4. In the Cartesian coordinate system $x O y$, points $A, B$ are two moving points on the right branch of the hyperbola $x^{2}-y^{2}=1$. Then the minimum value of $\overrightarrow{O A} \cdot \overrightarrow{O B}$ is
(Wang Huixing provided the problem)
4. 1 .

For a point $(x, y)$ on the right branch of the hyperbola, from $x^{2}-y^{2}=1(x>0)$, it is easy to know $|y|<x$.

Let $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$, then $x_{1}^{2}-y_{1}^{2}=1, x_{2}^{2}-y_{2}^{2}=1\left(x_{1}>0, x_{2}>0\right)$, i.e., $\left(x_{1}+y_{1}\right)\left(x_{1}-\right.$ $\left.y_{1}\right)=1,\left(x_{2}+y_{2}\right)\left(x_{2}-y_{2}\right)=1$, and $x_{i} \pm y_{i}>0(i=1,2)$. Therefore,
$$
\begin{aligned}
\overrightarrow{O A} \cdot \overrightarrow{O B} & =\left(x_{1}, y_{1}\right) \cdot\left(x_{2}, y_{2}\right)=x_{1} x_{2}+y_{1} y_{2} \\
& =\frac{\left(x_{1}+y_{1}\right)\left(x_{2}-y_{2}\right)+\left(x_{1}-y_{1}\right)\left(x_{2}+y_{2}\right)}{2} \\
& \geqslant \sqrt{\left(x_{1}+y_{1}\right)\left(x_{2}-y_{2}\right)\left(x_{1}-y_{1}\right)\left(x_{2}+y_{2}\right)} \\
& =\sqrt{\left(x_{1}^{2}-y_{1}^{2}\right)\left(x_{2}^{2}-y_{2}^{2}\right)}=1 .
\end{aligned}
$$

The equality holds when $A, B$ are symmetric with respect to the $x$-axis (i.e., $x_{1}=x_{2}, y_{1}=-y_{2}$).
Therefore, the minimum value of $\overrightarrow{O A} \cdot \overrightarrow{O B}$ is 1.