A Mathematicians' View of Geometrical Unification of General Relativity and Quantum Physics

TL;DR

This paper presents a purely geometric framework based on a high-dimensional pseudo-Riemannian manifold to unify general relativity and quantum physics without additional objects or principles.

Contribution

It introduces a geometric approach that derives physical laws from curvature properties of a manifold, avoiding traditional Lagrangian or Hamiltonian formalisms.

Findings

Derives Einstein and quantum equations from geometric constraints.

Provides a new perspective on quantum phenomena through geometry.

Avoids conventional physical objects like fields or particles.

Abstract

This document contains a description of physics entirely based on a geometric presentation: all of the theory is described giving only a pseudo-riemannian manifold (M, g) of dimension n > 5 for which the g tensor is, in studied domains, almost everywhere of signature (-, -, +, ..., +). No object is added to this space-time, no general principle is supposed. The properties we impose to some domains of (M, g) are only simple geometric constraints, essentially based on the concept of "curvature". These geometric properties allow to define, depending on considered cases, some objects (frequently depicted by tensors) that are similar to the classical physics ones, they are however built here only from the g tensor. The links between these objects, coming from their natural definitions, give, applying standard theorems from the pseudo-riemannian geometry, all equations governing physical phenomena usually described by classical theories, including general relativity and quantum physics. The purely geometric approach introduced hear on quantum phenomena is profoundly different from the standard one. Neither Lagrangian nor Hamiltonian are used. This document ends with a quick presentation of our approach of complex quantum phenomena usually studied by quantum field theory.