欧氏几何变形及其在微观时空几何中的应用
The Euclidean geometry deformations and capacities of their application to microcosm space-time geometry
摘要：
通常黎曼几何被认为是最一般的几何，它可以被用作时空几何。事实上，任何黎曼几何都是欧几里得几何变形的结果。这类黎曼变形受到一系列毫无根据的约束的限制。消除这些限制，我们得到了更广泛的可能时空几何（T-几何）。任何T-几何都完全由世界函数描述。T-几何是微观世界研究的有力工具，它有三个显著的特点：（1）任何几何对象在所有T-几何中同时被定义，因为它的定义不依赖于世界函数的形式; (2)T-几何的语言不使用外部的描述手段，如坐标和曲线;它只使用原始几何概念：子空间和世界函数。(3)没有必要构造T-几何的完备公理系统，因为它使用变形的欧几里德公理系统，并且人们只能研究有趣的几何关系。本文讨论了T-几何描述微观世界的能力。当世界函数是对称的且T几何是非简并的时，粒子质量被几何化，非相对论性量子效应被描述为几何效应，即不涉及量子理论的原理。当世界函数不对称时，未来与过去不是几何等价的，T-几何的容量成倍增加。世界函数的反对称分量产生了一些度量场，它们对几何性质的影响在微观世界中尤为强烈。
英文摘要：
Usually a Riemannian geometry is considered to be the most general geometry, which could be used as a space-time geometry. In fact, any Riemannian geometry is a result of some deformation of the Euclidean geometry. Class of these Riemannian deformations is restricted by a series of unfounded constraints. Eliminating these constraints, one obtains a more wide class of possible space-time geometries (T-geometries). Any T-geometry is described by the world function completely. T-geometry is a powerful tool for the microcosm investigations due to three its characteristic features: (1) Any geometric object is defined in all T-geometries at once, because its definition does not depend on the form of world function. (2) Language of T-geometry does not use external means of description such as coordinates and curves; it uses only primordially geometrical concepts: subspaces and world function. (3) There is no necessity to construct the complete axiomatics of T-geometry, because it uses deformed Euclidean axiomatics, and one can investigate only interesting geometric relations. Capacities of T-geometries for the microcosm description are discussed in the paper. When the world function is symmetric and T-geometry is nondegenerate, the particle mass is geometrized, and nonrelativistic quantum effects are described as geometric ones, i.e. without a reference to principles of quantum theory. When world function is asymmetric, the future is not geometrically equivalent to the past, and capacities of T-geometry increase multiply. Antisymmetric component of the world function generates some metric fields, whose influence on geometry properties is especially strong in the microcosm.
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