= Canonical identities for a bosonic Bogoliubov transformation
{title2=$AA^\dagger-BB^\dagger=I,\quad AB^T=BA^T$}
For mode mixing $\psi'_i=\sum_j(A_{ij}\psi_j+B_{ij}\overline\psi_j)$, the <Klein-Gordon inner product> extracts $a'_i=\sum_j(\overline A_{ij}a_j-\overline B_{ij}a_j^\dagger)$. The <canonical commutation relation> is preserved exactly when $AA^\dagger-BB^\dagger=I$ and $AB^T=BA^T$. These follow by computing $[a'_i,a_k'{}^\dagger]$ and $[a'_i,a'_k]$; they are also the positive- and negative-frequency mode orthonormality identities. In infinitely many modes, convergence and the existence of a common <bosonic Fock space> representation require additional analysis; the expected total particle number is finite only if $B$ is a <Hilbert-Schmidt operator>.
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