One-dimensional field momentum

[Independent]

How does one arrive at the formula 4.8?

The Lagrangian (one spatial dimension) is:

 

[vanhees71]

That's a special case of Noether's theorem for space-time translations, which is a symmetry of Minkowski space. The corresponding conserved quantities are energy and momentum. For fields it defines the canonical energy-momentum tensor
$$\Theta^{\mu \nu}=\frac{\partial \mathcal{L}}{\partial (\partial_{\nu} \phi)}\partial^{\mu} \phi-\mathcal{L} g^{\mu \nu}.$$
The momentum density components are given by $\Theta^{0j}$ ($j \in \{1,2,3 \}$). Now it should be easy to show the above formula.