In Lancaster and Blundell, *Quantum Field Theory for the Gifted Amateur,* p.99, the Hamiltonian density is
\begin{equation}
\mathcal{H}=\frac{1}{2}[\partial\_0\phi(x)]^2+\frac{1}{2}[\nabla\phi(x)]^2+\frac{1}{2}m^2[\phi(x)]^2,\tag{11.5}
\end{equation}
and it tells us that the energy has contributions from

1. a kinetic energy term reflecting changes in the configuration in time,
2. a 'shear term' giving an energy cost for spatial changes in the field, and
3. a 'mass' term reflecting the potential energy cost of there being a field in space at all.

In the equation above, i think the first term is the same as the classical mechanics. But i don't understand why second (shear) and third (mass) term are represent potential energy.