 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/3/e/solution

c = cos (2 α_{1})

and

s = sin (2 α_{1})

. Multiplying the matrices, or using the Pauli matrix multiplication law, gives

A_{1}^{†} τ_{1} A_{1} = τ_{1}, A_{1}^{†} τ_{2} A_{1} = c τ_{2} + s τ_{3}, A_{1}^{†} τ_{3} A_{1} = c τ_{3} - s τ_{2} .

i τ_{j}

M^{'}

yields

x_{1}^{'} x_{2}^{'} x_{3}^{'} = 100 0 c s 0 - s c x_{1} x_{2} x_{3} .

This is the right-handed active rotation by

2 α_{1}

about the positive

x_{1}

axis. Likewise, with

c_{j} = cos (2 α_{j})

and

s_{j} = sin (2 α_{j})

A_{2} : (x_{1}, x_{2}, x_{3}) \mapsto (c_{2} x_{1} + s_{2} x_{3}, x_{2}, - s_{2} x_{1} + c_{2} x_{3}),

A_{3} : (x_{1}, x_{2}, x_{3}) \mapsto (c_{3} x_{1} - s_{3} x_{2}, s_{3} x_{1} + c_{3} x_{2}, x_{3}) .

Thus the other axes also have rotation angle

2 α_{j}

. The factor of two is essential: replacing

A_{j}

- A_{j}

leaves the conjugation unchanged, illustrating the two-to-one relation between SU(2) and three-dimensional rotations.

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