Leech Lattice Extension of the Non-linear Schrodinger Equation Theory of Einstein spaces

TL;DR

This paper proposes extending the nonlinear Schrödinger equation approach to Einstein spaces by embedding it into a 25+1 dimensional Leech lattice, incorporating matter fields and symmetries to address issues in gravitational collapse and cosmology.

Contribution

It introduces a novel Leech lattice-based framework that generalizes the nonlinear Schrödinger equation for Einstein spaces to include matter and supersymmetry.

Findings

Wave function becomes an 11x11 complex matrix with matter fields.

Continuous Lie group symmetries emerge from the Leech lattice automorphisms.

Potential to resolve unphysical features in gravitational collapse and big bang models.

Abstract

Although the nonlinear Schrodinger equation description of Einstein spaces has provided insights into how quantum mechanics might modify the classical general relativistic description of space-time, an exact quantum description of space-times with matter has remained elusive. In this note we outline how the nonlinear Schrodinger equation theory of Einstein spaces might be generalized to include matter by transplanting the theory to the 25+1 dimensional Lorentzian Leech lattice. Remarkably when a hexagonal section of the Leech lattice is set aside as the stage for the nonlinear Schrodinger equation, the discrete automorphism group of the complex Leech lattice with one complex direction fixed can be lifted to continuous Lie group symmetries. In this setting the wave function becomes an 11x11 complex matrix which represents matter degrees of freedom consisting of a 2-form abelian gauge field and vector nonabelian SU(3)xE6 gauge fields together with their supersymmetric partners. The lagrangian field equations for this matrix wave function appear to be a promising starting point for resolving the unphysical features inherent in the general relativistic descriptions of gravitational collapse and the big bang.