Vacuum manifold

The vacuum manifold is the space of field configurations that minimize the energy. If a group $G$ acts transitively and a chosen vacuum has stabilizer $H$, its connected component is the homogeneous space $G/H$.

A spatially homogeneous field value minimizing the potential energy. For a finite-energy field configuration on an infinite domain, the potential is normally shifted so the relevant vacuum energy density is zero. Disconnected components of the vacuum manifold allow a scalar-field kink to connect distinct vacuum values.

A scalar-field vacuum is a constant field at a minimum of its potential energy, with the minimum shifted to zero when discussing finite-energy classical field-theory solitons. Distinct isolated vacua allow a scalar-field kink to approach different values at the two ends of space.

For $V(\varphi )=m^2{\varphi }^2/2-\mu {\varphi }^3/6+\lambda {\varphi }^4/24$, positive $\lambda$ makes the classical potential bounded below. For $m^2\ge 0$, completing the square givesThus zero is a global minimum precisely when ${\mu }^2\le 3\lambda m^2$. At a nontrivial equality there are degenerate minima at zero and $2\mu /\lambda$. If $m^2>0$ and the inequality fails, zero is a false vacuum: its Hessian is positive but another minimum has lower energy. A negative quartic coupling, or a nonzero cubic term without a quartic term, is unbounded below. These are classical or tree-level criteria.

A local minimum of the scalar potential that has higher energy than a global minimum. It can be stable against infinitesimal classical fluctuations but metastable to a transition to a lower-energy vacuum, including quantum tunnelling. The stability of a cubic-quartic scalar potential gives an elementary example distinguishing local curvature from global stability.