Momentum-pair counting in bosonic diagonalization

For a translation-invariant quadratic bosonic quantum field, anomalous terms couple $\mathbf{k}$ to $-\mathbf{k}$. A full-momentum sum uses half the pair-creation coefficient in each term, because both labels describe the same anomalous pair. After a canonical Bogoliubov transformation, however, the two members carry distinct quasiparticle occupations; each contributes its own oscillator energy. The vacuum correction is $\frac{1}{2}{\sum }_{k\ne 0}(E_k-A_k)$ in a full-zone sum. Switching to one representative from each pair requires changing both the occupation sum and constant consistently.