Irreducible Cartesian Tensors. II. General Formulation

Abstract

A general formulation is given of a method of reduction of Cartesian tensors, by Cartesian tensor operations, to tensors irreducible under the three-dimensional rotation group. The criterion of irreducibility is that a tensor be representable as a traceless symmetric tensor, its reduced or natural form, invariantly embedded in the space of appropriate order. The general formulation exploits the properties of invariant linear mappings between tensor spaces. Considered abstractly, such mappings bring out the structure of the theory and illuminate the relation to spherical tensor theory. On the other hand, any linear invariant mapping between tensor spaces is equivalent to a combination of operations with the elementary invariant tensors U and ε. The general abstract formation therefore has a direct operational representation in terms of the ordinary tensor operations of contraction and permutation of indices. An analogous formulation is given for spinors, and the relations between spinors, Cartesian tensors, and spherical tensors is discussed in the language of the present formalism. Lastly, several examples are given as to how the general formalism may be applied to groups other than the rotation group.