1. 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.

2. 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.

3. 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

4. 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 .

5. 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

6. 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.

7. U (  ) is unitary 
i.e.,  transforms like A ( Fermionic )
i can be generated from any n-particle function  using the projector method.

8. 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) = {1a, 2b, 3c }
p(ab) = { a  b }
Let 
Results already listed in E.g
Mathematica

9. 7. Continuous Groups Special Rotations in 2-D : Orthogonal
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 n21.
( indep. real parameters
in nn SU matrix )
Used in classification of elementary particles

10. 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.
&

11. Example 17.7.1. SO(2) Generator
active rep.
( eq is the passive version )
Rotations about a fixed axis : 
§ 2.2, Euler identity : 

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

13. 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.)

14. 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.

15. SO(2) & SO(3)
For SO(2)
For SO(3)    
§16.4 : see next page

16. Basis { x, y } Basis = { i , j } Using functions { x, y } as basis :
 generator for V is 
Alternatively,  

17. Example 17.7.2. Generators Depend on Basis

18. SU(2) & SU(2)-SO(3) Homomorphism

19. SU(3)

20. Example 17.7.3. Quantum Numbers of Quarks

21. Example Quark Ladders

22. Example 17.7.5. Generators for Direct Products

23. Example 17.7.6. Decomposition of Baryon Multiplets

24. 8. Lorentz Group

25. Homogeneous Lorentz Group

26. Example 17.8.1. Addition of Collinear Velocities

27. Minkowski Space

28. 9. Lorentz Covariance of Maxwell’s Equations

29. Lorentz Transformation of E & B

30. Example 17.9.1. Transformation to Bring Charge to Rest

31. 10. Space Groups

32. Example Tiling a Floor