Riemannian Geometry Framed as a Generalized Lie Algebra to Incorporate General Relativity with Quantum Theory 1

Abstract

This paper reframes Riemannian geometry as a generalized Lie algebra allowing the equations of both RG and then General Relativity to be expressed as commutation relations among fundamental operators. We begin with an Abelian Lie algebra of n operators, X, whose simultaneous eigenvalues, y, define a real n-dimensional space. Then with n new operators defined as independent functions, we define contravariant and covariant tensors in terms of their eigenvalues, on a Hilbert space representation. We then define n additional operators, D, whose exponential map is to translate X as defined by a noncommutative algebra of operators (observables) where the structure constants are shown to be the metric functions of the X operators thus allowing for spatial curvature resulting in a noncommutativity among the D operators. The D operators then have a Hilbert space position-diagonal representation as generalized differential operators plus an arbitrary vector function A(X), which, with the metric, written as a commutator, can express the Christoffel symbols, and the Riemann, Ricci and other tensors as commutators in this representation. Traditional RG and GR are obtained in a position diagonal representation of this noncommutative algebra of 2n+1 operators. We seek to provide a more general framework for RG to support an integration of GR, QT, and the SM by generalizing Lie algebras as described.