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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