Gary Gibbons, Claude Warnick and I just announced a new paper, in which we prove that Skyrmions are narcissistic (this colourful name is due to Gary). Basically what we have demonstrated is a version of the rule well-known to tods, that opposites attract and likes repel. (In this case Skyrmions carry parity: so its reflection in a mirror becomes its anti-particle.) A more precise version of the statement is that "finite energy solutions to the Skyrme model in Minkowski space that are symmetric or anti-symmetric across a mirror plane cannot remain in static equilibria."
Physically the intuition is simple: when something remains in static equilibrium, the total net force across any plane must vanish. This is just Newton's second law where the action of a force will produce an acceleration. So to show that such static equlibria cannot exist, it suffices to show that any configurations of these kinds must have a net force across the mirror surface.
For linear theories (for which the particles obey the superposition principle), the situation is simple. The linearity essentially implies that there cannot be internal structure to the particles which can provide a counter-balance to the interactions between the particles. A simple way to look at this is to hold up one's pinky and drop a bead of water on it. The top of your finger should be slightly curved, so parts of the bead of the water is sitting on a slope. But it doesn't flow down hill! Why? This is because water has internal structure (hydrogen bonds, van der Waals forces, surface tension, etc.) that holds the small bead together. When the bead of water is small enough, the internal energy is enough to overcome gravity on a gentle slope to prevent the drop from breaking up and flowing off. Now if you take some other liquid, say rubbing alcohol, with much less surface tension, and you take the same volume of liquid and try to bead it on your pinky, you'd find it much more difficult.
In the linear theory, without the internal structure, each infinitesimal volume acts independently of other infinitesimal volumes. So for a macroscopic configuration to remain in equilibrium, it is necessary that the potential energy everywhere is constant. And for most field theories this implies that the only finite energy solution must have zero energy, and thus the solution itself is trivial. (This is a reflection of the fact that in linear theories one typically do not expect the existence of solitons.)
In nonlinear theory, however, the fields can have internal structure. From these internal attractions come the possibility of solitary states, solutions which are concentrated spatially. A striking example is the phenomenon of tidal bores. For small amplitude surface waves on water, the equation of motion is well approximated by the linear wave equation. Hence we see the waves disperse as it propagates outwards in rings. For larger amplitude water waves in a narrow and shallow channel, however, the equation of motion is better described by the Kortewag-de Vries equation, whose nonlinearity better models the internal structure of the wave. The tidal bores observed in nature are reproducible theoretically as a soliton solution to the KdV equation.
Now, the soliton solutions to KdV are necessarily traveling waves. However, for other equations that are used to model nature, soliton solutions can be stationary or even static. Some examples are given by the focusing nonlinear Schrodinger equation, focusing semi-linear wave equation, the Yang-Mills instantons, and, in this particular case we are considering, the Skyrmions. The Skyrmions are used in nuclear physics to model baryons. Their equation of motion also has a nonlinearity that captures the presence of internal structure. The question we are interested in then is whether two such baryons can remain in static equilibrium.
Now, in the case of gravity, two astronomical bodies cannot remain in static equilibrium: this is because gravity is a purely attractive force. But the interaction of Skyrmions, like interaction of magnets, can be either attractive or repulsive. So one may try to look for situations where the attractive force exactly counter balances the repulsive force. In the case of the Maxwell theory of electromagnetism, because the theory is purely linear, the only way for two bodies to have exactly counter-balancing forces is for them to be without electromagnetic charge. (Whenever there is a potential gradient the charges will flow to even out the electromagnetic potential.) In the case of a non-linear theory like Skyrmions, it is possible for an extended body to have "positively charged" and "negatively charged" regions that are held apart by the internal structure and do not immediately cancel each other out, unlike the case for classical electromagnetism. Then it is conceivable that certain configurations can exist in which two such extended bodies have their various regions aligned just right so that the net force between them is zero.
And such configurations do exist under the name of Sphalerons.
What we prove in this paper is that in other types of arrangements unlike that of sphalerons, we can mathematically rigorously show that the two bodies cannot be kept at equilibrium.
The trick is one about symmetries. For a scalar valued function, there are basically two symmetry types you can have after reflection across a plane: even or odd. For a vector valued function, however, there are more allowed symmetries. A symmetry compatible with the reflection across a plane is just any symmetry that, when you do it twice, you recover the identity (what we call an action of $\mathbb{Z}_2$). If a function takes vector values, then besides the simple symmetries like the identity $x\to x$ and the complete negation $x\to -x$, we can also have reflections across vector-subspaces in the target. The sphaleron solution is exactly one such: it has a symmetry that is a nontrivial reflection in the target.
In our paper, we show that the two simplest symmetry types (identity and total negation) cannot lead to static equilibria. The proof essentially boils down to a statement about the internal structure of the Skyrmions, that in these types of symmetries, the complicated "region-by-region" interactions that may allow to two bodies to remain in equilibrium in fact completely cancel each other. So for these types of symmetries the interaction between two bodies is dictated only by the "total charge" of the each of them. And thus we again have that the opposites attract...
And now for the mathematics. What we exploit in the Skyrme model is a manifestation of the dominant energy condition: we consider is the internal stress of the solution. In general, the stress can take arbitrary sign, as long as they average out to zero so there is no bulk motion. But by imposing a symmetry condition, we require that on the mirror surface the solution has either a Dirichlet (negation symmetry) or Neumann (identity symmetry) boundary. The Skyrme model, along with some other Lagrangian field theories, has the property that on such boundaries the stress has a sign (positive if Dirichlet; negative in Neumann). Now, since there is no bulk motion in a static solution, the total stress across the mirror surface cannot be anything but zero. This in fact forces the solution to have both a Dirichlet and a Neumann boundary condition. Using certain properties associated to the ellipticity of the static solution (either the maximum principle or strong unique continuation), we can then conclude that the solution must then vanish everywhere.
