Bosonic mode mixing implementability (${\sum }_{i,j}|{\beta }_{ij}{|}^2<\infty$)

A bounded invertible bosonic Bogoliubov transformation is unitarily implementable in the usual bosonic Fock space precisely when its creation-mixing part is a Hilbert-Schmidt operator. Without this condition, the transformation still preserves the algebraic canonical commutation relations, but its two vacua need not be vectors in a common Fock space. Equal nonzero squeezing in countably many independent modes is a simple counterexample.

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.