  In the presence of topologically nontrivial bosonic field configurations, the
fermion number operator may take on fractional eigenvalues, because of the
existence of zero-energy fermion modes. The simplest examples of this occur in
1+1 dimensions, with zero modes attached to kink-type solitons. In the presence
of a kink-antikink pair, the two associated zero modes bifurcate into positive
and negative energy levels with energies $\pm ge^{-g\Delta}$, in terms of the
Yukawa coupling $g\ll 1$ and the distance $\Delta$ between the kink and
antikink centers. When the kink and antikink are moving, it seems that there
could be Landau-Zener-like transitions between these two fermionic modes, which
would be interpretable as the creation or annihilation of fermion-antifermion
pairs; however, with only two solitons in relative motion, this does not occur.
If a third solitary wave is introduced farther away to perturb the
kink-antikink system, a movement of the faraway kink can induce transitions
between the discrete fermion modes bound to the solitons. These state changes
can be interpreted globally as creation or destruction of a novel type of pair:
a half-fermion and a half-antifermion. The production of the half-integral
pairs will dominate over other particle production channels as long as the
solitary waves remain well separated, so that there is a manifold of discrete
fermion states whose energies are either zero or exponentially close to zero.
