Leech Lattice Extension of the Non-Linear Schrodinger Equation Theory of Einstein Spaces

George Chapline

doi:10.4236/jmp.2017.814134

Abstract

Although the scalar nonlinear Schrodinger equation has provided valuable insights into how quantum mechanics might modify the classical general relativistic description of space-time, a detailed understanding of space-times with matter has remained elusive. In this paper, we propose generalizing the nonlinear Schrodinger equation theory of Einstein spaces to include matter by transplanting the 3 + 1 dimensional theory to the 24-dimensional Leech lattice plus 1 time dimension. The scalar wave function and Chern-Simons gauge potential which encode the classical Kahler potential become 11 × 11 complex matrices belonging to a 195,442 dimensional representation of the Mathieu group M11. This theory describes gravity coupled to internal degrees of freedom which include a supersymmetric E6 × E6 Yang-Mills theory of matter.

Highlights

Among the known exact solutions of the Einstein’s general relativity equations self-dual Einstein spaces stand out as interesting [1]
We propose generalizing the nonlinear Schrodinger equation theory of Einstein spaces to include matter by transplanting the 3 + 1 dimensional theory to the 24-dimensional Leech lattice plus 1 time dimension
These include the physical nature of the compact objects formed by gravitational collapse, the physical nature of the big bang, and under what circumstances a vacuum energy may appear

Summary

Introduction

Among the known exact solutions of the Einstein’s general relativity equations self-dual Einstein spaces stand out as interesting [1]. Our interest is that these spaces were stimulated by the discovery that these solutions can be expressed as solutions of the nonlinear Schrodinger equation in 2 + 1 dimensions with a Chern-Simons gauge potential [2]. This in turn led to a “superfluid” representation for the vacuum state of a space-time with no matter [3] [4]. Of this development it had been suggested [5] [6] that a superfluid description of space-time might provide general insights into how quantum

Chapline DOI

Non-Linear Schrodinger Equation for Einstein Spaces

Extension to Include Matter

Conclusions