抽象代数：物理量的数学
Dimensioned Algebra: the mathematics of physical quantities
摘要：
也许令人惊讶的是，直到最近才发展出一种严格的量纲分析数学理论，系统地解释了科学和工程中物理量的使用。我们声称，这已经形成了目前的理论物理数学模型，通常缺乏任何明确的参考测量单位，我们提出了一个新的数学框架，以减轻这一点。我们的建议是用于制定物理理论（群，环，向量空间......）的代数结构的通常类别的推广，在此被称为维度，其可以自然地表达物理维度的结构。我们目前工作的目标并不是要定义一个物理量的代数理论--这已经完成了--而是要定义一个由物理量在实践中如何使用所提供信息的代数理论。最后，我们通过研究泊松代数的维数模拟在一定程度上，由于它们在雅可比几何和经典力学的相关性的细节。作者在后续论文中进一步探讨了这些主题。
英文摘要：
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. These topics are further explored in sequel papers by the author.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
