标题： 假设和公理：描述刚体物理的数学结构 显示英文标题

标题： Assumptions and Axioms: Mathematical Structures to Describe the Physics of Rigid Bodies

摘要： 本文挑战了用于描述物理世界的数学中的一些常见假设。 我们首先回顾了实物体、物理刚体以及这类刚体的平移和旋转性质背后的许多假设。 几乎所有初级和高级教材都做出了与我们的假设略有不同的物理假设，因此我们得出的数学描述也与标准数学结构略有不同。 利用空间的均匀性和各向同性，我们研究了二维和三维空间中刚体的平移和旋转特性。 我们发现，刚体的概念与空间均匀性的概念是内在联系的。 对刚体旋转的几何研究导致了直线和向量之间的几何积关系。 通过要求该乘积既满足结合律又满足毕达哥拉斯定理，我们得到了一种选择 Clifford 代数的方式。 我们将论证从空间扩展到时间。 通过假设 ct=l 并将其重写为洛伦兹不变形式 c^2t^2-x^2-y^2-z^2=0，我们得到了毕达哥拉斯定理在时空中的推广。 这直接引导我们确立 Clifford 代数 CL(1,3) 是描述时空的合适数学结构。 Clifford 代数不是除法代数。 我们证明了代数中非可逆元素的存在并不是该代数对物理学有用性的限制，而是准确反映了物理系统的时空特性。 显示英文摘要

摘要： This paper challenges some of the common assumptions underlying the mathematics used to describe the physical world. We start by reviewing many of the assumptions underlying the concepts of real, physical, rigid bodies and the translational and rotational properties of such rigid bodies. Nearly all elementary and advanced texts make physical assumptions that are subtly different from ours, and as a result we develop a mathematical description that is subtly different from the standard mathematical structure. Using the homogeneity and isotropy of space, we investigate the translational and rotational features of rigid bodies in two and three dimensions. We find that the concept of rigid bodies and the concept of the homogeneity of space are intrinsically linked. The geometric study of rotations of rigid objects leads to a geometric product relationship for lines and vectors. By requiring this product to be both associative and to satisfy Pythagoras' theorem, we obtain a choice of Clifford algebras. We extend our arguments from space to include time. By assuming that ct=l and rewriting this in Lorentz invariant form as c^2t^2-x^2-y^2-z^2=0 we obtain a generalization of Pythagoras to spacetime. This leads us directly to establishing that the Clifford algebra CL(1,3) is an appropriate mathematical structure to describe spacetime. Clifford algebras are not division algebras. We show that the existence of non-invertible elements in the algebra is not a limitation of the usefulness to physics of the algebra but rather that it reflects accurately the spacetime properties of physical systems.