# Can a perturbation add more symmetry?

Many textbooks of quantum mechanics argues that in presence of the ground degeneracies, at least some of them will be removed if some perturbation reduces the corresponding symmetry.

Then, is there any examples which a perturbation to a free theory can add new symmetries so that the new perturbed ground states possess more degeneracies?

In principle, I would say yes it is possible. But that would be totally theoretical.

Say you have some anisotropic potential to start with such as, $$V(x,y,z) = ax^2 + (a+\epsilon)y^2 + az^2$$, where, $$\epsilon$$ is a small parameter. Now if you add perturbation such that, the perturbation potential is $$V_{pert} = - \epsilon\, y^2$$, then after adding this potential you will have a spherically symmetric potential. Thus you have added more symmetry into the system by perturbing it.

But this is purely theoretical. I don't think there can be any real life example of this.

• Yes, I’m aware of that an inverse of perturbation or regarding the original theory to be already perturbed from free theory can reduce symmetry. I wonder if there exist such phenomena in a theoretically meaningful situation. Oct 1, 2020 at 18:55
• Right. I am not aware of any such perturbation theory as well. Oct 2, 2020 at 11:15

Suppose our system (which could be any kind of system) depends on some parameters in a topological space $$P$$ and there is a group $$G$$ which acts on $$P$$, such that for each $$p \in P$$, the theory described by the parameter value $$p$$ has symmetry whatever subgroup of $$G$$ fixes $$p$$. Let's assume that $$G$$ acts continuously on $$P$$. Then if $$S \subset P$$ is a subset fixed pointwise by some subgroup $$H, it is easy to see that the closure of $$S$$ is also fixed pointwise by $$H$$. This implies that the boundary of $$S$$ has at least as much symmetry as $$S$$ itself. Therefore, generic perturbations will always lower the symmetry.

(Note that RG flow can get around this by the appearance of emergent symmetries.)

• Regarding to your last sentence, do you mean RG flow from IR to UV? (recovery of broken symmetries) Oct 1, 2020 at 18:58
• @Amplituhedron Either way, after perturbing the theory can flow to a fixed point with more symmetry. Oct 1, 2020 at 23:14