Bosonic Bogoliubov diagonalization (${\omega }_k=\sqrt{A_k^2-B_k^2}$)

For real even coefficients with $A_k>|B_k|$, the quadratic Hamiltonianis diagonalized by the Bogoliubov transformation $a_k=\cosh {\theta }_k{\alpha }_k-\sinh {\theta }_k{\alpha }_{-k}^{†}$, with $\tanh 2{\theta }_k=B_k/A_k$. The canonical commutation relations follow from ${\cosh }^2{\theta }_k-{\sinh }^2{\theta }_k=1$. The diagonal form is ${\sum }_k[{\omega }_k{\alpha }_k^{†}{\alpha }_k+({\omega }_k-A_k)/2]$, 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.

A quadratic bosonic Hamiltonian contains terms of degree two in bosonic creation operators and bosonic annihilation operators, with an optional constant. The number-conserving matrix $A$ is Hermitian and the pair matrix $B$ may be taken symmetric. Positive stable forms can be diagonalized by a canonical Bogoliubov transformation; the strict stability condition matters at zero modes.

For $H=\alpha a^{†}a+\Delta (a^2+a^{†2})/2$, a finite real squeeze requires $\alpha >|\Delta |$. At $\alpha =\Delta >0$, the Hamiltonian instead equals $\alpha Q^2-\alpha /2$, with $Q=(a+a^{†})/\sqrt{2}$. The formal oscillator frequency is zero, but this operator is not constant: $⟨2|H|0⟩=\alpha /\sqrt{2}$. Thus a normalizable oscillator ground state and finite canonical squeeze cannot be obtained by simply substituting zero frequency into the positive-frequency formula.

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.

For a two-mode quadratic Hamiltonian, a transformation can remove $AB+A^{†}{B}^{†}$ while leaving $A^{†2}$ or $B^{†2}$ 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.

Canonical diagonalization of $A_k{b}_k^{†}{b}_k$ plus an anomalous pair term gives the vacuum shift $(E_k-A_k)/2$ per momentum. Thus $H=C+\frac{1}{2}{\sum }_k(E_k-A_k)+{\sum }_k{E}_k{\alpha }_k^{†}{\alpha }_k$. The subtraction of $A_k/2$ accompanies the quasiparticle zero-point term; omitting it changes the ground-state energy.