Here, the Klein–Gordon equation in natural units, ( ◻ ◻ + m 2 ) ψ ψ ( x ) = 0 , with the metric signature η η μ μ ν ν = diag ( + 1 , − − 1 , − − 1 , − − 1 ) is solved by Fourier transformation. Inserting the Fourier transformation ψ ψ ( x ) = ∫ ∫ d 4 p ( 2 π π ) 4 e − − i p ⋅ ⋅ x ψ ψ ( p ) and using orthogonality of the complex exponentials gives the dispersion relation p 2 = ( p 0 ) 2 − − p 2 = m 2 This restricts the momenta to those that lie on shell, giving positive and negative energy solutions p 0 = ± ± E ( p ) where E ( p ) = p 2 + m 2 . For a new set of constants C ( p ) , the solution then becomes ψ ψ ( x ) = ∫ ∫ d 4 p ( 2 π π ) 4 e i p ⋅ ⋅ x C ( p ) δ δ ( ( p 0 ) 2 − − E ( p ) 2 ) . It is common to handle the positive and negative energy solutions by separating out the negative energies and work only with positive p 0 : ψ ψ ( x ) = ∫ ∫ d 4 p ( 2 π π ) 4 δ δ ( ( p 0 ) 2 − − E ( p ) 2 ) ( A ( p ) e − − i p 0 x 0 + i p i x i + B ( p ) e + i p 0 x 0 + i p i x i ) θ θ ( p 0 ) = ∫ ∫ d 4 p ( 2 π π ) 4 δ δ ( ( p 0 ) 2 − − E ( p ) 2 ) ( A ( p ) e − − i p 0 x 0 + i p i x i + B ( − − p ) e + i p 0 x 0 − − i p i x i ) θ θ ( p 0 ) → → ∫ ∫ d 4 p ( 2 π π ) 4 δ δ ( ( p 0 ) 2 − − E ( p ) 2 ) ( A ( p ) e − − i p ⋅ ⋅ x + B ( p ) e + i p ⋅ ⋅ x ) θ θ ( p 0 ) In the last step, B ( p ) → → B ( − − p ) was renamed. Now we can perform the p 0 -integration, picking up the positive frequency part from the delta function only: