Symmetries in Physics

Transcription

1 Symmetries in Physics September 23, 2009

2 a) The Unitary Symmetry group, SU(2) b) The Group SU(3) c) SU(N) tesnors and Young Tableaux. a) The SU(2) group Physical realisation is e.g. electron spin and isospin of up and down quarks. The vector space consists of wavefunctions given by a two-component spinor ψ = ( ψu ψ d ), where ψ u and ψ d refer to up and down components of say spin or isospin. Transformations through an angle θ effected by a unitary matrix U generated by the Pauli spin matrices : U = exp [ i2 ] σ.nθ.

3 Here σ is the vector of Pauli matrices (σ x, σ y, σ z ) and n is a unit vector. Note that the algebra of the group is same as for SU(3) since: [ Xi, X j ] = iǫijk X k, the commutation relation between generators is satisfied by X i = 1 2 σ i. Relation between SO(3) and SU(2) Since these two groups share the same algebra and the transformation matrices take the same general form U = exp[ ix.nθ] we are led to consider a homomorphism or mapping between them. While mapping elements U of SU(2) into those of SU(3) we see that the mapping is not 1 1. Two distinct elements U(2π) = 1 and U(0) = 1 of SU(2) are mapped onto the identity of SO(3), since in the SO(3) case U(0) = U(2π) = 1. Note that we had disallowed non-integral values of the label m for irreps of SO(3) on classical grounds. The irreps of SU(2) are the same but with halfinteger m allowed.

4 The homomorphism between SU(2) and SO(3) thus has a non-trivial Kernel,K, since two elements are mapped onto the identity of SO(3). The relation can be summarised as SO(3) = SU(2) Z 2 where Z 2 is a normal subgroup representing the Kernel of the mapping from SU(2) to SO(3). The Z 2 group is isomorphic to C 2 and can be represented by the matrices : ( ), ( ). Irreps. of SU(2) and Clebsch-Gordan series: We work in terms of the vector-space (ie wavefunctions on which the irreps. act) rather than in terms of group elements or matrices themselves. Looking for invariant subspaces of the vector-space is the same as working out the irreps. of the algebra. Fundamental building block, transforming under U, is two component spinor ψ a, a = 1,2 where we now

5 use 1 and 2 instead of up and down. Transformation is as below ψ a = U ab ψ b, U SU(2). Antiparticles transform under the complex-conjugate representation denoted by an upper index ψ a : ψ a = ( U ) ab ψ b, U SU(2). However the two representations are equivalent and hence not distinguished : U = CUC 1. The matrix C can be expressed as C ab = ǫ ab, where ǫ is antisymmetric rank 2 tensor. C is invariant under SU(2) and is a raising operator or metric ψ a = C ab ψ b. Clebsch-Gordan series for SU(2) We now want to combine the fundamental spinors to build compund objects or wavefunctions. We wish to examine how these compound objects transform under SU(2). Thus we want to work out the

6 decomposition of a composite of spinors into invariant subspaces. Key observation : The symmetric and antisymmetric parts of a rank 2 tensor transform independently (i.e do not mix) and hence form invariant subspaces. Thus for combination of two spinors ψ i, φ j we have three symmetric combinations : ψ 1 φ 1, ψ 2 φ 2, 1 2 (ψ 1φ 2 + ψ 2 φ 1 ) and a single antisymmetric one, 1 2 (ψ 1φ 2 ψ 2 φ 1 ). The antisymmetric tensor is obviously an invariant scalar and does not transform under SU(2). The symmetric tensors form a 3d invariant subspace. Hence one has the decomposition 2 2 = 3 1. This corresponds to the familiar law for combining electron spins below : = 1 0. One can diagramatically express the process of symmetrisation and antisymmetrisation as below : =

7 where two boxes arranged vertically represents antisymmetrisation and those horizontally represent the symmetric tensors. We shall later show that the dimensionalities of these diagrams can be computed to be 1 (antisymmetric) and 3 (symmetric), as required. Similarly one has the decomposition = corresponding to the addition of three spins = b) The Unitary Group SU(3) The fundamental object is a three-component wavefunction ψ i, i = 1,2,3. Physically realised by, for instance, the colour component of the quark wavefunction. The transformation matrices that act on this are 3 3 unitary matrices. A major difference from SU(2) is that the complexconjugate representation (carried e.g by the antiparticles) is not equivalent to the fundamental representation.

8 The conjugate wavefunctions are represented as usual by an upper spinor but this is equivalent to (transforms as) an antisymmetric pair of lower indices : ψ a = ǫ abc φ [bc]. The ǫ tensor is once again the invariant metric that changes an antisymmetric pair of lower indices [bc] to an upper index. Once again the process of symmetrisation and antisymmetrisation of indices is used to construct the invariant subspaces. However we shall from now on use the diagrammatic rules provided by the boxes or Young Tableaux. For more details on the general tensorial method see S.Coleman s Aspects of Symmetry. c)young tableaux method The Young tableaux can be used to easily construct the Clebsch-Gordan decomposition for SU(N). Each tableau represents a specific process of symmetrisation and anti-symmetrisation, of a general SU(N) tensor obtained by combining objects that transform in a definite representation of SU(N).

9 For instance combining two objects that transform in the fundamental triplet representation of SU(3) will yield the decomposition : 3 3 = 6 3, ie the 9 dimensional composite tensor can be broken into sextet and triplet (but complex conjugate) invariant subspaces. We need to know how to calculate the dimensionality of a legal Young tableau as well as how to combine tableaux. Rules for dimensionality of a Young tableau A legal Young tableau is one where each row contains at least as many boxes as the ones immediately below it. The diagram must be convex downwards and to the right. Thus is not a valid diagram while is. The rules for calculating the dimensionality of a Young Tableau are : Calculate a numerator n by filling up the boxes starting with N (for SU(N)) in the top left-hand

10 corner and increasing by 1 for each successive column and decreasing by 1 for each succesive row. For example for SU(3) the diagram is filled up as The numerator is the product of all entries above. Calculate the denominator d by filling for each box the number of boxes below it in the same column and to the right of it in the same row + 1 for the box itself. So the diagram above has denominator given by The ratio n/d gives the dimension of the diagram. In the example above we have n/d = 27. Note that the irreps. of SU(2) are carried by a Young Tableau consisting of a single row with n boxes, which has dimensionality n+1. Why is there no second row? (refer to your lecture notes!)

11 The irreps of SU(3) are carried by a tableau with m+n boxes in the first row and n boxes in the second row. The dimensionality of this irrep (tableau) shd be calculated by you to be 1 2 (m+1)(n+1)(m+ n + 2) (check with above example). Forming the Clebsch-Gordan series We need to combine two tableaux each representing an irrep. of SU(N) and work out the resultant in terms of a sum of irreps. (i.e further tableaux). To do this a) Write down the tableaux (say T 1 and T 2 ) to be combined and label successive rows of T 2 with indices a, b, c : aaa c Add the boxes of T 2 one at a time (first adding all boxes labeled a, then all those labeled b etc.) such that the augmented T 1 diagram is a valid

12 Young Tableau at every stage. Additionally one must make sure that boxes containing the same label (e.g a) never appear in the same vertical column of the augmented diagram. At any given box position, one defines n a to be the number of a s above and to the right of it. Similarly we define n b, n c etc. Then one must satisfy n a n b n c etc. Two final tableaux that have the same shape are counted as different only if the indices a, b, c etc are distributed differently in them. From any final Young tableau one shd remove columns of N boxes in SU(N), since they correspond to a singlet or a scalar. For an explicit example 8 8 in SU(3) consult your lecture notes. E.12 Use the form of the Pauli matrices and the definition of a unit vector to show that an SU(2) matrix U = exp [ i2 ] σ.n

13 can be written as ( ) 1 cos 2 σ.n isin ( 1 2 σ.n ) E.13 Calculate the dimensionality of the following Young Tableaux for SU(6) (Answers : 35, 21, 15 and 20 respectively) P.6 Calculate the Clebsch-Gordan decomposition for the direct product 6 8 in SU(3) by considering the depiction below, in terms of Young Tableaux :