3Skyrmions

III Classical and Quantum Solitons



3 Skyrmions
We now move on to one dimension higher, and study Skyrmions. In recent years,
there has been a lot of interest in what people call “Skyrmions”, but what they
are studying is a 2-dimensional variant of the original idea of Skyrmions. These
occur in certain exotic magnetic systems. But instead, we are going to study the
original Skyrmions as discovered by Skyrme, which have applications to nuclear
physics.
With details to be filled in soon, hadronic physics exhibits (approximate)
spontaneously broken chiral symmetry
SU(2)
L
×SU(2)
R
Z
2
=
SO
(4), where the
unbroken group is (diagonal)
SO
(3) isospin, and the elements of
SO
(3) are
(g, g) SU(2) × SU(2).
This symmetry is captured in various effective field theories of pions (which
are the approximate Goldstone bosons) and heavier mesons. It is also a feature
of QCD with very light u and d quarks.
The special feature of Skyrmion theory is that we describe nucleons as solitons
in the effective field theory. Skyrme’s original idea was that nucleons and bigger
nuclei can be modelled by classical approximations to some “condensates” of
pion fields. To explain the conservation of baryon number, the classical field
equations have soliton solutions (Skyrmions) with an integer topological charge.
This topological charge is then identified with what is known, physically, as the
baryon number.

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