Physics 745 - Group Theory
Homework Set 22
Due Monday, March 30
1. It is rare we will actually use the representation matrices Γ( j ) ( R ) , but occasionally it
( )
is useful. We want to work out explicitly Γ 2 ( R ( x ) ) , generated by the generators
1
Ta = 12 σ a ,
⎛0 1⎞
⎛ 0 −i ⎞
⎛1 0 ⎞
where σ 1 = ⎜
⎟, σ2 = ⎜
⎟, σ3 = ⎜
⎟.
⎝1 0⎠
⎝i 0 ⎠
⎝ 0 −1 ⎠
We will write x in the form x = xrˆ , where r̂ is a unit vector, and x is the magnitude
of x.
2
(a) Show that ( rˆ ⋅ σ ) = 1 , where 1 is the unit matrix, for any unit vector r̂ .
Furthermore, find a simplification for ( rˆ ⋅ σ ) for arbitrary positive integer n when
n
n is even or odd.
(b) Expand out the representation
Γ
( 12 )
∞
( R ( x ) ) = exp ( iT ⋅ x ) = ∑
( iT ⋅ x )
n
n!
n=0
dividing it into even and odd terms. Simplify as much as possible
(c) Show that
Γ
( 12 )
( R ( x ) ) = cos ( x ) + i sin ( x ) σ ⋅ rˆ
1
2
1
2