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The equal-time canonical commutation relation is $[\varphi (t,\mathbf{x}),\pi (t,\mathbf{y})]=i{\delta }^3(\mathbf{x}-\mathbf{y})$ in units $\hslash =1$, with the two field-field and momentum-momentum commutators zero. The quadratic Hamiltonian operator gives $\dot{\varphi }=\pi$ and $\dot{\pi }={\nabla }^2\varphi -m^2\varphi$, hence the Klein-Gordon equation as an operator identity.

For the global phase transformation $\delta \varphi =-i\alpha \varphi$, the Noether current is $j^{\mu }=i({\varphi }^{†}{\partial }^{\mu }\varphi -{\partial }^{\mu }{\varphi }^{†}\varphi )$. The Klein-Gordon equation gives ${\partial }_{\mu }{j}^{\mu }=0$. With zero vacuum charge, its normal-ordered charge is $Q=\int d{\Pi }_p(a^{†}a-b^{†}b)$. Thus $[Q,a^{†}]=a^{†}$, $[Q,b^{†}]=-b^{†}$ and $[Q,\varphi ]=-\varphi$. Particle and antiparticle creation carry opposite charges while both increase energy.

On a scalar, the covariant wave operator is $\square \varphi =|g{|}^{-1/2}{\partial }_a(|g{|}^{1/2}{g}^{ab}{\partial }_b\varphi )$. For the metric signature $(-+++)$ it has the flat-space form $-{\partial }_t^2+{\nabla }^2$. This divergence expression is convenient for separation of variables and for integrating the Klein-Gordon equation by parts.

For the dilaton equation from contracted Bianchi identity, put $F=e^{-2\phi }$. The chain rule gives ${\square }_{g}F=F(-2{\square }_g\phi +4|\nabla \phi {|}^2)=-2CF$. Thus the exponential obeys a linear Klein-Gordon equation on the given background, although the original dilaton equation is nonlinear and the string-frame metric remains coupled to the dilaton. With the opposite named-field convention $\phi =-\Phi$, this variable is $F=e^{2\Phi }$.

For the free real scalar field with signature $(+,-,-,-)$,The canonical stress-energy tensor is $T^{\mu \nu }={\partial }^{\mu }\varphi {\partial }^{\nu }\varphi -{\eta }^{\mu \nu }\mathcal{L}$. Its divergence vanishes by the Klein-Gordon equation, giving constant integrated charges when integrals and boundary flux are well behaved. The minus sign in contravariant spatial momentum is from ${\partial }^i=-{\partial }_i$; lower-index spatial charges have the opposite sign. A plane wave $\cos (Et-\mathbf{k}\cdot \mathbf{x})$ has momentum density pointing along $\mathbf{k}$.

For complex solutions of the Klein-Gordon equation,The associated current is conserved, so the inner product is independent of the Cauchy hypersurface when boundary flux vanishes.

For $p^0=\sqrt{{\mathbf{p}}^2+m^2}>0$ in Minkowski spacetime, these modes obey $i{\partial }_t{\psi }_{\mathbf{p}}=p^0{\psi }_{\mathbf{p}}$ and have Klein-Gordon inner product $({\psi }_{\mathbf{p}},{\psi }_{\mathbf{q}})={\delta }^3(\mathbf{p}-\mathbf{q})$. Their conjugates have the opposite norm and are orthogonal to the positive-frequency modes. Plane waves are a generalized, delta-normalized basis; normalizable wave packets have positive norm $\int |a(\mathbf{p}){|}^2{d}^{3}p$. Completeness follows by applying the spatial Fourier transform to the Klein-Gordon equation, which gives the two branches $e^{\mp i{p}^{0}t}$ for each momentum.

If ${\partial }_{-}$ is invertible, the divergence constraint of a massive two-form field makes $A_{-i}$ and $A_{ij}$ independent. It gives $A_{+-}=-{\partial }_{-}^{-1}{\partial }_i{A}_{-i}$ and $A_{+i}={\partial }_{-}^{-1}(-{\partial }_{+}{A}_{-i}+{\partial }_j{A}_{ji})$. All independent components obey the Klein-Gordon equation. Their count is $(D-2)+\left(\genfrac{}{}{0ex}{}{D-2}{2}\right)=\left(\genfrac{}{}{0ex}{}{D-1}{2}\right)$.

With $d{s}^2=-2d{x}^{+}d{x}^{-}+d{x}_i^2$, $x^{+}=t$, and $p_{-}\ne 0$, the mass-shell condition gives $H=-p_{+}=-(p_i^2+{\mu }^2)/(2p_{-})$. In the future-directed sector $p^{+}=-p_{-}>0$, this is $(p_i^2+{\mu }^2)/(2p^{+})$. Its Schrodinger equation is equivalent to the Klein-Gordon equation after multiplication by $2p_{-}$.

A free massive spin-two field is a symmetric tensor obeying a massive Klein-Gordon equation, vanishing divergence, and vanishing trace. Its rest-frame polarizations form the symmetric traceless square of the $SO(D-1)$ vector, with $(D-2)(D+1)/2$ degrees of freedom.

An antisymmetric tensor $A_{mn}$ with $F=dA$ obeys ${\partial }^m{F}_{mnp}={\mu }^2{A}_{np}$ for $\mu \ne 0$. Taking a divergence yields ${\partial }^m{A}_{mn}=0$, and the remaining equation is the Klein-Gordon equation for each component. The physical particle polarizations form the exterior square of the massive little group vector representation, of dimension $(D-1)(D-2)/2$. Unlike a two-form gauge field, it has physical longitudinal polarizations.

The FP ghost square vanishes, so this BRST charge is nilpotent. On a ghost-number-zero wavefunction, BRST cohomology implements the Klein-Gordon equation. Closure of an unrestricted ghost-extended wavefunction does not independently constrain every component: a pure FP ghost component can lie in the kernel without satisfying the same scalar equation. The FP ghost sector is part of the physical prescription.