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The Euler-Lagrange equation of each component of the string embedding map is the wave equation $({\partial }_{\tau }^2-{\partial }_{\sigma }^2)X^{\mu }=0$. Its left- and right-moving solutions are identified by the endpoint Neumann boundary conditions, so the resulting open-string mode expansion isIndeed, the Fourier cosine series makes ${\partial }_{\sigma }{X}^{\mu }$ vanish at $\sigma =0,\pi$. The canonical momentum density is ${\Pi }_{\mu }=(2\pi {\alpha }^{\prime }{)}^{-1}{\dot{X}}_{\mu }$. Imposing the equal-time canonical commutation relations $[X^{\mu }(\sigma ),{\Pi }_{\nu }({\sigma }^{\prime })]=i{\delta }_{\nu }^{\mu }\delta (\sigma -{\sigma }^{\prime })$ gives the covariant quantization of the bosonic stringwith ${\alpha }_0^{\mu }=\sqrt{2{\alpha }^{\prime }}{p}^{\mu }$.

It is cleaner to use the action of the charge than to expand every double sum. The commutators from part a give, for $v^{\rho }=x^{\rho },p^{\rho }$, or any ${\alpha }_n^{\rho }$,Thus $M^{\mu \nu }$ acts on every mode in the vector representation of the Lorentz algebra. Apply the Jacobi identity to the action of $[M^{\mu \nu },M^{\kappa \lambda }]$ on every mode. The result agrees with the action ofThere is no extra scalar term: the orbital and oscillator pieces commute with one another, and direct use of their canonical commutation relations gives no central contribution. Hence the displayed operators obey precisely the standard Lorentz algebra.

Choose complete orthonormal sets of complex Klein-Gordon solutions $\{u_i^{\mathrm{in}}\}$ and $\{u_i^{\mathrm{out}}\}$ that have positive frequency with respect to the asymptotic timelike Killing fields in the remote past and future. Their normalization uses the conserved Klein-Gordon inner product:The real quantum field has either mode expansion of a free field

Completeness relates the mode bases by the Bogoliubov transformationand hencePreservation of the canonical commutation relations requiresThe past and future notions of positive frequency therefore define the generally different in-vacuum and out-vacuum. Nonzero $\beta$ means that the in-vacuum is a squeezed state containing out-particles.

The in-vacuum satisfies $a_j^{\mathrm{in}}|0_{\mathrm{in}}⟩=0$. For the out-mode number operator $N_i^{\mathrm{out}}=a_i^{\mathrm{out}†}{a}_i^{\mathrm{out}}$, substitute the operator transformation and use the canonical commutation relations. Only the $a_j^{\mathrm{in}}{a}_k^{\mathrm{in}†}$ contraction survives, giving the particle number from Bogoliubov coefficientsFor a continuous mode label, the sum becomes the corresponding integral with Dirac-delta normalization.

The equal-time canonical commutation relation for the transverse-traceless graviton and its canonical momentum is the transverse-traceless projector. At zero momentum the supplied polarization completeness relation giveswhere symmetry and tracelessness of ${\omega }_{ij}$ remove the trace term. Thus the charge generates precisely the transformation in part i:This is the soft, field-independent part of the Noether charge associated with the large diffeomorphism.

For a test wavefunction $f$,Thus the canonical commutation relation is

The canonical commutation relation is

Expand the quantized field using the normalized modeThe canonical commutation relation fixes the Wronskian $M_{\mathrm{Pl}}^2{z}^2(u_k{u}_k^{\ast \prime }-u_k^{\ast }{u}_k^{\prime })=i$. Writing $x=k\eta$, the primed correlators areEach unprimed expression includes $(2\pi {)}^3{\delta }^{(3)}(\mathbf{k}+{\mathbf{k}}^{\prime })$. The imaginary part of the mixed correlator givesConsequently the classicality parameter of a cosmological perturbation isIt tends to zero on superhorizon scales, so the perturbation becomes effectively classical even while its amplitude grows as $a^3$. In ordinary slow-roll inflation, $\mathcal{R}$ instead freezes to a constant; its analogous normalized commutator also vanishes, but behaves as $|k\eta |$ for the leading de Sitter mode rather than $|k\eta {|}^3$.

The conserved number operator isSubstitution of $\Psi =\sqrt{\rho }{e}^{-i\theta }$ into the Landau-Ginzburg theory givesThe imaginary term is a total derivative, soThus $\rho$ is the number density and is the canonical momentum conjugate to $\theta$. The canonical commutation relation is $[\theta (\mathbf{x}),\rho (\mathbf{y})]=i{\delta }^{(d)}(\mathbf{x}-\mathbf{y})$. Consequently, for ${\theta }_0=\int d^{d}x\theta$ in volume $V$,Equivalently, the averaged phase $\bar{\theta }={\theta }_0/V$ obeys $\Delta \bar{\theta }\Delta N\ge 1/2$. This number-phase conjugacy means that a state of sharp $N$ has no sharp phase, whereas a phase-selected state exhibiting spontaneous symmetry breaking must superpose different number sectors. Such sectors become effectively degenerate in the thermodynamic limit.

Let ${\Sigma }_t$ be a Cauchy hypersurface with induced metric $h_{ij}$, lapse $N$, shift $N^i$, and future unit normal $n^a$. For the normalization of the action in the question, the canonical momentum density isThe equal-time canonical commutation relations areandWith the conventional extra factor $1/2$ in the action, $\Pi$ loses the factor two.

For complex classical solutions, the Klein-Gordon inner product isThe integrand is a conserved current because both fields obey the Klein-Gordon equation. Applying the divergence theorem between two Cauchy hypersurfaces shows that the value is independent of the foliation, provided there is no boundary flux.

For the real scalar field expansion $\varphi =\int d^{3}p[a_{\mathbf{p}}{e}^{-ipx}+a_{\mathbf{p}}^{†}{e}^{ipx}]/((2\pi {)}^3\sqrt{2E_{\mathbf{p}}})$, the equal-time fields extract $a_{\mathbf{p}}=e^{iEt}\int d^{3}x{e}^{-i\mathbf{p}\cdot \mathbf{x}}[\sqrt{E/2}\varphi +i\pi /\sqrt{2E}]$. The two mixed canonical commutation relation terms yield $[a_{\mathbf{p}},a_{\mathbf{q}}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$, while the other oscillator commutators vanish. This verifies the normalization of the mode expansion directly from the canonical fields.

The reduced canonical commutation relation for a transverse field uses a transverse delta function instead of an unconstrained componentwise delta. Its Fourier transform is the spatial transverse projector. It can be derived from the two-polarization oscillator expansion or from the Dirac bracket after gauge fixing.

An unnormalized bosonic coherent state is $|\psi ⟩=e^{\psi a^{†}}|0⟩$ for a complex label $\psi$. The canonical commutation relation gives $a|\psi ⟩=\psi |\psi ⟩$ and overlap $⟨\psi |{\psi }^{\prime }⟩=e^{\bar{\psi }{\psi }^{\prime }}$. Multiplication by $e^{-|\psi {|}^2/2}$ produces a normalized coherent state. The unnormalized form makes the coherent-state resolution of identity and thermal time slicing particularly simple.

A fermionic mode with canonical anticommutation relations $\{b,b^{†}\}=1$ has the symmetrically ordered Hamiltonian operator $H=\frac{E}{2}(b^{†}b-b{b}^{†})=E(b^{†}b-1/2)$. Its empty-state zero-point contribution is $-E/2$. A quantum harmonic oscillator with bosonic canonical commutation relations contributes $+E/2$ instead. For a Dirac field, sum over both particle and antiparticle modes and over spin: there are four independent oscillators per spatial momentum in four spacetime dimensions. Comparing bosonic and fermionic vacuum energies requires a common regularization in quantum field theory, matching degrees of freedom and a stated vacuum-energy subtraction convention.

The canonical commutation relation fixes the antisymmetric product of a mode function $f$ and its canonical-momentum mode $g$. In the convention $[a,a^{†}]=1$, one commonly has $f{g}^{\ast }-f^{\ast }g=i$.