  Some time ago Rubakov and Shaposhnikov (RS) suggested that elementary
particles might be excitations trapped on a soliton in a flat higher
dimensional space. They gave as an example phi-4 theory in five dimensions with
a bosonic excitation on a domain wall in the fifth dimension. They also trapped
a chiral Dirac particle on the domain wall. We extend the RS field equation for
the Dirac particle to eight flat dimensions (M4 X R4) and replace the
one-dimensional domain-wall potential with a harmonic- oscillator "potential"
in the four higher dimensions. This generates bound states in approximate SU(4)
X SU(2) representations. These representations have corresponding "orbital"
times "spin" quantum numbers and bear some resemblance to the quarks and
leptons of the Standard Model. Soliton-theory suggests that the
harmonic-oscillator potential should rise only a finite amount. This will then
limit the number of generations in a natural way. It will also mean that
particles with sufficient energy might escape the well altogether and propagate
freely in the higher dimensions. It may be worth- while to search for a soliton
(instanton?) which can confine Dirac particles in the manner of the
harmonic-oscillator potential.
