I am following along Marc Casal's lecture slides "Quantum Field Theory in Curved Spacetime". For scalar functions $f$ and $g$ we define the Klein-Gordon inner product as follows: $$ \langle f,g \rangle \ = \ i \int_{\Sigma} d^{3}\mathbf{x}\ \left[ f^{\ast}(t,\mathbf{x}) \frac{\partial g(t,\mathbf{x})}{\partial t} - \frac{\partial f^{\ast}(t,\mathbf{x})}{\partial t} g(t,\mathbf{x}) \right] $$
Where we take $\Sigma$ to a 3D hypersurface of constant $t = t_{0}$. If $f$ and $g$ are the solutions to the Klein-Gordon then the above is independent of the choice of $t_0$ used to integrate it.
One can use this to show that the modes $u_{\mathbf{k}} \propto e^{-i \sqrt{\mathbf{k}^2 + m^2} t + i \mathbf{k} \cdot \mathbf{x}}$ are orthogonal to one another.
My question is; where does this inner product come from? I under stand that $i \phi^{\ast} \frac{\partial \phi}{\partial t} - i \phi^{\ast} \frac{\partial \phi}{\partial t} $ is a conserved current, but why would you put a conserved current under and integral sign to define an inner product? It seems arbitrary. What if one came up with a different inner product? What benefits does the KG inner product have?
