CHAPTER 6 - TENSORS

Abstract

Publisher Summary The chief reason for the introduction of irreducible tensors stems from the Wigner–Eckart theorem. The Wigner–Eckart theorem separates the features of a physical process that depend on geometry or symmetry properties from those that depend on the detailed physical interactions. Under a general linear coordinate transformation, second rank tensors decompose into symmetric and antisymmetric parts that are irreducible, provided no additional restrictions are imposed on the transformations. However, further reduction is possible if the coordinate transformation is orthogonal. The transformation law is derivable from the commutation relations. Permutation operators, acting on the indices of the tensor components, commute with the linear transformation. If the indices of a tensor are permuted, the new tensor obeys the same transformation law as the original tensor. Therefore, tensor components may be organized into linear combinations that form bases for irreducible representations of the permutation groups. Such combinations transform among themselves.