Dimensioned Algebra: Mathematics with Physical Quantities

Abstract

A rigorous mathematical theory of dimensional analysis, systematically accounting for the use of physical quantities in science and engineering, perhaps surprisingly, was not developed until relatively recently. We claim that this has shaped current mathematical models of theoretical physics, which generally lack any explicit reference to units of measurement, and we propose a novel mathematical framework to alleviate this. Our proposal is a generalization of the usual categories of algebraic structures used to formulate physical theories (groups, rings, vector spaces...), herein dubbed dimensioned, that can naturally articulate the structure of physical dimension. Our goal in the present work is not so much to define an algebraic theory of physical quantities—this has already been done—but to define a theory of algebra informed by how physical quantities are used in practice. We conclude by studying the dimensioned analogue of Poisson algebras in some detail due to their relevance in Jacobi geometry and classical mechanics.