2. The line $l$ passing through the origin intersects the hyperbola $x y=-2 \sqrt{2}$ at points $P$ and $Q$, with point $P$ in the second quadrant. If the lower half-plane is folded along the $x$-axis to form a right dihedral angle with the upper half-plane, then the shortest length of segment $P Q$ is ( ).
(A) $2 \sqrt{2}$
(B) $3 \sqrt{2}$
(C) $4 \sqrt{2}$
(D) 4
2. D.

Let point $P(x, y)$. By symmetry, point $Q(-x, -y)$. Draw perpendiculars from points $P$ and $Q$ to the $x$-axis, with feet at $P_{1}(x, 0)$ and $Q_{1}(-x, 0)$, respectively.
After folding into a right dihedral angle,
$$
\begin{array}{l}
|P Q|^{2}=\left|P P_{1}\right|^{2}+\left|P_{1} Q_{1}\right|^{2}+\left|Q_{1} Q\right|^{2} \\
=y^{2}+(2 x)^{2}+(-y)^{2} \\
=4 x^{2}+2\left(\frac{-2 \sqrt{2}}{x}\right)^{2} \\
=4 x^{2}+\frac{16}{x^{2}} \geqslant 2 \sqrt{64}=16 .
\end{array}
$$

When $x=\sqrt{2}$, $|P Q|_{\text {min }}=4$.