This is the Euclidean classical action ##S_{cl}[\phi]=\int d^{4}x\ (\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi))##.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Classical Euclidean Action

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