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Some Basic Aspects of Fractional Quantum Numbers

I review why and how physical states with fractional quantum numbers can occur, emphasizing basic mechanisms in simple contexts. The general mechanism of charge fractionalization is the passage from states created by local action of fields to states having a topological character, which permits mixing between local and topological charges. The primeval case of charge fractionalization for domain walls, in polyacetylene and more generally, can be demonstrated convincingly using Schrieffer's intuitive counting argument, and derived formally from analysis of zero modes and vacuum polarization. An important generalization realizes chiral fermions on high-dimensional domain walls, in particular for liquid He3 in the A phase. In two spatial dimensions, fractionalization of angular momentum and quantum statistics occurs, for reasons that can be elucidated both abstractly, and specifically in the context of the quantum Hall effect.