Families of Spinor Fields

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Abstract
It is shown that a system of spinor fields can be classified uniquely into several sets, called families, when the interaction is given. This is made possible by noting that an interaction imposes some restriction on the commutation relation between different spinor fields. The fermion system is then classified, by a method of construction, into a set of families characterized by the following conditions. (1) Spinor fields of the same family anticommute with each other. (2) Commutation relations between any pair of spinor fields of different families are completely unspecified. To this end, we use a mathematical device which is derived from the self-evident requirement that equations of motion for fermions are determined unambiguously by the variation principle. We have applied the present method to the classification of observed spinor fields and discussed the structure of families in some detail.

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