# Violation of Derrick's theorem for finite energy, time independent solutions?

How are vortices the finite energy time independent solutions for 2+1 dimensions abelian higgs model? Doesn't it violate derricks theorem that there are no finite energy time independent solutions in dimensions higher than 1+1?

No, it's not a violation. The conclusion of Derrick's theorem is modified in the presence of gauge fields, cf. e.g. this Phys.SE post.

No, it does not violate. Derrick's theorem is actually very restricted. It says that there is no finite energy time independent solution for $$L=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-V(\phi),$$ in dimensions larger than $1+1$. The vortice in the Abelian-Higgs model appears for the lagrangian $$L=-\frac 14F_{\mu\nu}F^{\mu\nu}+(D_\mu\phi)^*D^\mu\phi-\frac{\lambda}{4}(|\phi|^2-v^2)^2,$$ in $2+1$ or $3+1$. This is a gauge theory and the gauge field provides "more structure" to the theory, allowing for finite time independent solutions.

The intuition is that the for the first theory, there is a term $\int rdr\partial_i\phi\partial_i\phi$ in the energy density of the possible vortex but this diverges logarithmically. The corresponding term for the second theory is $\int rdr (D_i\phi)^*D_i\phi$ and the presence of the gauge field $A$ turns it possible that $D_i\phi=\partial_i\phi+ieA_i\phi$ decays sufficiently fast.