Possible value of angles in parametrization of $2\times 2$ unitary matrices

Question

I am doing parametrization of $2\times 2$ matrices from book 'The Unitary and Rotation Groups' by F.D. Murnaghan. Writing the unitary matrix as $U = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$ and applying the condition $U^\dagger U = I_{2\times 2}$ we get equations like $$a^*a + b^*b = 1$$ $$c^*c + d^*d = 1$$ $$a^*c + b^*d = 0$$ where $a, b, c, d \in \mathbb{C}$.

So, we can write for the first equation: $$ a = e^{\iota \alpha_1}\cos\phi$$ and $$ b = e^{\iota \alpha_2}\sin \phi$$ Then what are the bounds on the values of $\alpha_1, \alpha_2 \ \& \ \phi ?$ I can understand that $\phi$ can vary from $-\pi$ to $\pi$, but so should $\alpha_1$ and $\alpha_2$. But the book says that the latter angles can have values between $-\pi/2$ to $\pi/2$ and terms them as latitude angles, while $\phi$ as longitude angle. Why is there a constraint on the value of $\alpha$'s?

Answer 1

The problem with taking $a = e^{i \alpha_1} \cos \phi$ where $\alpha_1$ and $\phi$ are both taken from $-\pi$ to $\pi$ is that it's a redundant parameterization; that is, the same $a$ corresponds to multiple possible combinations of $\phi$ and $\alpha_1$. For example, $a = -1$ can be reached either by using $\alpha_1 = 0$ and $\phi = \pi$ or $\alpha_1 = \pi$ and $\phi = 0$.

In order to avoid this redundancy, we need to either restrict the possible values of $\phi$ or the possible values of $\alpha_1$. In this case, the author has decided to restrict the possible values of $\alpha_1$. A similar situation arises with $b = e^{i \alpha_2} \sin \phi$.

Another context where this kind of redundancy in parameterization arises is in the definition of spherical coordinates.