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## Summary:

- Derrick scaling argument and topological soliton energy bound seem to contradict each other?

## Main Question or Discussion Point

I have been reading Manton & Sutcliffe for some time now and can't quite wrap my head around something.

If you take the Hopf invariant N of a topological soliton ϕ then its Skyrme-Faddeev energy (which I hope I've gotten right up to some constants)

satisfies the bound E>c|N|

However, using the idea behind Derrick's theorem, I can rescale x→μx for some μ∈ℝ. Then the energy for the re-scaled soliton is something like E(μ(x))∽μE(x). Re-scaling space doesn't affect the Hopf invariant though (since it's a topological property) so you can make E arbitrarily small without changing N just by re-scaling the topological soliton. So I don't see how the inequality can hold since E can be made arbitrarily small.

What is wrong with my intuition here?

If you take the Hopf invariant N of a topological soliton ϕ then its Skyrme-Faddeev energy (which I hope I've gotten right up to some constants)

E=∫∂

_{i}ϕ⋅∂_{i}ϕ+(∂_{i}ϕ×∂_{j}ϕ)⋅(∂_{i}ϕ×∂_{j}ϕ) d^{3}xsatisfies the bound E>c|N|

^{3/4}(as per page 423). This seems like a really nice result. (BTW ϕ here is a vector field in ℝ^{3})However, using the idea behind Derrick's theorem, I can rescale x→μx for some μ∈ℝ. Then the energy for the re-scaled soliton is something like E(μ(x))∽μE(x). Re-scaling space doesn't affect the Hopf invariant though (since it's a topological property) so you can make E arbitrarily small without changing N just by re-scaling the topological soliton. So I don't see how the inequality can hold since E can be made arbitrarily small.

What is wrong with my intuition here?

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