Different representations of Euclidean geometry and their application to the space-time geometry

TL;DR

This paper explores three different ways to represent Euclidean geometry, emphasizing the sigma-representation's suitability for modifying geometry to model multivariant space-time, which may explain quantum effects.

Contribution

It introduces and compares three representations of Euclidean geometry, highlighting the sigma-representation's advantages for modeling multivariant space-time geometry.

Findings

Sigma-representation effectively models multivariant geometry.

Multivariance in space-time geometry explains quantum effects.

The real space-time geometry is inherently multivariant.

Abstract

Three different representation of the proper Euclidean geometry are considered. They differ in the number of basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point, segment, angle) and no additional structures. V-representation contains two basic elements (point, vector) and additional structure: linear vector space. In sigma-representation there is only one basic element and additional structure: world function \sigma =\rho^{2}/2, where \rho is the distance. The concept of distance appears in all representations. However, as a structure, determining the geometry, the distance appears only in the sigma-representation. The sigma-representation is most appropriate for modification of the proper Euclidean geometry. Practically any modification of the proper Euclidean geometry turns it into multivariant geometry, where there are many vectors Q_0Q_1, Q_0Q_1^{\prime},..., which are equal to the vector P_0P_1, but they are not equal between themselves, in general. Concept of multivariance is very important in application to the space-time geometry. The real space-time geometry is multivariant. Multivariance of the space-time geometry is responsible for quantum effects.