= Bosonic mode mixing implementability
{title2=$\sum_{i,j}|\beta_{ij}|^2<\infty$}
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.
Back to article page
