= Canonical transformation of mode operators
{title2=$a'_i=\sum_j(U_{ij}a_j+V_{ij}a_j^\dagger)$}
A linear change of <creation operators> and <annihilation operators> is canonical when it preserves their defining brackets. For finitely many bosonic modes, the <canonical commutation relations> require $UU^\dagger-VV^\dagger=I$ and $UV^T-VU^T=0$. For fermions, the <canonical anticommutation relations> instead require $UU^\dagger+VV^\dagger=I$ and $UV^T+VU^T=0$. A <Bogoliubov transformation> uses such mixing to diagonalize a quadratic <Hamiltonian> or relate different vacuum descriptions. Infinite-mode <unitary> implementation has further conditions, as in <bosonic mode mixing implementability>.
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