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

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 M 11 . This theory describes gravity coupled to internal degrees of freedom which include a supersymmetric E 6 × E 6 Yang-Mills theory of matter.