 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2014/paper-1/6/b/solution

At first order write

α = 1 + i z

and

β = x + i y

with real

x, y, z

. The constraint forces the real part of the change in

α

to vanish. Then

m = (i z x - i y x + i y - i z) = 2 x m_{1} - 2 y m_{2} + 2 z m_{3} .

Thus the three supplied generators span the tangent space over the reals. Multiplying their matrices gives

[m_{1}, m_{2}] = m_{3}, [m_{2}, m_{3}] = - m_{1}, [m_{3}, m_{1}] = - m_{2} .

For example,

m_{1} = σ_{x} /2

m_{2} = σ_{y} /2

m_{3} = i σ_{z} /2

makes the noncompact signs explicit.

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