Klein-Gordon inner product

For complex solutions of the Klein-Gordon equation,The associated current is conserved, so the inner product is independent of the Cauchy hypersurface when boundary flux vanishes.

For a free real scalar field expanded with measure $d^{3}p/[(2\pi {)}^{3}2E_p]$ and $f_p=e^{-ip\cdot x}$, the displayed Klein-Gordon inner product extracts its annihilation operator. Indeed $(f_p,f_q)=(2\pi {)}^{3}2E_p{\delta }^3(\mathbf{p}-\mathbf{q})$ and $(f_p,f_q^{\ast })=0$; spatial Fourier inversion removes the measure. Taking the Hermitian adjoint gives $a^{†}(\mathbf{p})=-i\int d^{3}x{f}_p\overleftrightarrow{{\partial }_t}\varphi$. The equal-time canonical commutation relation then gives $[a(\mathbf{p}),a^{†}(\mathbf{q})]=(2\pi {)}^{3}2E_p{\delta }^3(\mathbf{p}-\mathbf{q})$.