关于张量分布乘法的新观点
New ideas about multiplication of tensorial distributions
摘要：
在广义相对论中，需要一个一致的和有用的数学理论，以几何（同构不变量）的方式定义张量分布的乘法。通过Colombeau代数的概念，以及任意张量微分流形上的完全Colombeau代数的构造，已经取得了重大进展。尽管这一目标已经实现，但它没有足够清楚地纳入协变导数的概念，因此用途有限。我们采取不同的方法：我们认为对光滑分布（在光滑流形上）的任何类型的偏好都是非直观的，这意味着我们所有的方法都必须完全基于Colombeau等价关系作为理论的基本特征。采取这种方法后，我们很自然地得到一个规范和几何理论定义张量操作与张量分布，包括协变导数。这也是因为我们不再需要Colombeau代数的任何明确的标准几何构造。我们的方法的最大优势还在于，它为所使用的数学概念带来了物理见解，并自然导致在（我们称之为）分段光滑流形上而不是光滑流形上制定物理学。这给物理学语言带来了更高的对称性（就像从庞加莱不变性转向自同构不变性一样），并且与我们的直觉相容，即在某种隐喻意义上的“逐点”性质“无关紧要”。
英文摘要：
There is a need in general relativity for a consistent and useful mathematical theory defining the multiplication of tensor distributions in a geometric (diffeomorphism invariant) way. Significant progress has been made through the concept of Colombeau algebras, and the construction of full Colombeau algebras on differential manifolds for arbitrary tensors. Despite the fact that this goal was achieved, it does not incorporate clearly enough the concept of covariant derivative and hence is of a limited use. We take a different approach: we consider any type of preference for smooth distributions (on a smooth manifold) as nonintuitive, which means all our approach must be based fully on the Colombeau equivalence relation as the fundamental feature of the theory. After taking this approach we very naturally obtain a canonical and geometric theory defining tensorial operations with tensorial distributions, including covariant derivative. This also happens because we no longer need any explicit canonical geometric construction of Colombeau algebras. The big advantage of our approach lies also in the fact that it brings a physical insight into the mathematical concepts used and naturally leads to formulation of physics on (what we call) piecewise smooth manifolds, rather than on smooth manifold. This brings to the language of physics much higher symmetry (in the same way as turning from Poincare invariance to diffeomorphism invariance), and is compatible with our intuition, that "pointwise" properties in some metaphorical sense "do not matter".
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。
