The Expansion of Physical Quantities in Terms of the Irreducible Representations of the Scale-Euclidean Group and Applications to the Construction of Scale-Invariant Correlation Functions. Part II. Three-Dimensional Problems; Generalizations of the Helmholtz Vector Decomposition Theorem

摘要

The irreducible representations of the scale-Euclidean group in three dimensions are introduced, and the general tensor is expanded in terms of these representations. The cases of zero-rank tensor (scalar), rank-1 tensor (vector), and rank-2 tensor, are studied in detail. The expansion is shown to be a generalization of the Helmholtz expansion of a vector into rotational and irrotational parts. As in Part 1 of th work (Concepts: One-Dimensional Problems), the correlations that are introduced are invariant under changes of frames of reference. Correlations are set up between tensors of different ranks and dimensions. A correlation that measures a degree of isotropy is defined. (Author)