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For independent bosonic modes, the canonical commutation relation isFor independent fermionic modes, the canonical anticommutation relations arewhere $[A,B]=AB-BA$ and $\{A,B\}=AB+BA$. The creation operator is the adjoint of the annihilation operator. In either case the mode's number operator iswith $a_i=b_i$ or $c_i$ as appropriate. It measures the mode's occupation number. Its possible eigenvalues areFor fermions, $c_i^2=0$ implies $n_i^2=c_i^{†}(1-c_i^{†}{c}_i)c_i=n_i$, so the number operator is a projection. This expresses the exclusion of double occupation of one fermionic mode. The total particle number is $N={\sum }_i{n}_i$.

Using $[x,p]=i\hslash$, the Heisenberg picture equations for the quantum harmonic oscillator areSolving these coupled linear equations givesThe oscillator annihilation operator and creation operator areThey obey $[a,a^{†}]=1$ and $H_0=\hslash \omega (a^{†}a+1/2)$. Substitution of the position and momentum solutions, or the corresponding commutators with $H_0$, yields

For a local spin quantized along its positive $\tilde{z}$ axis, the exact Holstein–Primakoff transformation isThe bosonic operators obey $[a_j,a_k^{†}]={\delta }_{jk}$. The physical local subspace is $0\le n_j\le 2S$; the larger unconstrained boson Fock space is not itself the finite spin Hilbert space. The ordering of the square roots matters, since $n_j$ does not commute with $a_j$.

This represents spin lowering using a bosonic creation operator, making a fully polarized spin state the boson vacuum. On the physical subspace it reproduces the spin commutators and the Casimir ${\tilde{\mathbf{S}}}_j^2=S(S+1)$. Expanding the square roots for small occupation relative to $S$ organizes a controlled large-$S$ linear spin-wave approximation; it is not an assumption that a spin has an infinite number of physical levels.

The Holstein–Primakoff transformation replaces each finite-dimensional spin algebra by a bosonic creation operator and annihilation operator, while restricting the local occupation number to $0\le n_j\le 2S$. The number $n_j=a_j^{†}{a}_j$ counts deviations from the fully polarized reference quantum state. Its zero-boson Fock vacuum has ${\tilde{S}}^z=S$.

The representation is exact on this physical subspace: the ordered square root gives ${\tilde{S}}^{+}|n⟩=\sqrt{n(2S-n+1)}|n-1⟩$ and ${\tilde{S}}^{-}|n⟩=\sqrt{(n+1)(2S-n)}|n+1⟩$, reproducing the spin commutation relations. For $n_j\ll S$,so the leading Hamiltonian is quadratic in bosons and describes spin waves.

The square root generates boson interactions beyond leading order and its operator ordering matters. Truncation also enlarges the Hilbert space to unphysical occupations unless the low-density approximation remains self-consistent. Large fluctuations, particularly near a gap closing or in low dimensions, can invalidate the linear spin-wave approximation even when its algebraic manipulation is straightforward.

Promote the mode expansion of a free field coefficients to annihilation operators and creation operators, replacing $a^{\ast }$ by $a^{†}$. The canonical commutation relations areTogether with the normalization in part a, they give $[\varphi (t,\mathbf{x}),\pi (t,\mathbf{y})]=i{\delta }^3(\mathbf{x}-\mathbf{y})$ and vanishing equal-time commutators between two fields or two conjugate momenta.

Insert the mode expansion of a free field in the Hamiltonian. The spatial integral produces $(2\pi {)}^3{\delta }^3(\mathbf{p}\pm \mathbf{q})$. The $aa$ and $a^{†}{a}^{†}$ coefficients cancel because ${\omega }_{\mathbf{p}}^2={\mathbf{p}}^2+m^2$. The remaining terms giveThis is a continuum of quantum harmonic oscillators. In a finite periodic volume $V$, $(2\pi {)}^3{\delta }^3(0)=V$. The Fock vacuum satisfies $a_{\mathbf{p}}|0⟩=0$, so its vacuum energy density isNormal ordering would subtract this constant; here it must be retained because it is the quantity being calculated.^[ref]

Write $d{\Pi }_p=d^{3}p/[(2\pi {)}^{3}2{\omega }_p]$, $p^{\mu }=({\omega }_p,\mathbf{p})$ and ${\omega }_p=\sqrt{{\mathbf{p}}^2+m^2}$. The Fourier expansion in the source is compatible with the oscillator canonical commutation relationsFor the free real scalar field, $\pi =\dot{\varphi }$ andSubstitution of the Fourier expansion produces spatial Dirac delta functions. Terms with two annihilation operators or two creation operators cancel: in $P_0$ their coefficient is $-{\omega }_p^2+{\mathbf{p}}^2+m^2=0$, and in $P_i$ their integral is odd under $\mathbf{p}\mapsto -\mathbf{p}$. The two mixed terms combine, with normal ordering removing the vacuum constant, to giveIn particular, $P_i$ contains the lower component $p_i=-{\mathbf{p}}_i$ in this Minkowski metric convention.

To check commutativity without manipulating a square of a Dirac delta function, define $N_f=\int d{\Pi }_{p}f(p)a_p^{†}{a}_p$. The oscillator algebra gives $[N_f,a_q]=-f(q)a_q$ and $[N_f,a_q^{†}]=f(q)a_q^{†}$. Thus $[N_f,a_q^{†}{a}_q]=0$ and $[N_f,N_g]=0$. Taking $f=p_{\mu }$, $g=p_{\nu }$ provesThese identities hold on the usual dense domain of finite-particle states, or can first be proved in a finite box and then continued to the continuum.

For a free real scalar field in four spacetime dimensions, $\mathcal{L}=(\partial \varphi {)}^2/2-m^2{\varphi }^2/2$ and $\pi =\dot{\varphi }$. Canonical quantization promotes these canonical fields to Hermitian operator-valued distributions with equal-time canonical commutation relationsExpand the solution of the Klein-Gordon equation into positive- and negative-frequency modes:Hermiticity pairs the two coefficients as adjoints. The field commutators are equivalent toFor example, inserting the expansion in $[\varphi ,\dot{\varphi }]$ yields two equal Fourier integrals, each with coefficient $i/2$, and hence $i{\delta }^{(3)}$. The vacuum obeys $a(\mathbf{k})|0⟩=0$, and repeated creation operators generate the symmetric bosonic Fock space. Normal ordering of the quadratic Hamiltonian removes its vacuum constant and gives $H=\int d{\Pi }_k{\omega }_k{a}^{†}a$; the same computation for the spatial momentum gives the stipulated four-momentum $P^{\mu }=\int d{\Pi }_k{k}^{\mu }{a}^{†}a$.

The oscillator algebra now impliesConsequentlyIndeed, ${\partial }^{\mu }{e}^{\mp ikx}=\mp i{k}^{\mu }{e}^{\mp ikx}$. For $\mu =0$, the result is $[H,\varphi ]=-i\dot{\varphi }$, which is the Heisenberg equation of motion with the correct sign. As usual, the field identities are interpreted after smearing the operator-valued distributions, or on a suitable common finite-particle domain.

Ideal conditional application of a creation operator, followed by normalization. It reweights the input number distribution as well as raising each occupied number by one. A heralded physical implementation has a success probability; the algebraic creation operator alone is not a trace-preserving deterministic quantum operation.

A quantum field is an operator-valued distribution on spacetime: smearing $\Phi (x)$ against a smooth test function gives an operator, whereas pointwise products can require renormalization. For example, a free complex scalar field contains particle annihilation operators and antiparticle creation operators. Its microcausality condition makes field commutators vanish at spacelike separation. For fermions, the corresponding locality condition uses anticommutators. This combines quantum operators with the spacetime locality of a quantum field theory.

Second quantization describes identical-particle quantum states in Fock space using creation operators and annihilation operators. Relative to an orthonormal basis of one-particle states, a one-body operator becomes ${\sum }_{\alpha \beta }{h}_{\alpha \beta }{a}_{\alpha }^{†}{a}_{\beta }$. The canonical commutation relation implements bosonic statistics, while the canonical anticommutation relations implement fermionic statistics. Expanding the quantized field in a complete basis gives a delta-function equal-time bracket; restricting the expansion to finitely many modes replaces that delta function by the corresponding projection operator kernel.

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.