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In Minkowski spacetime with signature $(-+++)$, $p\cdot x=-{\omega }_{\mathbf{p}}t+\mathbf{p}\cdot \mathbf{x}$, where ${\omega }_{\mathbf{p}}=p^0=\sqrt{|\mathbf{p}{|}^2+m^2}>0$. Hence $i{\partial }_t{\psi }_{\mathbf{p}}={\omega }_{\mathbf{p}}{\psi }_{\mathbf{p}}$: these are positive-frequency solutions relative to the inertial time translation.

On a constant-time Cauchy hypersurface, the Klein-Gordon inner product and the spatial Fourier transform identity giveAlso $({\overline{\psi }}_{\mathbf{p}},{\overline{\psi }}_{\mathbf{q}})=-{\delta }^3(\mathbf{p}-\mathbf{q})$ and $({\psi }_{\mathbf{p}},{\overline{\psi }}_{\mathbf{q}})=0$: the latter is proportional to $({\omega }_{\mathbf{p}}-{\omega }_{\mathbf{q}}){\delta }^3(\mathbf{p}+\mathbf{q})$, which vanishes. This is the Klein-Gordon plane-wave normalization.

For completeness, spatial Fourier transformation of the field equation gives $\ddot{\hat{\Phi }}(t,\mathbf{p})+{\omega }_{\mathbf{p}}^2\hat{\Phi }(t,\mathbf{p})=0$. Its solutions have the two branches $e^{\mp i{\omega }_{\mathbf{p}}t}$. Selecting the positive-frequency branch leaves precisely the Fourier superpositions of ${\psi }_{\mathbf{p}}$. These plane waves form a generalized basis, normalized with the Dirac delta distribution, rather than individually normalizable vectors. For a normalizable packet,