Intrinsic geometry and constructivity methods for Hilbert's 6th problem

The main mathematical work of this paper is to establish a theoretical framework based on a unique basic principle or axiom, so that the major components of theoretical physics can be constructed, and finally the redundant principles and postulates of traditional fundamental physics, as well as artificially introduced equations, can all be turned into theorems which hold automatically in the theory of this paper.

The key ideas are as following. (1)Improve the expression form of Erlangen program, and then generalize Riemannian manifold to geometric manifold. On geometric manifold, bring Riemannian geometry into the geometric framework of improved Erlangen program. (2)Strictly define the general concept of reference-system and generalize the concept of intrinsic geometry, so that the traditional intrinsic geometry based on the first fundamental form becomes a subgeometry of the intrinsic geometry of this paper. (3)Define the concept of simple connection and use it to describe those bending properties that cannot be described by Levi-Civita connection.

Other important ideas are as following. (1)Time metric is defined as the total metric of space. (2)Actual evolution direction is defined as the gradient direction of geometric quantity. (3)Gauge potential is defined as simple connection. (4)Gauge transformation is defined as the transformation of general reference-systems. (5)Energy-momentum of general charge is defined as the absolute derivative of charge tensor, and canonical energy-momentum is defined as the normal derivative. (6)Feynmann propagator and wave function are expressed as the distribution density of actual evolution direction field, which are defined as functions related to measure and become probability after normalization.

The idea of symmetry emphasized in traditional theoretical physics is more convenient to be expressed in the viewpoint of geometry. Concretely, (1)the traditional theory starts from a very large symmetry group, and reduces symmetries in the way of some kind of breaking to approach the target geometry; (2)the theory of this paper starts from the smallest symmetry group $\{e\}$, and adds symmetries in the way of some kind of symmetry conditions to approach the target geometry. These two ways must lead to the same destination. They both go towards the same specific geometry. The way of this paper has more advantages.

Based on these ideas, the concepts of charged lepton, neutrino, down-type color charge, up-type color charge and various gauge potentials are all distinguished by constructive definitions, so that the asymmetric characteristic of chirality of weak interaction, the MNS mixing of leptons and the CKM mixing of color charges hold automatically. There is no need to artificially set up these postulates like the standard model.