⟨ p ∣ q ⟩ = 2 E_{p} 2 E_{q} ⟨ 0∣ a_{p} a_{q}^{†} ∣0 ⟩ = 2 E_{p} (2 π)^{3} δ^{3} (p - q) .

(1)

Equality of the square-root product and

2 E_{p}

holds on the support of the Dirac delta function.

The positive-energy mass shell has the Lorentz invariant measure

d^{4} p θ (p^{0}) δ (p^{2} - m^{2}) = \frac{d ^{3} p}{2 E _{p}} .

(2)

Here both

p^{2}

and the four-dimensional volume element are invariant, and positive energy is preserved by a proper orthochronous Lorentz transformation. Consequently the three-dimensional Jacobian determinant is

J = E_{Λ p} / E_{p}

. Transforming the Dirac delta function yields

δ^{3} (Λ p - Λ q) = \frac{E _{p}}{E _{Λ p}} δ^{3} (p - q),

(3)

so its factor exactly cancels that of

2 E_{Λ p}

. Thus the displayed inner product is invariant under simultaneous transformation of both momenta. This is also why the one-particle completeness measure is