In this chapter, we studied the Lie algebra of rotations in great detail, but didn't construct the Lie group through exponentiation, except in some limited examples. In this problem, we'll study the structure of the honest rotation group, focusing on its two-dimensional representation.

(a) A general element of the two-dimensional representation of the rotation group \(\mathbb{R}\) can be expressed through exponentiation of the spin-1/2 matrices as

\[\begin{equation*}\mathbb{U}=e^{\frac{i}{\hbar}\left(\theta_{x} \hat{S}_{x}+\theta_{y} \hat{S}_{y}+\theta_{z} \hat{S}_{z}\right)} \tag{8.140}\end{equation*}\]

where \(\theta_{x}, \theta_{y}, \theta_{z}\) are real-valued angles of rotation about their respective axis. By construction, such a matrix is unitary, as the spin operators are Hermitian. Show that the determinant of this matrix is 1 .

Remember, the determinant of a matrix is basis-independent.

(b) With part (a), this shows that the group of rotations is isomorphic to the group of \(2 \times 2\) unitary matrices with unit determinant, which is called \(\mathbf{S U ( 2 )}\). We can construct a general complex-valued \(2 \times 2\) unitary matrix in the following way. Let's first construct two orthonomal complex vectors, \(\vec{v}_{1}\) and \(\vec{v}_{2}:\)

\[\vec{v}_{1}=\left(\begin{array}{c}e^{i \xi_{1}} \cos \theta \tag{8.141}\\e^{i \xi_{2}} \sin \theta\end{array}\right), \quad \vec{v}_{2}=\left(\begin{array}{c}-e^{i \xi_{3}} \sin \theta \\e^{i \xi_{4}} \cos \theta\end{array}\right)\]

for some real angle \(\theta\) and real phases \(\xi_{1}, \xi_{2}, \xi_{3}, \xi_{4}\). Show that these vectors are unit normalized and determine the constraint on the phases such that they are orthogonal: \(\vec{v}_{1}^{\dagger} \vec{v}_{2}=0\). With this constraint on \(\xi_{1}, \xi_{2}, \xi_{3}, \xi_{4}\) imposed, show that the matrix formed from these vectors as columns is unitary. That is, show that

\[\mathbb{U}=\left(\begin{array}{ll}\vec{v}_{1} & \vec{v}_{2} \tag{8.142}\end{array}\right)\]

is a unitary matrix.

(c) Next, impose the unit determinant constraint on the matrix \(\mathbb{U}\). What further restriction does this impose on the phases \(\xi_{1}, \xi_{2}, \xi_{3}, \xi_{4}\) ?

(d) A general complex-valued \(2 \times 2\) matrix can be represented as

\[\mathbb{U}=\left(\begin{array}{ll}a_{11}+i b_{11} & a_{12}+i b_{12} \tag{8.143}\\a_{21}+i b_{21} & a_{22}+i b_{22}\end{array}\right)\]

where \(a_{i j}\) and \(b_{i j}\) are real numbers. Determine all values of the \(a_{i j} \mathrm{~s}\) and \(b_{i j} \mathrm{~S}\) for the \(2 \times 2\) unitary matrix with unit determinant that you constructed in part (c).

(e) In terms of \(a_{11}, b_{11}, a_{21}\), and \(b_{21}\), express the normalization of vector \(\vec{v}_{1}, \vec{v}_{1}^{\dagger} \vec{v}_{1}=1\). What space does this describe? The parametrization of this

space with the vectors as in Eq. (8.141) is called the Hopf fibration \({ }^{4}\) and demonstrates that the group \(\mathrm{SU}(2)\) has a very beautiful geometric structure.