In Coulomb gauge, the equation of motion for \( \vec{A} \) is: \( \partial_{\mu} \partial^{\mu} \vec{A}=0 \), which we can solve in the usual way: \( \vec{A}=\int \frac{d^{3} p}{(2 \pi)^{3}} \vec{\xi}(\vec{p}) e^{i p \cdot x} \) with \( p_{0}^{2}=|\vec{p}|^{2} \). The constraint \( \nabla \cdot \vec{A}=0 \) tells us that \( \vec{\xi} \) must satisfy: \( \cdot \vec{p}=0 \), which means that \( \vec{\xi} \) is perpendicular to the direction of motion \( \vec{p} \). We can pick \( \vec{\xi}(\vec{p}) \) to be a linear combination of two orthonormal vectors \( \vec{\epsilon}_{r}, r=1,2, \) each of which satisfies \( \vec{\epsilon}_{r}(\vec{p}) \cdot \vec{p}=0 \) and \( \vec{\epsilon}_{r}(\vec{p}) \cdot \vec{\epsilon}_{s}(\vec{p})=\delta_{r s} , \; r, s=1,2 \).
