= Bosonic Bogoliubov diagonalization
{title2=$\omega_k=\sqrt{A_k^2-B_k^2}$}
For real even coefficients with $A_k>|B_k|$, the quadratic <Hamiltonian>
$$
H=\sum_k\left[A_ka_k^\dagger a_k+\frac{B_k}{2}(a_ka_{-k}+a_k^\dagger a_{-k}^\dagger)\right]
$$
is diagonalized by the <Bogoliubov transformation> $a_k=\cosh\theta_k\,\alpha_k-\sinh\theta_k\,\alpha_{-k}^\dagger$, with $\tanh2\theta_k=B_k/A_k$. The <canonical commutation relations> follow from $\cosh^2\theta_k-\sinh^2\theta_k=1$. The diagonal form is $\sum_k[\omega_k\alpha_k^\dagger\alpha_k+(\omega_k-A_k)/2]$, with $\omega_k=\sqrt{A_k^2-B_k^2}$. The original-mode vacuum depletion is $\sinh^2\theta_k=(A_k/\omega_k-1)/2$. At $A_k=|B_k|$, the exact zero mode requires separate infrared treatment.
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