Canonical identities for a bosonic Bogoliubov transformation ($A{A}^{†}-B{B}^{†}=I,A{B}^T=B{A}^T$)

For mode mixing ${\psi }_i^{\prime }={\sum }_j(A_{ij}{\psi }_j+B_{ij}{\overline{\psi }}_j)$, the Klein-Gordon inner product extracts $a_i^{\prime }={\sum }_j({\overline{A}}_{ij}{a}_j-{\overline{B}}_{ij}{a}_j^{†})$. The canonical commutation relation is preserved exactly when $A{A}^{†}-B{B}^{†}=I$ and $A{B}^T=B{A}^T$. These follow by computing $[a_i^{\prime },a_k^{\prime }{}^{†}]$ and $[a_i^{\prime },a_k^{\prime }]$; 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.