Exact mathematical models of a unified quantum theory; Expanding and static micro universes

TL;DR

This paper develops exact mathematical models using Riemannian and Lorentzian metrics on nilpotent Lie groups to unify fundamental forces and describe microscopic expanding universes consistent with Hubble's law.

Contribution

It introduces a novel geometric framework that unifies fundamental interactions and models microscopic expanding universes with relativistic properties.

Findings

Unified description of electromagnetic, weak, and strong forces.

Models exhibit microscopic expansion obeying Hubble's law.

Provides a relativistic quantum theory with exact mathematical models.

Abstract

In this paper such Riemann metrics are established whose Laplace-Beltrami operators are identical to familiar Hamilton operators of elementary particle systems. Such metrics are the natural positive definite invariant metrics defined on two-step nilpotent Lie groups. The corresponding wave and Schroedinger operators emerge in the Laplacians of the static resp. solvable extensions of these nilpotent groups. The latter manifolds are endowed with natural invariant indefinite metric of Lorentz signature. Thus, these new exact mathematical models provide a relativistic theory for elementary particles. On the solvable extensions these models look like Friedmann's expanding universe being adopted to the microscopic level. Like the macroscopic one, also these microscopic expanding models obey Hubble's law. The microscopic models, however, offer much more complex structures with much more subtle explanations for some of those phenomena which have originally been clarified by the Friedmann model. One of the most important new features of this theory is that the electromagnetic, the weak-nuclear, and the strong-nuclear forces emerge in a unified way.