Canonical transformation of mode operators ($a_i^{\prime }={\sum }_j(U_{ij}{a}_j+V_{ij}{a}_j^{†})$)

A linear change of creation operators and annihilation operators is canonical when it preserves their defining brackets. For finitely many bosonic modes, the canonical commutation relations require $U{U}^{†}-V{V}^{†}=I$ and $U{V}^T-V{U}^T=0$. For fermions, the canonical anticommutation relations instead require $U{U}^{†}+V{V}^{†}=I$ and $U{V}^T+V{U}^T=0$. A Bogoliubov transformation uses such mixing to diagonalize a quadratic Hamiltonian or relate different vacuum descriptions. Infinite-mode unitary implementation has further conditions, as in bosonic mode mixing implementability.

A Bogoliubov transformation is a canonical transformation of mode operators mixing creation operators and annihilation operators. It diagonalizes quadratic Hamiltonians or relates different vacuum descriptions. Bosonic and fermionic canonical bracket constraints differ. A nonzero creation-mixing coefficient makes its vacuum contain particles relative to the original mode decomposition.

A Nambu spinor groups an annihilation operator with an opposite-momentum creation operator to express pairing in a quadratic Hamiltonian as a matrix problem. Its components are not independent: the spinor at $-k$ is related to conjugated components at $k$. This redundancy must be accounted for when summing energies and normal ordering constants. Component phases are a convention and change the displayed off-diagonal matrix entries.

In a unit-spacing periodic spinless chain, self-paired momenta are $k=0$ and, when available, $k=\pi$. An anomalous pair of the same creation operator vanishes by the canonical anticommutation relations. Such a mode is a single normal number level, filled or empty according to its energy sign, rather than a two-distinct-mode Bogoliubov transformation. This excludes the paired-angle construction at those special momenta.

In a bosonic Nambu spinor expression, the second diagonal entry commonly contains $a_{-k}{a}_{-k}^{†}$. The canonical commutation relations add one constant per mode when this is normal ordered. For an antiferromagnetic chain coefficient $JS$, rewriting the quadratic Hamiltonian as a full Nambu sum therefore changes the exterior constant from $-NJ{S}^2$ to $-NJS(S+1)$. Omitting the shift corrupts the ground-state energy even when the dispersion is unchanged.

With Fourier convention $c_n=N^{-1/2}{\sum }_k{e}^{-ikn}{c}_k$, the Transverse-field Ising model has $H=J{\sum }_k{\Psi }_k^{†}[(g-\cos k){\sigma }_z-\sin k{\sigma }_x]{\Psi }_k$. Expanding the spinor retains the constant $-JgN$ and pairs $k$ with $-k$. The physical quasiparticle energy is twice the positive matrix eigenvalue times $J$, because both momentum partners occur in the Nambu sum.

A bounded invertible bosonic Bogoliubov transformation is unitarily implementable in the usual bosonic Fock space precisely when its creation-mixing part is a Hilbert-Schmidt operator. Without this condition, the transformation still preserves the algebraic canonical commutation relations, but its two vacua need not be vectors in a common Fock space. Equal nonzero squeezing in countably many independent modes is a simple counterexample.

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.

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.

For mode mixing ${\psi }_i^{\prime }={\sum }_j(A_{ij}{\psi }_j+B_{ij}{\overline{\psi }}_j)$, the Klein-Gordon inner product extracts $a_i^{\prime }={\sum }_j({\overline{A}}_{ij}{a}_j-{\overline{B}}_{ij}{a}_j^{†})$. The canonical commutation relation is preserved exactly when $A{A}^{†}-B{B}^{†}=I$ and $A{B}^T=B{A}^T$. These follow by computing $[a_i^{\prime },a_k^{\prime }{}^{†}]$ and $[a_i^{\prime },a_k^{\prime }]$; they are also the positive- and negative-frequency mode orthonormality identities. In infinitely many modes, convergence and the existence of a common bosonic Fock space representation require additional analysis; the expected total particle number is finite only if $B$ is a Hilbert-Schmidt operator.

If a background is stationary in the remote past and future, positive-frequency modes in those two regions define an in-vacuum and an out-vacuum. The intervening time dependence generally makes the two mode bases differ by a Bogoliubov transformation.

The state annihilated by every annihilation operator of the positive-frequency mode basis chosen in the asymptotic future. It need not coincide with the in-vacuum on a time-dependent spacetime.

The state annihilated by every annihilation operator of the positive-frequency mode basis chosen in the asymptotic past. A Bogoliubov transformation can make it contain particles when tested by future out-mode number operators.

Ifthen the expected out-particle number in mode $i$ in the in-vacuum is

For an oscillator whose positive frequency jumps from ${\omega }_{-}$ to ${\omega }_{+}$, continuity of the normalized mode and its first derivative gives the Bogoliubov coefficientsThe initial vacuum therefore contains $|\beta {|}^2$ final particles in that mode.