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A (generalized) 't Hooft-Polyakov monopole and a Dirac monopole with a Dirac string attached are two types of magnetic monopoles, which differ in several ways, as OP and user ACuriousMind correctly states. On one hand, (generalized) 't Hooft-Polyakov monopoles are regular, soliton-like, finite-energy solutions to the classical Euler-Lagrange field ...


2

Here we assume that OP's question asks about $\phi^4$-theory in 1+1D, where the lagrangian density reads $$\tag{1} {\cal L}~=~\frac{1}{2}\dot\phi^2 -{\cal U}, \qquad {\cal U}~:=~ \frac{1}{2} \phi^{\prime 2} + {\cal V},\qquad \phi \in C^1(\mathbb{R}^2),$$ where the $\phi^4$-potential density $$\tag{2} {\cal V}(\phi)~\propto~(\phi^2-v^2)^2~ \geq~ 0$$ ...


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The energy density of the state $\pm v$ is going to be something like $\propto μ^4$, if you are using the basic $\varphi^4$ theory. While the energy of the domain wall is finite, the energy of the vacuum state is not, and so the transition to the vacuum state iver all space will be infinite.



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