When applying a spacetime translation $x^\mu\rightarrow x^\mu+a^\mu$ the KG lagrangian density changes by -
$$\mathcal{L} \rightarrow \mathcal{L} + a^\nu \partial_\mu \delta^\mu_{\;\nu} \mathcal{L}$$ giving rise to the momentum-energy tensor and the corresponding conserved charges.
The first one is the energy -
$$Q^0 =\intop d^3x (\frac{\partial\mathcal{L}}{\partial\dot\phi}\dot\phi-\mathcal{L})=E$$ this is obvious since the integrand is simply the Legendre transform of the Lagrangian density.
But the the other charges, associated with the spatial momentum are less clear to me -
$$Q^i =\intop d^3x \frac{\partial\mathcal{L}}{\partial\dot\phi}\partial^i\phi=p^i$$
What is the intuitive meaning of the integrand here? $\pi=\frac{\partial\mathcal{L}}{\partial\dot\phi}$ is already the field's momentum density. And what is the meaning of the spatial derivative?
