HAIRS ON THE UNICORN Fine structure of monopoles and other solitons

Intrinsically stable or ‘fundamental’ solitons may be decorated with conserved charges which are pieces of those carried by elementary particles in the same medium. These ‘hairs’ are always significant in principle, and in the strong-coupling regime (where solitons and particles exchange roles) may become major factors in dynamics. ∗To appear in Proceedings of the CRM-FIELDS-CAP Workshop “Solitons” at Queen’s University, Kingston, Ontario, July 1997 (Springer, New York 1998). I. Monopoles and unicorns Long ago my developing fascination with magnetic monopoles led to an awareness of a kinship between the monopole and another mythical beast, the unicorn. As we shall see shortly, the monopole (which was my personal entrée into the subject of solitons) occupies a very special place, as the only ‘fundamental’ soliton able to move freely in our (3+1)-dimensional world. Before looking at the parallel between monopole and unicorn, let us go into another aspect of the monopole. From Dirac’s quantization argument[1] we know that the magnetic charge g of a pole is quantized, and has a superstrong value. Strong coupling is incompatible with point structure of the coupled particle, and so the pole must have internal geometrical dimension greater than its Compton wavelength by a factor of order g [2]. Because the natural quantum length scale is small compared to the size of the object, this is a powerful indication that one should be able to describe the pole as a classical field configuration, so that quantum consistency conditions have brought us back to classical physics, and hence to view the monopole as a soliton. However, the quantum consistency conditions for pure QED, or even SU(2)×U(1) electroweak theory, still require quantized g, so clearly the charge cannot be spread out over a length scale larger than the minimum for which the standard model is accurate. That is the basis for the claim below that existing or planned particle accelerators won’t have the energy to produce monopoles. Now to the title: The unicorn is a wonderful creature. It has many features in common with the monopole, including 1. Origin in medieval Europe The concept of a single pole, and the recognition that north and south magnetic poles always come paired together, goes back at least to Peter the Pilgrim in 1269. I don’t know a first date for publication on unicorns, but it cannot have been much later. In both cases, one can find earlier but vaguer antecedents in Asia.