5 Direct Products The basis of a system

5. Direct Products The basis of a system may be the direct product of other basis { j } if 1. The system consists of more than one particle. 2. More than one degrees of freedom of a single particle are considered. Group representation U w. r. t. is therefore a i. e. direct product of representations {Uj } w. r. t. { j }. U is in general reducible. Taking the trace of U(g) gives Decomposition of U is easily done using the charcter table of G.

Example 17. 5. 1. Even-Odd Symmetry Consider system of n particles in potential where Let there be k particles in Au , then i. e. , is always irreducible.

Example 17. 5. 2. 2 Particles with D 3 Symmetry D 3 E. g. , 2 valence electrons in a molecule with D 3 symmetry. Let both particles be in states of E symmetry ( basis = ). Let The direct product basis is Projectors Mathematica

6. Symmetric Groups Group of permutations of n objects = Symmetric group Sn. Consider a system of n identical (indistinguishable) particles. Let Pi j be operator that interchanges the positions of the i & j particles. Bosons Fermions P Sn For scalar , only 1 -D representations of Sn are needed. Group treatment is essential only for spinor .

Example 17. 6. 1. 2 & 3 Fermions Ground state of the 2 electrons of He : Space part even Spin part odd Streamlined notation: S 2 ~ C 2 Ground state of the 3 electrons of Li : S 3 ~ D 3 E: Mathematica see Eg. 17. 6. 2

Contruction of Let the spin part transforms like th IR of Sn with basis . ( This fixes the multiplicity of ) where { i } is a basis for representation U ( ) with Let i. e. Furthermore, let i. e. U ( ) is the dual of U ( ) is an IR since U ( ) is. .

U ( ) is unitary i. e. , transforms like A 2. ( Fermionic ) i can be generated from any n-particle function using the projector method.

Example 17. 6. 2. Construction of Many. Body Spatial Functions S 3 ~ D 3 C 2 D 3 I C 3 2 C 2 S 3 I P(312) P(231) P(132) P(321) P(213) = p(32) = p(31) = p(21) P(abc) = {1 a, 2 b, 3 c } p(ab) = { a b } Let Results already listed in E. g. 17. 6. 1. Mathematica

7. Continuous Groups Special Orthogonal Rotations in 2 -D : Rotations in 3 -D : Rotations in n-D : SO(n) = Lie group of order n(n 1)/2. ( indep. elements in n n SO matrix ) { R( ) } = Fundamental representation Generalization to complex vector space: Special Unitary = Lie group of order n 2 1. ( indep. real parameters in n n SU matrix ) Used in classification of elementary particles

Lie Groups & Their Generators Lie group of order n = group that is also an n-D differentiable manifold. ( group elements have local 1 -1 map to region in Rn. ) ~ group with continuous parameters over finite n-D region(s). For elements close to I, Sj = generators for elements “connected” to I. &

Example 17. 7. 1. SO(2) Generator active rep. Rotations about a fixed axis : ( eq. 17. 38 is the passive version ) § 2. 2, Euler identity :

SO(n) & SU(n) U unitary Sj are hermitian Let i be the eigenvalues of U : Sj are traceless

Let & multiplication is closed Set f j k l = structure constants Can be used to define “identity component” of G. ( f j k l is antisymmetric in its indices. )

rank of G = max # of mutually commuting independent generators. Basis of IRs of G are labelled using the eigenvalues of such set of generators. rank of G = # of indices needed to label the basis of an IR. E. g. , SO(n) & SU(n) ~ generated by generalized angular momenta For SO(3), rank = 1 IR label = ML. For SU(2), rank = 1 IR label = MS. For SU(3), rank = 2 IR label = ( I 3 , Y ). Casmir operator = operator that commutes with all generators of G. IRs of G are labelled using the eigenvalues of the Casmir operator(s). For SO(3), L 2 is the Casmir operator.