1 − 2 − r − For a given g G,thesets g G,iscalledg{-conjugate to H or simply conjugate} to H. ∈ ∈ gH = gh1,gh2,...,ghr ,Hg= h1g, h2g, . , hrg Normal Subgroup H of G.IfH is a subgroup of G and { } { } 1 H = gHg− g G,thenH is called a normal subgroup ∀ ∈ are called the left and right cosets of H. of G. Groups which contain no proper normal subgroups are termed Lagrange’s Theorem.Ifg1H and g2H are two (left) cosets simple. of H,theneither g H = g H or g H g H = ∅. 1 2 1 ∩ 2 Groups which contain no proper normal Abelian subgroups Coset Decomposition.IfH is a proper subgroup of G,then are called semi-simple. G can be decomposed into a sum of (left) cosets of H: Quotient Group G/H.LetG/H = H, g1H, . , gν 1H { − } G = H g1H g2H gν 1H, be the set of all distinct cosets of a normal subgroup H of G, ∪ ∪ ··· ∪ − with the multiplication law: where g G, g / H; g / H, g / g H,etc. 1,2,... ∈ 1 ∈ 2 ∈ 2 ∈ 1 (giH) (gjH)=(gi gj) H, The number ν is called the index of H in G. · · where giH, gjH G/H.Then,itcanbeshownthat(G/H, ) The set of all distinct cosets, H, g1H, . , gν 1H ,isa ∈ · manifold, the coset space, and is{ denoted by G/H−. } is a group and is termed quotient group. Note that G/H is not a subgroup of G.(Why?) 7 8 – Morphisms between Groups 2. Group Representations (Reps) Group Homorphism.If(A, ) and (B,⋆) are two groups, then group homorphism is a functional· mapping f from the – Definition of a Vector Space V set A into the set B, i.e. each element of a A is mapped AvectorspaceV over the field of complex numbers C is into a single element of b = f(a) B, such that∈ the following ∈ a set of elements vi , endowed with two operations (+, ), multiplication law is preserved: satisfying the following{ } properties: · f(a1 a2)=f(a1) ⋆ f(a2) . (A0) Closure. u + v V u, v V . · ∈ ∀ ∈ In general, f(A) = B,i.e.f(A) B. (A1) Commutativity. u + v = v + u u, v V . ̸ ⊂ ∀ ∈ (A2) Associativity. u+(v+w)=(u+v)+w u, v, w V Group Isomorphism. Consider a 1:1mapping f of (A, ) ∀ ∈ onto (B,⋆), such that each element of a A is mapped· ∈ (A3) The identity (null) vector. 0 V ,suchthat into a single element of b = f(a) B, and conversely, each ∃ ∈ ∈ v + 0 = v , v V . element of b B is the image resulting from a single element ∀ ∈ of a A.Ifthisbijectiv∈ 1:1mapping f satisfies the (A4) Existence of inverse. v V , ( v) V ,suchthat composition∈ law: v +( v)=0. ∀ ∈ ∃ − ∈ − f(a1 a2)=f(a1) ⋆ f(a2) , (B0) λ u V λ C, u V . · · ∈ ∀ ∈ ∀ ∈ it is said to define an isomorphism between the groups A and (B1) λ (u + v)=λ u + λ v. · · · B, and is denoted by A ∼= B. (B2) (λ1 + λ2) u = λ1 u + λ2 u. A group homorphism of A into itself is called endomorphism. · · · (B3) λ (λ u)=(λ λ ) u. A group isomorphism of A into itself is called automorphism. 1 · 2 · 1 2 · (B4) 1 u = u. · 9 10 – Definition of a Group Rep. – Reducible and Irreducible Reps. Group Rep.Agroup representation T , Reducible rep. A group rep. T (g) is said to be (completely) reducible, if there exists a non-singular matrix M GL (N,C) T : g T (g) GL (N,C) g G, independent of the group elements, such that ∈ → ∈ ∀ ∈ is a homomorphism of the elements g of a group (G, ) into T1(g)0 0 C · ··· . the group GL(N, ) of non-singular linear tranformations of 1 0 T2(g) . MT(g) M − = ⎛ . ⎞ g G. avectorspaceV of dimension N, i.e. the set of N N- . ... 0 ∀ ∈ × dimensional invertible matrices in C. ⎜ 0 0 T (g) ⎟ ⎜ ··· r ⎟ In addition, homomorphism implies that the group ⎝ ⎠ multiplication is preserved: T1(g), T2(g),...,Tr(g) divide T into reps. of lower r dimensions, i.e. dim (T )= i=1 dim (Ti), and is denoted T (g1 g2)=T (g1) T (g2) . by the direct sum: · ! ··· T (g)=T1(g) T2(g) Tr(g)= T(i) . Two reps. T1 and T2 are equivalent if there exists an ⊕ ⊕ ··· ⊕ ⊕ isomorphism (1:1correspondance) between T1 and T2. ( Such an equivalence is denoted as T = T ,orT T . 1 ∼ 2 1 ∼ 2 Two equivalent reps may be related by a similarity trans. S: Irreducible rep (Irrep). A group rep. T (g) which cannot be 1 T (g)=ST (g)S− g G and S independent of g. written as a direct sum of other reps. is called irreducible. 1 2 ∀ ∈ Character χ of a rep T of··· a group G is defined as the set of all traces of the matrices T (g): χ = χ(g)/χ(g)= [T (g)] g G . { i ii ∧ ∈ } Corollary! :Equivalentrepshavethesame character. Conversely, if two reps have the same character, they are equivalent. 11 12 – Direct Products and Clebsch–Gordan Series Clebsch–Gordan Series Direct Product of Groups.If(A, )=( a ,a,..., a , ) If g = g = g, then the symmetry of the product group · { 1 2 n} · 1 2 and (B, ⋆)=(b1,b2,..., bm , ⋆) are two groups with G G is reduced to its diagonal G,i.e.G G G. composition laws{ and ⋆,respectively,} × × → · In this case, D(a)(g) D(b)(g) may not be an irrep and can then anewdirect-product group (G, )=(A B, ) can be ⊗ uniquely defined with elements g = a⊙ b. The× multiplication⊙ be further decomposed into a direct sum of irreps of G: law in G is defined as ⊗ ⊙ D(a)(g) D(b)(g)= a D(c)(g) .