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 Bogoliubov vacuum is annihilated by the quasiparticle linear operators that diagonalize a stable quadratic bosonic Hamiltonian. If $a_k=u_k{b}_k-v_k{b}_{-k}^{†}$, its original-mode occupation number is $|v_k{|}^2$. It is generally a multimode squeezed vacuum rather than the original Fock vacuum. In infinitely many modes, existence in the original Fock space requires a separate convergence analysis.

For a spinless Bogoliubov transformation $c_p=\cos {\theta }_p{\alpha }_p+i\sin {\theta }_p{\alpha }_{-p}^{†}$, consistency of the doubled Nambu spinor requires $\cos {\theta }_{-p}=\cos {\theta }_p$ and $\sin {\theta }_{-p}=-\sin {\theta }_p$. Thus choose ${\theta }_{-p}=-{\theta }_p$ modulo $2\pi$. The cross anticommutator from the first-row formulas is $\{c_p,c_{-p}\}=i\sin ({\theta }_p+{\theta }_{-p})$; a nontrivial even angle would violate the canonical anticommutation relations. A self-paired zero mode must instead have vanishing mixing in this convention. Real-angle unitarity within each two-by-two block alone does not enforce consistency between duplicated momentum labels.

A real two-mode Bogoliubov transformation preserving the full canonical anticommutation relations. The self anticommutators become $\{a,a^{†}\}=\{b,b^{†}\}=u^2+v^2$, while $\{a,b\}=-uv+uv=0$ and every other cross anticommutator vanishes. Thus its matrix is an orthogonal matrix. For the two-orbital fermionic pairing Hamiltonian, let $E=\sqrt{{\epsilon }^2+{\Delta }^2}$, take $u=\sqrt{(E+\epsilon )/(2E)}$ and $v=\sqrt{(E-\epsilon )/(2E)}$, and obtain $H=E(c^{†}c+d^{†}d)+\epsilon -E$. Inverting gives $c=ua-v{b}^{†}$, $d=v{a}^{†}+ub$.

The normalized vacuum state annihilated by $c=ua-v{b}^{†}$ and $d=v{a}^{†}+ub$, for positive $u,v$ with $u^2+v^2=1$. Since $(a^{†}{b}^{†}{)}^2=0$, it can also be written $u\exp [(v/u)a^{†}{b}^{†}]|0⟩$. Acting with either annihilation operator makes the two terms cancel. Its original occupation numbers each have expected value $v^2$ and its pair amplitude is $⟨ab⟩=-uv$. It has even fermion parity although the original total number operator is not sharp. For the two-orbital fermionic pairing Hamiltonian, it is the unique ground state of energy $\epsilon -\sqrt{{\epsilon }^2+{\Delta }^2}$.

An ordered-reference spin-wave calculation has zero-energy collective rotation modes. Their squeezing parameters become singular; a finite-size or symmetry-breaking regulator is required rather than treating them as ordinary normalized oscillator vacua.

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.

For real even coefficients, ${\Psi }_k=(a_k,a_{-k}^{†}{)}^{\mathsf{T}}$ and a full-zone sum, this expression equals $C+{\sum }_k[A_k{a}_k^{†}{a}_k+B_k(a_k{a}_{-k}+a_k^{†}{a}_{-k}^{†})/2]$. The constant subtraction follows from $a_{-k}{a}_{-k}^{†}=1+a_{-k}^{†}{a}_{-k}$. Pairing redundancy and normal ordering determine both factors together. Switching to half a zone requires grouping both members of each pair; changing only a diagonal factor is inconsistent.

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.

If the traceless matrix $h_k$ has eigenvalues $\pm d_k$ and the Nambu spinor includes an opposite-momentum creation operator, a full-zone sum contains both partners. Its physical quasiparticle energy is $2J{d}_k$, not $J{d}_k$. The negative term becomes a vacuum constant only after normal ordering. In the Ising-chain Nambu spinor, $d_k=\sqrt{1+g^2-2g\cos k}$. A half-zone calculation instead groups one complete two-mode block; changing the momentum range without changing the normalization is inconsistent.

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 a nonzero instantaneous homogeneous vector-potential step, scalar-field mixing has $|v_{\mathbf{k}}{|}^2\sim [e{\mathbf{A}}_0\cdot \hat{\mathbf{k}}{]}^2/(4{\hslash }^2{k}^2)$ at large wavevector. Its three-dimensional momentum integral diverges. Per-mode squeezed-state formulas remain meaningful, but at finite quantization volume the mode product needs an ultraviolet regulator or a sufficiently smooth physical pulse. Infinite-volume representations require a separate limit.

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.

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.

For $a_{\mathrm{out}}=\alpha a_{\mathrm{in}}+\beta b_{\mathrm{in}}^{†}$ with real canonical coefficients, the inverse is $a_{\mathrm{in}}=\alpha a_{\mathrm{out}}-\beta b_{\mathrm{out}}^{†}$. The in-vacuum annihilation condition therefore has coefficient $v=-\beta$ when written as $(u{a}_{\mathrm{out}}+v{b}_{\mathrm{out}}^{†})|0_{\mathrm{in}}⟩=0$. Forward and inverse maps must not be confused.

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

If $b_s={\sum }_j(A_{sj}{a}_j+B_{sj}{a}_j^{†})$ and the state is the in-vacuum, put $n_s={\sum }_j|B_{sj}{|}^2$ and $c_s={\sum }_j{A}_{sj}{B}_{sj}$. The canonical commutation relations give $N_s^2=b_s^{†2}{b}_s^2+N_s$. Acting with $b_s^2$ on the vacuum produces $c_s|0⟩$ and a two-particle vector whose squared norm is $2n_s^2$, because its two possible vacuum contractions have equal contributions. This proves the displayed second moment and $\mathrm{Var}(N_s)=n_s(n_s+1)+|c_s{|}^2$. A thermal-mode formula requires $c_s=0$; a general squeezed vacuum state need not satisfy that restriction.

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.