Orthogonality and Normalization

$\int d^3x\phi^*\stackrel{\leftrightarrow}{\partial}_0\phi$ is time independent. If $\phi$ is a Lorentz scalar than so is $\int d^3x\phi^*\stackrel{\leftrightarrow}{\partial}_0\phi$ and it can be used for normalization. Since $\int d^3x\phi^*\stackrel{\leftrightarrow}{\partial}_0\phi$ is purely imaginary we can define the normalization as

$$\pm i\int d^3x \phi^*(x) \stackrel{\leftrightarrow}{\partial}_0 \phi(x) = 1 .$$ (4.55)

If we have a set of wave functions $\phi_a(x), a=1,2,3,...$, they may be orthonormalized via

$$\int d^3x {\phi_a^{(\pm)}}^*(x) i\stackrel{\leftrightarrow}{\partial}_0 \phi_b^{(\pm)}(x) = \pm\delta_{ab} ,$$ (4.56)

$$\int d^3x {\phi_a^{(\pm)}}^*(x) i\stackrel{\leftrightarrow}{\partial}_0 \phi_b^{(\mp)}(x) = 0 .$$ (4.57)

We can also require that the general solution (for wavepackets) satisfies

$$\int d^3x{f{_{p^\prime}}^{(\pm)}}^*(x) i \stackrel{\leftrightarrow}{\partial}_0 f_p^{(\pm)}(x) = \pm\delta^3(\vec{p}-\vec{p}^{\:\prime}) ,$$ (4.58)

$$\int d^3x{f{_{p^\prime}}^{(\pm)}}^*(x) i \stackrel{\leftrightarrow}{\partial}_0 f_p^{(\mp)}(x) = 0.$$ (4.59)