Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 5 / a / Solution

With the Minkowski metric $(+---)$, the momentum conjugate to the real field vector is $\mathbf{\pi }=\dot{\mathbf{\varphi }}$. The Hamiltonian density isAssume the potential has a global minimum $V_{\min }$ and the boundary conditions permit a constant minimizing field. ThenThe bound is attained byThe homogeneous vacuum minimum criterion proves existence of a homogeneous lowest-energy configuration under those assumptions. On infinite space one subtracts the constant vacuum energy before comparing finite energy differences. This identifies the classical vacuum configuration; it does not assert that the quantum field has no fluctuations.

The claim does not follow from “no derivative dependence in $V$” alone. For example, the allowed functional form $V(\varphi )=-{\varphi }^4$ is unbounded below, so there is no state of lowest energy to identify. Boundary conditions forcing different vacua at opposite ends can instead require an inhomogeneous minimum, such as a domain wall. The positive quartic potential used in the following parts does have homogeneous minima, so these caveats do not obstruct its vacuum calculation.