I'm considering D=1+1 kink solution here.
Given a D=2 theory with $\mathbb{Z}_2$ symmetry, there are 4 different mappings (or 2 distinct sectors---trivial and kink) from spacetime manifold (or just a spacelike hypersurface as a section of it) to vacuum manifold: $$ \phi_{\text{Domain wall}}(x=\pm \infty)=\pm v\quad \text{and} \quad\phi_{\text{Domain wall}}(x=\pm \infty)=\mp v. $$ Normally, we say that transition between configurations is not allowed because it takes infinite amount of energy. I know that degenerate vacuum allows topologically stable field configuration and explicitly broken symmetry would lead to unstable configuration.
Question 1: Is the "transition'' we talk about here is spatial version of the similar "tunneling'' of spacetime in instanton physics (semi-classical solutions in quantum cosmology)? After all, instantons are interpreted as "localized in time'' (thus like a particle in some sense) and solitons are localized in space.
If so, basically, transition between different field configurations is tunneling process.
Question 2: How do I verify the claim "one needs infinite amount of energy to transient from one configuration to another"? I can imagine the tunneling exponent is proportional to the action. Infinite amount of energy means infinite action. Doesn't this mean that the tunneling possibility is zero?
