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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.

Bosonic exchange symmetry is the symmetric exchange statistics of identical bosons. Commuting creation operators build the symmetric bosonic Fock space; a normalized one-particle mode has occupation states $(a^{†}{)}^n|0⟩/\sqrt{n!}$ for every nonnegative integer $n$. At thermal equilibrium this symmetry leads to Bose-Einstein statistics and the Bose-Einstein distribution. Exchange symmetry itself does not require thermal equilibrium.

For two equal-energy bosonic species with annihilation operators $a,b$, set $c_{-}=(a-b)/\sqrt{2}$ and let $N_{-}$ be its number operator. Since $[N_{-},c_{-}]=-c_{-}$, conjugation by $e^{i\pi N_{-}}$ reverses the antisymmetric combination while fixing $(a+b)/\sqrt{2}$, and therefore exchanges $a,b$ and their creation operators. For the particle and antiparticle modes of a complex scalar field this realizes charge conjugation, fixes the Fock vacuum, and squares to one. The normal-ordered generator $\int (a^{†}-b^{†})(a-b)$ equals $2N_{-}$. Writing the annihilation factors first adds an infinite constant in the continuum, which must be subtracted or regulated before defining its exponential.

Let $b_k=\cosh {\theta }_k{a}_k+\sinh {\theta }_k{a}_{-k}^{†}$ for real even ${\theta }_k$, and let every $b_k$ annihilate a normalized vacuum state. Inverting the Bogoliubov transformation gives $a_k=\cosh {\theta }_k{b}_k-\sinh {\theta }_k{b}_{-k}^{†}$. Only $⟨b_k{b}_q^{†}⟩={\delta }_{kq}$ survives, proving the displayed vacuum expectation values. Taking the Hermitian adjoint gives the same real negative pair amplitude for two creation operators. The normal occupation number is quadratic at small ${\theta }_k$, whereas the pair amplitude is linear. For one opposite-momentum pair the two-mode squeezed vacuum is $\mathrm{sech}\theta {\sum }_{n\ge 0}(-\tanh \theta {)}^n|n,n⟩$, fixing the sign convention directly.

Commuting creation operators make multiparticle states invariant under exchange of their labels. With $[a,a^{†}]=1$, the normalized number states $(a^{†}{)}^r|0⟩/\sqrt{r!}$ exist for all $r\ge 0$, giving unrestricted bosonic occupation numbers. The single-mode thermal sum is $Z={\sum }_{r\ge 0}{e}^{-\beta Er}=(1-e^{-\beta E}{)}^{-1}$ and the mean occupation is $(e^{\beta E}-1{)}^{-1}$. Thus the exchange symmetry and the Bose-Einstein distribution are two expressions of the same bosonic algebra.

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.

For the internal rotation symmetry of two real scalar fields with $\delta {\varphi }_1=\alpha {\varphi }_2$ and $\delta {\varphi }_2=-\alpha {\varphi }_1$, the Noether charge is $Q=i\int (a_1^{†}{a}_2-a_2^{†}{a}_1)d^{3}p/(2\pi {)}^3$. Its commutators satisfy $[Q,b_{\pm }^{†}]=\pm b_{\pm }^{†}$, so these creation operators produce charge eigenstates in the one-particle state sector. Substitution gives $Q=N_{+}-N_{-}$, converting the rotation of real components into opposite charges in a complex basis.

Suppose linear operators $A,B$ obey $[A,B]=cB$, where $c$ is a scalar, on a common invariant operator domain. The commutator product rule givesFor $n=0$ both sides vanish. This identity explains how powers of a creation operator shift the number operator eigenvalue.

With $d{\Pi }_p=d^{3}p/[(2\pi {)}^{3}2E_p]$, a free real scalar field has mode expansion of a free field $\varphi =\int d{\Pi }_p(a{e}^{-ipx}+a^{†}{e}^{ipx})$ and commutator $[a(p),a^{†}(q)]=(2\pi {)}^{3}2E_p{\delta }^3(\mathbf{p}-\mathbf{q})$. Substitution into the quadratic Hamiltonian density gives $H=\frac{1}{2}\int d{\Pi }_p{E}_p(a^{†}a+a{a}^{†})$. The terms with two annihilation operators or two creation operators cancel by $E_p^2={\mathbf{p}}^2+m^2$. Normal ordering removes the constant vacuum energy. The commutators $[H,a(p)]=-E_{p}a(p)$ and $[H,a^{†}(p)]=E_p{a}^{†}(p)$ give the boson energy without changing under this subtraction. Replacing $a(p)$ by $\sqrt{2E_p}c(p)$ recovers the noncovariant normalization $[c(p),c^{†}(q)]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$.

The historical interpretation fills the negative-energy electron states of the Dirac equation and uses the Pauli exclusion principle to prevent additional electrons from falling into them. Removing an electron of energy $-E$ and charge $-e$ gives a hole of excitation energy $E$ and charge $+e$, interpreted as a positron. A quantum field theory instead treats negative-frequency field coefficients as independent antiparticle creation operators in a Fock space, with positive excitation energies after normal ordering. No literal infinite material sea is needed. Bosonic antiparticles also exist, although a filled-sea exclusion argument cannot stabilize bosonic negative-energy levels.

The antisymmetric exchange statistics of identical fermions. Interchanging two one-particle labels reverses the total state, and fermionic creation operators anticommute. Occupation of a one-particle mode is therefore zero or one, giving the Pauli exclusion principle; thermal equilibrium leads to the Fermi-Dirac distribution. Color, flavor, spin and orbital factors must together have the required antisymmetry, even if one factor is symmetric.

For a normalized spinless paired ground state $u|0⟩+iv{c}_p^{†}{c}_{-p}^{†}|0⟩$, its mean fermion number is $2v^2$. The canonical Bogoliubov quasiparticle creation operator produces $c_p^{†}|0⟩$, containing exactly one fermion. With BCS coherence factors $v^2=(1-ϵ/E)/2$, the change in mean fermion number is $1-2v^2=ϵ/E$. This need not be an integer because the pairing Hamiltonian preserves fermion parity while failing to conserve the original fermion number operator. Every measurement of that number operator still has integer outcomes.

For a one-particle Hilbert space $H$, the bosonic or fermionic Fock space is ${\mathcal{F}}_{\pm }(H)={⨁}_{n=0}^{\infty }{H}^{{\otimes }_{\pm }n}$, using symmetric or antisymmetric tensor powers. Its zeroth summand is the one-dimensional vacuum sector. Creation operators and annihilation operators change particle number. The bosonic Fock space and fermionic Fock space implement the corresponding exchange statistics.

The oscillator vacuum is annihilated by all annihilation operators. Acting with creation operators constructs Fock states. In single-string quantization this is the unexcited string, not the empty spacetime state.

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.

For transverse coupling with equal mode amplitudes $\lambda$, Fermi's golden rule and the oscillator creation operator and annihilation operator givewhere the first index is the final spin and $g$ is the density of states per unit angular frequency. Decay creates a quantum and carries the factor $n_{\Omega }+1$; excitation annihilates one and carries $n_{\Omega }$. For nonuniform resonant occupations, $n_{\Omega }$ is their density-weighted mean.

With a thermal Bose-Einstein distribution, $n_{\Omega }/(n_{\Omega }+1)=e^{-\beta \hslash \Omega }$. The two-state master equation therefore has the Boltzmann distribution as its unique stationary populations when ${\Gamma }_0>0$. The population deviation relaxes at rate ${\Gamma }_0(2n_{\Omega }+1)$. If there is no resonant coupling, both rates vanish and stationarity does not determine the populations.

For a Neumann-Dirichlet open-string boundary condition, the transverse modes have frequencies $r\in \mathbb{Z}+1/2$. The canonical commutation relations give $[{\alpha }_r^i,{\alpha }_s^j]=r{\delta }^{ij}{\delta }_{r+s,0}$ and ${\alpha }_r^{†}={\alpha }_{-r}$. One derivation expands $X-y={\sum }_{r>0}{q}_r\cos (r\sigma )$: spatial orthogonality reduces the Polyakov action to $(8{\alpha }^{\prime }{)}^{-1}\int {\sum }_r({\dot{q}}_r^2-r^2{q}_r^2)d\tau$. These are harmonic oscillators of mass $1/(4{\alpha }^{\prime })$, whose normalized creation operators satisfy ${\alpha }_{-r}=i\sqrt{r}{a}_r^{†}$ in the conventional phased expansion.

For a localized quantum harmonic oscillator along $z$, let the transverse magnetic vector potential be $A_z(t)=2A_0\cos (\theta -\omega t)$ for $t>0$, with zero potential before the switch. The Coulomb gauge form of minimal coupling gives $H_1=e{A}_z{p}_z/m$. The creation operator matrix element $⟨1|p_z|0⟩=i\sqrt{m\hslash {\omega }_0/2}$ means that only $|1⟩$ is reached at first order from the ground state. Its interaction picture amplitude isIn particular $F_0=t$, so resonant finite-time expressions are nonsingular. The induced electric dipole moment is $-2e\sqrt{\hslash /(2m{\omega }_0)}\mathrm{Re}(e^{-i{\omega }_{0}t}{b}_1)$. It contains a driven component with the dynamic electric polarizability and free-frequency transients required by the initial conditions. The discontinuity in $A_z$ gives an impulsive electric field and an initial mechanical velocity $e{A}_z(0^{+})/m$ even though the initial canonical momentum has zero mean.

For a number operator eigenstate $|n⟩$, the annihilation operator and creation operator satisfy $⟨a⟩=⟨a^{†}⟩=⟨a^2⟩=⟨(a^{†}{)}^2⟩=0$, $⟨a^{†}a⟩=n$ and $⟨a{a}^{†}⟩=n+1$. Inserting $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$ and $p=i\sqrt{m\hslash \omega /2}(a^{†}-a)$ gives the displayed second moments and zero means.

Thus the kinetic energy and potential energy each have mean $\hslash \omega (n+1/2)/2$, and the uncertainty product is $\Delta x\Delta p=\hslash (n+1/2)$. Among number states only the ground state saturates the Heisenberg uncertainty relation. A quantum superposition can have nonzero position and momentum means despite each constituent number state's zero means.

The Holstein–Primakoff transformation represents a spin-$S$ operator algebra with a bosonic creation operator and annihilation operator:The order of the square root and oscillator operator matters. On the physical Fock states $|n⟩$, $0\le n\le 2S$, the raising and lowering matrix elements are $\sqrt{n(2S-n+1)}$ and $\sqrt{(n+1)(2S-n)}$. Their squared difference gives $[S^{+},S^{-}]|n⟩=2(S-n)|n⟩$, proving the spin commutation relations. Expanding the square root yields the linear spin-wave approximation and its interaction corrections.