Field configuration minimizing the Higgs potential Why is it that the field configuration that minimizes the Higgs potential is said to be a constant one (meaning $\phi(x) = \phi_0$)? Is it a physical reasoning, which kind of makes sense, since a non flat ground state of the field would be rather strange? Or is it a mathematical reasoning that the field configuration which minimizes the potential must be a constant one? If so how is it done? (It is not obvious to me right now, since the fields are functions of space-time (no quantization yet)).
I have read this and links therein but it did not really answered my question.
 A: The ground state (or vacuum state) minimizes the energy, not just the potential.  A solution of the field equations with non-constant field $\phi$ (satisfying $|\phi(x)|=\phi_{0}$, so that the potential is minimized) will still have kinetic energy associated with it—making it automatically not the vacuum.
That’s all there is to say about finding the vacuum state of a scalar field.  However, things get more complicated for fields with higher spin, like gauge fields.  For a gauge field $A^{\mu}$, there are certain kinds of spacetime dependence that do not contribute to the energy, because they the energy only depends on the field strength $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$.  A gauge transformation can make $A^{\mu}(x)$ spacetime dependent, without affecting $F^{\mu\nu}=0$.  In certain cases, such as with nonabelian gauge fields, this can mean that there are different vacuum states, with different spacetime-dependent $A^{\mu}$, which nonetheless all minimize the energy.
