Connection between Spin, Statistics, and Kinks

Abstract

Sufficiently nonlinear classical fields admit modes called kinks, whose number is strictly conserved in virtue of boundary conditions and continuity of the field as a function of space and time. In a quantum theory of such fields, with canonical commutation (not anticommutation) relations, kinks and their conservation still persist, and even if the intrinsic angular momentum is an integer, a rotating kink can have half‐odd angular momentum, if double‐valued state functionals are admitted. We formulate a natural concept of exchange appropriate for kinks. The principal result is that for fields with integer‐valued intrinsic angular momentum, the observed relation between spin and (exchange) statistics follows from continuity alone, parastatistics being excluded. It is likely that in the theories with even (odd) exchange statistics, suitable creation operators will commute (anticommute). We show that, while the rotational spectrum of a kink will in general possess both integer and half‐odd spin states, in fields with integer‐valued intrinsic angular momentum only one of these two possibilities will ever be observed for each kind of kink, and that there is a nonzero ``particle number'' (strictly conserved, additive, scalar quantum number) attached to half‐odd‐spin kinks of each kind. It then follows that a boson and a fermion kink will always differ in at least one particle number, as well as in spin, and that, in particular, every fermion kink will have some nonzero particle number. These results are consistent with the hypothesis that the spinor fields usually employed to describe half‐odd‐spin quanta are not fundamental, but are useful ``point‐limit'' approximations to operators creating or annihilating excitations in a nonlinear field of particular kinds of kinks in particular internal states.