On Peskin & Schroder's QFT Book, page 19, they give us the conserved charge associated with spatial translations (equation 2.19): $$P^i=\int T^{0i}d^3x=-\int\pi \partial_i\phi d^3x$$ where $T^\mu_{\ \ \ \nu}$ is the energy momentum tensor: $$T^\mu_{\ \ \ \nu}=\frac{\partial L}{\partial(\partial_\mu\phi)}\partial_\nu \phi-L\delta^\mu_\nu$$ Where $L$ is the lagrangian density. Now, the book doesn't actually show the intermediate steps, so I decided to try to do them myself. And although I think I am close, there is something that eludes me.
My attempt is: Start with the Lagrangian density for the Klein-Gordon Field: $$L=\frac{1}{2}\partial_\mu \phi \partial^\mu \phi-\frac{1}{2}m^2\phi^2$$ Now I plug this into the energy-momentum tensor: $$T^\mu_{\ \ \ \nu}=\partial^\mu\phi\partial_\nu\phi-\delta^\mu_\nu(\frac{1}{2}\partial_\mu \phi \partial^\mu \phi-\frac{1}{2}m^2\phi^2)$$ Now, I know that momentum corresponds to $T^{0i}$, so I raise the $\nu$ everywhere and set $\mu=0$.(Recognizing it as the time derivative) $$T^{0\nu}=\partial_t\phi\partial^\nu\phi-(\frac{1}{2}\partial_t \phi \partial_t \phi-\frac{1}{2}m^2\phi^2)$$ Now if I were to take $\nu=i$, I would find $$T^{0i}=\partial_t\phi\partial^i\phi-(\frac{1}{2}\partial_t \phi \partial_t \phi-\frac{1}{2}m^2\phi^2)$$ Lowering the index on the first term: $$T^{0i}=-\partial_t\phi\partial_i\phi-(\frac{1}{2}\partial_t \phi \partial_t \phi-\frac{1}{2}m^2\phi^2)$$ Now, the first term resembles what I am looking for, keeping in mind that $\pi=\partial_t\phi$. But I don't know how to get rid of the $1/2$ or the 2nd and third term. What is the way forward here?
