Euclidean classical action This is the Euclidean classical action $S_{cl}[\phi]=\int d^{4}x\ (\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi))$.
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?
 A: 
This is the Euclidean classical action $S_{cl}[\phi]=\int d^{4}x (\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi))$.
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?

The integration over time, is from the definition of the action:
$$
S=\int dt L\;,
$$
where $L$ is the Lagrangian.
The integration over space is there because (by your choice) you are considering a Lagrangian that is equal to the integral over all space of a Lagrangian density
$$
L=\int d^3x \mathcal{L}\;,
$$
where
$$
\mathcal{L}\equiv (\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi))\;.
$$
The function $U$ is unspecified in your question. It could have pretty much any functional form you want. For example, it could be simple a quadradic function:
$$
U(\phi)=\alpha\phi^2+\beta\phi+\gamma\;,
$$
For example, with $\alpha=k/2$, $\beta=0$, and $\gamma=0$ you would have
$$
S=\int dt d^3x (\frac{1}{2}(\partial_{\mu}\phi)^{2}+\frac{k}{2}\phi^2)\;.
$$
The field $\phi(\vec r)$ is treated as the variable when calculating the equations of motion because that is what you are ostensibly interested in (this is a field theory)... there are no "particles" in this theory because (as you stated) this is a classical action and the fields are classical fields (i.e., just regular functions, not operator-valued functions).
