Monopole, 't Hooft-Polyakov

The SO(3) Yang-Mills theory in 3 + 1 dimensions with a Higgs field in adjoint representation:

where a = 1, 2, 3. It is necessary to introduce a Higgs field (or anyway a matter field coupling): there is the Deser theorem (1976) stating the impossibility of static solitonic solutions in pure Yang-Mills theories except the case with four space dimensions. Moreover, as we will see, the 't Hooft (http://www.phys.uu.nl/∼thooft/ap.html)-Polyakov solution requires that the ϕ field is a Lorentz scalar and transforms in the adjoint representation. The energy density is

One has always θ₀₀₀ ≥ 0; θ₀₀ = 0 if and only if the following statements are simultaneously valid:

We remind that a vacuum configuration satisfies θ₀₀ = 0 over the entire universe (a gauge invariant condition), while a Higgs vacuum configuration satisfies the second and third equations above, but may fail to satisfy the first one. The invariance group for the minima of V (and as well for any minimum of V) is the U(1) rotation...