Introduction to Grou Theory Note July 7, 009 Contents INTRODUCTION. Examles OF Symmetry Grous in Physics................................. ELEMENT OF GROUP THEORY. De nition of Grou................................................ Rearrangement Theorem............................................ 5. Alications of Rearrangement Theorem................................... 5.4 Grou Induced Transformations........................................ 5.5 Transformation of Functions.......................................... 6.6 Coset...................................................... 7.7 Class....................................................... 8.8 Class Multilication.............................................. 9 INTRODUCTION Grou theory is the framework for studying hysical system with symmetry. In articular, the reresentation theory of the grou simli es the hysical solutions to the systems which have symmetries. For examle, suose that an one-dimensional Hamiltonian has the symmetry x x; i.e. Then from the time-indeendent Schrödinger s equation, H(x) = H( x) H(x) (x) = E (x) we get which means that ( these two states, H( x) ( x) = H(x) ( x) = E ( x) x) is also an eigenstate with same eigenvalue E. Thus we can form the linear combinations of = ( (x) ( x)) which are arity eigenstates and are either symmetric or anti-symmetric under x x. These are the consequences of symmetry. Note that this means only that the eigenstates can be chosen to be either symmetric or antisymmetric and does not imly that the system has degenerate eigenstates. This is because the even or odd state can be identically zero. For examle, in the case of one-dimensional harmonic oscillator otential, the enegy eigenstate is either symmetric or antisymmetric but not both REMARK: The symmetry of H does not necessarily imly the symmetry of the eigenfunctions. It only says that given an eigenfunction, the symmetry oeration will generate other solutions which may or may not be indeendent of the original eigenfunction. As we will discuss later, in fact eigenfunctions form irreducible reresentations of the symmetry grou.
. Examles OF Symmetry Grous in Physics. Finite grous (a) Crystallograhic grous (symmetry grou of crystals) translations - eriodic Symmetry oerations rotations - sace grou Examle: i. NaCl Molecules (b) Permutation grou In quantum mechanics, the wavefunctions of identical articles are required to be either symmetric of antisymmetric under the ermutation of the coordinates. These ermutations form a grou, ermutation grou S n: Permutation grou also lays an imortant role in the study of reresentations of unitary grous.. Continuous grous (a) O() or SU() Rotation grou in dimension This grou forms the basis for the theory of angular momentum in quantum mechanics. The structure of this grou also rovides the foundation for describing other more comlicated grous. (b) SU() Secial unitary matrices This tye of grou has been used to describe the sectrum of hadrons in terms of quark model. But the symmetry here is only aroximate. SU() has also been used to formulate the theory of strong interaction, QCD. Here the symmetry is exact but has the eculiarity of con nement. (c) SU() L U() Y This is the symmetry grou used to describe the standard model of electromagnetic and weak interactions. However the symmetry here is also broken (sontaneously). (d) SU(5); SO(0); E(6) These symmetries unify electromagnetic, weak and strong interactions, grand uni ed theories (GUT). Of course these grous are badly broken.
(e) SL(; C) Lorentz grou This is the grou which rovides the descrition of sace-time structure in secial relativity. It is a central dogma to any formulation of relativistic system. In articular, it lays a crucial role in the relativistic eld theory. ELEMENT OF GROUP THEORY. De nition of Grou A grou G is set of elements (a; b; c : : :) with an oerstion * which satis es following roerties: (i) Closure : If a; b;, this imlies c = a b is also in G. (ii) Associative: a (b c) = (a b) c (iii) Identity : 9 an element e such that a e = e a = a 8 a G (iv) Inverse : for every a G; 9 an element a such that a a = a a = e To simlify the notation, we will denote the grou oeration a b by the simle roduct ab: Examles of Grou. All real numbers under \ + ". All real numbers without \0" under \ ". All integers under \ + " 4. All rotations in -dimensional sace: O() 5. All n n matrices under \ + " 6. All non-singular n n matrices under \ ": GL(n) (General Linear Grou in n-dimension) 7. All n n matrices with determinant : SL(n) (Secial Linear Grou in n-dimension) 8. All n n unitary matrices under \ ": U(n) (Unitary Grou in n-dimension) 9. All n n unitary matrices with determinant : SU(n) (Secial Unitary Grou in n-dimension) 0. All n n orthogonal matrices: O(n). All n n orthogonal matrices with determinant : SO(n). Permutations of n objects: S n Abelian Grou - If the grou multilication is commutative, i.e. ab = ba 8 a; b G. Finite Grou - If the number of elements in G is nite. Order of the Grou - # of elements in the grou. Subgrou - a subset of the grou which is also a grou. e.g. SL(n) is a subgrou of GL(n). Simle Examle : Symmetry of a regular triangle ( called D grou)
Oerations A: rotation y by 0 in the lane of triangle B: rotation y by 40 in the lane of triangle K: rotation y by 80 about zz 0 L: rotation y by 80 about yy 0 M: rotation y by 80 about xx 0 E: no rotation Grou Multilication : consider the roduct KA Thus we have KA = L This way we can work out the multilication of any grou elements and summarize the result in multilication table. E A B K L M E E A B K L M A A B E M K L B B E A L M K K K L M E A B L L M K B E A M M K L A B E Clearly fe; A; Bg ; fe; Lg ; fe; Kg ; fe; Mg are subgrous Isomorhism: Two grous G = fx ; x ; : : :g and G 0 = fx 0 ; x 0 ; : : :g are isomorhic if 9 a one-to-one maing x i x 0 i such that x i x j = x k =) x 0 ix 0 j = x 0 k In other words, the grous G and G 0 ; which might oerate on di erent hysical system, have the same structure as far as grou theory is concerned. 4
Symmetry grou S : ermutation symmetry of objects. S has 6 grou elements. ; ; ; ; ; We can show that S is isomorhic to D by associate the vertices of the triangle with ; and :. Rearrangement Theorem Theorem : Each element of G aears exactly once in each row or column of the multilication table. Proof: Take grou elements to be E; A ; A ; : : : ; A h. Multily by arbitrary grou element A k to this set roducing A k E; A k A ; A k A ; : : : ; A k A h. Suose elements in this set are the same, e.g. A k A i = A k A j where A i 6= A j. Since A k exists, we can multily this by A k to get the result A i = A j which contradicts the initial assumtion. Hence all elements in each row after multilication are di erent. But there are exactly h elements in each row. Therefore, each grou element occurs only once in each row. In other words, multilication of the grou by a xed element of the grou, simly rearrange them. This is why it is called the rearrangement theorem.. Alications of Rearrangement Theorem. Suose we are summing over the grou elements of some functions of grou elements, Ai f(a i ). Then rearrangement theorem imlies that f(a i ) = f(a i A k ) A i Ai for any A k G. This result is central to many imortant result of the reresentation theory of nite grous. The validity of this theorem for the case of continuous grou is then an imortant requirement in taking over the results from the nite grous to continuous grous.. Using this theorem, we can show that there is only one grou of order. This can be seen by using multilication table E A B E E A B A A B E B B E A In fact, this grou is of the form A; B = A ; E = A. This is an examle of cyclic grou of order. Cyclic Grou of order n; is of the form, Z n = A; A A ; : : : A n = E Clearly, all cyclic grous are Abelian. Examles of cyclic grous:. 4th roots of unity; ; ; i; i = i; i = ; i = i; i 4 = ;. Benzene molecule Z 6 = A; A : : : A 6 = E A = rotation by.4 Grou Induced Transformations For many grous in hysics, the grou transfomations are geometric in nature. In these cases, the grou transfomations can be reresented as oerations in the coordinate sace. As an examle, take a coordinate system for the triangle as shown, 5
If we kee the triangle xed and rotate the coordinate system, we get the relations between the old and new coordinates as or x 0 y 0 = x 0 = cos x + sin y = x + y y 0 = sin x + cos y = x y x or ~x 0 = A~x A = y ; ~x = Thus grou element A can be reresented by matrix A acting on the coordinate system (x; y). We can do this for other grou elements to get, x y B = E = 0 0 K = 0 0 L = M = The roduct of the grou elements can also be exressed in terms of matrices, e.g K = 0 0 ~x 0 = A~x ~x 00 = K~x 0 =) ~x 00 = KA~x A = KA = Thus, the matrix multilication gives the same result as the multilication table, isomorhism between the symmetry grou and the set of 6 matrices. This is an examle of reresentation - grou elements are reresented by a set of matrices (does not have to be - corresondence). From now on, for the simlicity of notation, we will denote the matrices by the same symbols as the grou elements..5 Transformation of Functions We can generalize the transformations of coordinates to functions of (x; y); f(x; y). (i) Take any grou element A of G, which generate matrix A on (x; y). (ii) Relace ~x by A x 0. This de nes a new function g(x 0 ; y 0 ). We will denote g(x; y) by g(x; y) = P A f(x; y) or more simly P A f (x) = f A x : = L Examle: f(x; y) = x y ; take A = 6
then and x 0 = x + y y 0 = x y or x = x0 y0 y = x0 y0 f(x; y) = x0 + y0 x0 y0 = x 0 y 0 + x 0 y 0 = g (x 0 ; y 0 ) Or g (x; y) = x y + xy = P A f (x; y) Symbolically, we have f (~x) f A ~x = g (~x) or P A f (~x) = g (~x) = f A ~x Theorem: If A s form a grou G, the P A s de ned on certain function f (~x) also form a grou G P. Proof: P A f (~x) = f A x = g (~x) P B P A f (~x; ) = P B g (~x) = g B ~x = f A B ~x = f A B ~x = f (BA) ~x = P BA f (~x) Thus we have P B P A = P BA =) G is homomorhic to G P : But the corresondence A P A is not necessarily one-to-one. Note that we de ne P A in terms of A in order to get the homomorhism P B P A = P BA : For examle if we have the function f (x; y) = x + y =) P A = P B = P E = : : : =..6 Coset Coset of a grou is a useful tool to decomose the grou into disconnect sets. Let H = fe; S ; S : : : S g gbe a subgrou of G: For x G but = H; if we multily the whoe subgrou by x on the right we get fex; S x; S x; : : : S g xg right coset of x, denoted by Hx and the left multilication gives, fxe; xs ; xs ; : : : xs g g left coset of x, denoted by xh Note that a coset can not form a grou, because identity is not in the set. Proerties of Cosets (i) Hx and H have no elements in common. Suose there is one element in common, S k = S j x; where x = H then x = S j S k H This is a contradiction because x = H by construction. (ii) Two right (or left) cosets either are identical or have no element in common. Consider Hx and Hy; with x 6= y:suose there is one element in common between these cosets then S k x = S j y xy = S k S j H But Hxy = H by rearrangement theorem which imlies that Hx = Hy 7
Theorem: If H is a subgrou of G, then the order of H is a factor of order of G. Proof: Consider all distinct right cosets H; Hx ; Hx ; : : : ; Hx l Each element of G must aear in exactly one of these cosets. Since there are no elements in common among these cosets, we must have g = l h where h the order of H, l some integer and g order of G. Remark: This theorm severely limits the ossible subgrou of a nite grou. For examle, a grou of order 6, like D ; the only non-trivial subgrous are those with orde or : Conjugate: B and A are conjugate to each other if 9 x G such that xax = B (similarity transformation) Remark: Relacing each element by its conjugate under some xed element x is an isomorhism under x. This can be seen as follows. From A 0 = xax B 0 = xbx A 0 B 0 = xax xbx = xabx = (AB) 0 we see that g i g 0 i = xg ix is an isomorhism because the corresondence is one-to-one. In coordinate transformations, similarity transformations corresond to change of basis and do not reresennt a intrinsically di erent oeration. Coset Sace G=H = fcosets Hx, x G but not in Hg Roughly seaking, coset sace is obtained by grouing together elements which are related by left (or right) multilication of elements in the subgrou H. This decomosition is useful in reducing the structure of the grou to a smaller structure..7 Class All grou elements which are conjugate to a given element is called a class. Roughly seaking these are grou elements which are essentially the same oeration with di erenent basis and these basis can be tranformed into one another by the grou transformation. Denote the grou elements by G = fe; x ; : : : ; x n g Take A G, 9 then EAE x Ax >= class (all grou elements conjugate to A). Note that these elements are not necessarily all di erent.. >; x h Ax h Examle: symmetry grou of triangle D From the multilication table, we see that AAA = A BAB = A KAK = B LAL = B MAM = B AKA = L BKB = M KKK = K LKL = M MKM = L Hence the classes are feg ; fa; Bg ; fk; L; Mg. Note: E is always in a class by itself because A i EA i = EA i A i = E; 8A i G. In this examle, fa; Bg rotations by and fk; L; Mg rotations by. This is a very general feature all elements in the same class have same angle of rotation. Thus roughly seaking elements in the same class have the same hysical oeration. Invariant Subgrou: If a subgrou H of G consists entirely of comlete classes. For examle H = fe; A; Bg is an invariant subgrou while fe; Kg is not. Invariant subgrou is also called normal subgrou or normal divisor. Symbolically for invariant subgrous, we have xhx = H for any x G. which imlies xh = Hx; i.e. left cosets are the same as right cosets. 8
For every grou G, there is at least two trivial invariant subgrous, feg and the grou G itself. If a grou only has these two invariant subgrous, then it is called a simle grou. Examles of simle grous are cyclic grous of rime order. Factor Grou (or Quotient Grou) Consider the invariant subgrou H = fe; h : : : h`g of G and the collection of all distinct left (or right) cosets [a] ah; [b] bh; : : : (where in this notation [E] = H). De ne the multilication of cosets as follows: Suose r [a] ; r [b] and r r = R 0. Then we de ne [a] [b] = [R 0 ]. Let a; b G but not in H, then the roduct of elements from these two cosets can be written as (ah i ) (bh j ) = ab b h i b h j = ab (h k h j ) coset containing ab where we have set h k = b h i b. Notation: If C and C are two classes, then C = C means that C and C have same collection of grou elements. Thus the coset multilication is well-de ned and is analogous to multilication of the grou elements. It is not hard to see that the collection of these cosets of H forms a grou, called the Quotient Grou and is denoted by G=H. Theorem: If C is a class, then for any x G, we have x Cx = C. Proof: Write C = fa ; A ; : : : ; A j g then x Cx = xa x ; xa x ; : : : ; xa j x. Take any element in x Cx say xa i x. This is an element related to A i by conjugation. xa i x is in the class containing A i and hence xa i x C. Thus, each element in x Cx must aear in C because C is a class. But all elements in x Cx are di erent. Therefore x Cx = C. Theorem: Any collection C which satis es x Cx = C for all x G consists wholly of comlete classes. Proof: First we can subtract out all comlete classes from both sides of the equation. Denote the remainder by R. So we have xrx = R. Suose R is not a comlete class. This means that there exists some element A i which is related to some element R j R by conjugate and A i = R; i.e. A i = yr j y and A i = R But this violates the assumtion xrx = R for all x G..8 Class Multilication For classes C i; C j in G; we have from the revious theorems C i C j = x C i x x C j x = x (C i C j ) x 8 x G: Thus C i C j consists of comlete classes, and we can write C i C j = X k c ijk C k where c ijk are some integers. Direct Product of Two Grous Given two grous G = fx i ; i = ; : : : ; ng ; G 0 = fy j ; j = ; : : : ; mg, the direct roduct grou is de ned as with grou multilication de ned by It is clear that G G 0 forms a grou. Note that G G 0 = f(x i ; y j ) ; i = ; : : : n; j = ; : : : ; mg (x i ; y j ) (x i 0; y j 0) = (x i x i 0; y j y j 0) (E; y j ) (x i ; E 0 ) = (x i ; y j ) = (x i ; E 0 ) (E; y j ) In some sense, this means that G and G 0 are subgrous of G G 0 with the roerty that grou elements from G commutes with grou elements from G 0. We can generalize this to de ne direct roduct of subgrous. Let S and T be subgrous of G such that S and T commute with each other, s i t j = t j s i 8 s i S; t j T: 9
Then we can de ne the direct roduct S T as Examle: S T = fs i t j j s i S; t j T g Z = f; g Z = n; e i= ; e 4i=o Clearly, this is isomorhic to Z 6. Z Z = n; e i= ; e 4i= ; ; e i= ; e 4i=o = n; e i= ; e 4i= ; e i ; e 5i= ; e i=o There is an interesting connection between direct roduct of grous and quotient grou. Sometimes it is ossible to reconstruct a grou G by taking the direct roduct of G=H and H (where H is an invariant subgrou of G). The question of whether this reconstruction is ossible or not is called the extension roblem. 0