lagrangian formalism - Determination of the ground state of a field theory

1. 
   

   Consider the Spontaneous symmetry breaking in the theory

   $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{\mu^2}{2}\phi^2+\frac{\lambda}{4!}\phi^4.$$ By the ground state of a classical field theory we mean the minimum value of the full Hamiltonian. Then in studying spontaneous symmetry breaking why do we only minimize the potential $V(\phi)=\frac{\mu^2}{2}\phi^2-\frac{\lambda}{4!}\phi^4$ and do not pay attention to the energy contributed by the gradient term $\partial_\mu\phi\partial^\mu\phi$? Certainly, if in the ground state, $\phi(x,t)$ has a spatial variation then $(\nabla\phi)^2$ term would contribute to the energy density. So why do we only minimize $V(\phi)$?
   
   


2. 
   

   Is it assumed that $\phi(x,t)=\phi_0=\text{a constant independent of space-time}$ in the ground state? Is it necessary in all theories that $\phi(x,t)=\text{a constant}$ in ground state of the system?

   
   

Answer

With a Lagrangian like: $\mathcal{L} = \partial_\mu \phi^\dagger \, \partial^\mu \phi - V(\phi) = \mathring{\phi^\dagger} \mathring{\phi} + \partial_i \phi^\dagger \, \partial^i\phi - V(\phi)$, the Hamiltonian is:

$$\begin{equation*} \mathcal{H} = \frac{\partial \mathcal{L}}{\partial \mathring{\phi}} \mathring{\phi} + \mathring{\phi^\dagger} \frac{\partial \mathcal{L}}{\partial \mathring{\phi^\dagger}} - \mathcal{L} \end{equation*}$$ which gives: $$\begin{equation*} \begin{array}{ll} \mathcal{H} & = \mathring{\phi^\dagger} \mathring{\phi} + \mathring{\phi^\dagger} \mathring{\phi} - ( \mathring{\phi^\dagger} \mathring{\phi} + \partial_i \phi^\dagger \, \partial^i\phi - V(\phi) ) \\ & = \mathring{\phi^\dagger} \mathring{\phi} - \partial_i \phi^\dagger \, \partial^i\phi + V(\phi)\\ & = \mathring{\phi^\dagger} \mathring{\phi} + \vec{\nabla} \phi^\dagger . \vec{\nabla} \phi + V(\phi) \\ & = |\mathring{\phi}|^2 + |\vec{\nabla} \phi|^2 + V(\phi) \end{array} \end{equation*}$$ We notice that the first 2 terms of the Hamiltonian are positively defined and vanish when $\phi$ is a constant field (not depending on space-time). Therefore, the minimum of $\mathcal{H}$ is reached for a constant field $\phi_0$ that minimizes the last 2 terms i.e. the potential $V(\phi_0)$.