I One-dimensional field momentum

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1. Apr 21, 2016

Independent

How does one arrive at the formula 4.8?

The Lagrangian (one spatial dimension) is:

2. Apr 23, 2016

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.