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 =.
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.
Cyclic group Z n -ThesetofintegernumbersG =(,,,..n ) with addition mod n. Itisequivalenttocyclicpermutationsofn objects. Symmetric group S n - All possible permutations of n objects.
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
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
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 = B @ 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
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
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
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
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.