Ultraviolet divergence of an instantaneous vector-potential quench ($|v_{\mathbf{k}}{|}^2=O(k^{-2})$)

For a nonzero instantaneous homogeneous vector-potential step, scalar-field mixing has $|v_{\mathbf{k}}{|}^2\sim [e{\mathbf{A}}_0\cdot \hat{\mathbf{k}}{]}^2/(4{\hslash }^2{k}^2)$ at large wavevector. Its three-dimensional momentum integral diverges. Per-mode squeezed-state formulas remain meaningful, but at finite quantization volume the mode product needs an ultraviolet regulator or a sufficiently smooth physical pulse. Infinite-volume representations require a separate limit.