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It would be nice if somebody could explain the structure of the potential.

I don't understand why ##\phi## is used instead of a position vector ##\textbf{r}##. Also, how can ##(\frac{1}{2}(\partial_{\mu}\phi)^{2}## be interpreted as the kinetic energy of the particle? I have integrated the Lagrangian over three spatial coordinates before, but why can the temporal coordinate be integrated over in this expression?