Relativistic normalization of a one-particle state ($⟨\mathbf{p}|\mathbf{q}⟩=2E_{\mathbf{p}}(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$)

If creation operators obey $[a_{\mathbf{p}},a_{\mathbf{q}}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$, set $|\mathbf{p}⟩=\sqrt{2E_{\mathbf{p}}}{a}_{\mathbf{p}}^{†}|0⟩$. Their inner product then has the displayed value. The associated completeness measure is $d^{3}p/((2\pi {)}^{3}2E_{\mathbf{p}})$, the invariant positive-energy mass shell measure. Specifying this normalization prevents hidden external energy factors in a scattering amplitude.