5. Direct Products The basis of a system may be the direct product of other basis { j } if The system consists of more than one particle. More than one degrees of freedom of a single particle are considered. Group representation U w.r.t. is therefore a direct product of representations {Uj } w.r.t. { j }. i.e. 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 D3 Symmetry E.g., 2 valence electrons in a molecule with D3 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: S2 ~ C2 Ground state of the 3 electrons of Li : S3 ~ D3 E : Mathematica see Eg.17.6.2
Contruction of Let the spin part transforms like the th IR of Sn with basis . ( This fixes the multiplicity of ) Let where { i } is a basis for representation U ( ) with . i.e. Furthermore, let i.e. U ( ) is the dual of U () U ( ) is an IR since U () is.
U ( ) is unitary i.e., transforms like A2 . ( Fermionic ) i can be generated from any n-particle function using the projector method.
Example 17.6.2. Construction of Many-Body Spatial Functions S3 ~ D3 D3 I C3 C32 C2 C2 C2 S3 P(312) P(231) P(132) = p(32) P(321) = p(31) P(213) = p(21) P(abc) = {1a, 2b, 3c } p(ab) = { a b } Let Results already listed in E.g.17.6.1. Mathematica
7. Continuous Groups Special Rotations in 2-D : Orthogonal Rotations in n-D : SO(n) = Lie group of order n(n1)/2. ( indep. elements in nn SO matrix ) { R() } = Fundamental representation Generalization to complex vector space: Special Unitary = Lie group of order n21. ( indep. real parameters in nn 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. ( eq.17.38 is the passive version ) Rotations about a fixed axis : § 2.2, Euler identity :
SO(n) & SU(n) Sj are hermitian U unitary Let i be the eigenvalues of U : Sj are traceless
Let & multiplication is closed f j k l = structure constants Set 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 = ( I3 , 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), L2 is the Casmir operator.
SO(2) & SO(3) For SO(2) For SO(3) §16.4 : see next page
Basis { x, y } Basis = { i , j } Using functions { x, y } as basis : generator for V is Alternatively,
Example 17.7.2. Generators Depend on Basis
SU(2) & SU(2)-SO(3) Homomorphism
SU(3)
Example 17.7.3. Quantum Numbers of Quarks
Example 17.7.4. Quark Ladders
Example 17.7.5. Generators for Direct Products
Example 17.7.6. Decomposition of Baryon Multiplets
8. Lorentz Group
Homogeneous Lorentz Group
Example 17.8.1. Addition of Collinear Velocities
Minkowski Space
9. Lorentz Covariance of Maxwell’s Equations
Lorentz Transformation of E & B
Example 17.9.1. Transformation to Bring Charge to Rest
10. Space Groups
Example 17.10.1. Tiling a Floor