4. If the line $x=\frac{\pi}{4}$ is intercepted by the curve $C$:
$$
(x-\arcsin \alpha)(x-\arccos \alpha)+(y-\arcsin \alpha)(y+\arccos \alpha)=0
$$

to form a chord of length $d$, the minimum value of $d$ as $\alpha$ varies is
(A) $\frac{\pi}{4}$;
(B) $\frac{\pi}{3}$;
(C) $\frac{\pi}{2}$;
(D) $\pi$.
4. (C)

From the problem, we know that the curve $C$ is a circle with the endpoints of its diameter being $P_{1}(\arcsin \alpha, \arcsin \alpha)$ and $P_{2}(\arccos \alpha, \arccos \alpha)$. The x-coordinate of the center of the circle is
$$
x_{0}=\frac{\arcsin \alpha+\arccos \alpha}{2}=\frac{\pi}{4},
$$

Thus, the line $x=\frac{\pi}{4}$ passes through the center of the circle, and $d$ is the diameter of the circle. Also,
$$
d^{2}=2\left[(\arcsin \alpha)^{2}+(\arccos \alpha)^{2}\right] \geqslant(\arcsin \alpha+\arccos \alpha)^{2}=\frac{\pi^{2}}{4} .
$$
$\therefore d \geqslant \frac{\pi}{2}$, and when $\arcsin \alpha=\arccos \alpha$, the corresponding $d=\frac{\pi}{2}$. Therefore, the answer is (C).