# Accession Number:

## AD0749855

# Title:

## 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 2. Three-Dimensional Problems; Generalizations of the Helmholtz Vector Decomposition Theorem

# Descriptive Note:

## Physical sciences research papers

# Corporate Author:

## AIR FORCE CAMBRIDGE RESEARCH LABS HANSCOM AFB MA

# Personal Author(s):

# Report Date:

## 1972-04-26

# Pagination or Media Count:

## 75.0

# Abstract:

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 the 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.

# Descriptors:

# Subject Categories:

- Statistics and Probability
- Quantum Theory and Relativity