If we have a Lagrangian $\mathcal L$ that depends on some scalar field $\phi$, we define the momentum as $\pi \doteqdot {\partial \mathcal L \over \partial \dot \phi}$. The Hamiltonian then is $\mathcal H \doteqdot \pi \dot \phi - \mathcal L$.
This rubs me the wrong way. It seems weird to be defining a quantity that puts special emphasis on time if you want a relativistic field theory. I would expect something like defining various momenta as $\pi^\mu \doteqdot {\partial \mathcal L \over \partial (\partial_\mu \phi)}$ and the Hamiltonian as $\mathcal H \doteqdot \pi^\mu \partial_\mu \phi - \mathcal L$, but I've never seen anyone use this.
I get that the Hamiltonian is the generator of time translations, but why should I care about it in the context of a relativistic field theory?
