The algebraic structure of physical quantities (Q1878879)

The quantities appearing in physical or engineering computations are not pure numbers but include a label namely their dimension. The algebraic structure of these labels has been indicated in more or less extensive ways in books on dimension analysis. The aim of this paper is to explore this structure in a more detailed way. Although there have been some attempts in the past to investigate the structure of the labels from an abstract algebraic point of view this paper is the first in which advanced group theoretic concepts are used. This allows also to introduce fractional powers of a basic unit. The author establishes the algebraic structure of infinite sets of labels that represent the ``units'' as infinite Abelian multiplicative groups with a finite basis. The author states that the paper is written for non-mathematicians. However the advanced mathematical language making strongly use of group theoretic concepts makes one wonder whether the interesting message of the paper will be accessible even for mathematically well trained engineers. The abstract general exposition is well complemented with several examples from physics and chemistry. Moreover it is shown that the developed properties of labeled quantities lead naturally to the concept of well posed relations and to Buckingham's theorem of dimensional analysis. Reading of this paper is strongly recommended for all scientists using dimensional analysis.