Bosonic Bogoliubov zero-point shift
= Bosonic Bogoliubov zero-point shift
{title2=$\Delta E_0=\tfrac12\sum_k(E_k-A_k)$}
Canonical diagonalization of $A_kb_k^\dagger b_k$ plus an anomalous pair term gives the vacuum shift $(E_k-A_k)/2$ per momentum. Thus $H=C+\frac12\sum_k(E_k-A_k)+\sum_kE_k\alpha_k^\dagger\alpha_k$. The subtraction of $A_k/2$ accompanies the <quasiparticle> zero-point term; omitting it changes the ground-state energy.