= 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 $\frac12\sum_{k\ne0}(E_k-A_k)$ in a full-zone sum. Switching to one representative from each pair requires changing both the occupation sum and constant consistently.
Back to article page
