Cross-pair cancellation need not diagonalize a bosonic Hamiltonian
= Cross-pair cancellation need not diagonalize a bosonic Hamiltonian
For a two-mode quadratic <Hamiltonian>, a transformation can remove $AB+A^\dagger B^\dagger$ while leaving $A^{\dagger2}$ or $B^{\dagger2}$ terms. Such terms create quanta from the proposed vacuum, so their absence must also be checked before identifying that vacuum with an <energy> <eigenstate>. Coordinate coupling $kxy$ contains both pair and number-conserving terms and supplies an explicit example.