Riemannian geometry reframed as a generalized lie algebra to integrate general relativity with the standard model

Abstract

This paper is based upon the observation that the translation operator D in a curved space–time must depend upon the position of the particle and thus one must allow the [D, X] commutator to be a function of position X in a generalized Lie algebra. This work consists of two parts; In a purely mathematical development, we first reframe Riemannian geometry (RG) as a Generalized Lie algebra (GLA) by allowing the structure constants to be functions of an Abelian subalgebra as is necessary when translations in a space of n variables depend upon the position in the space. In the second part we show that Einstein’s equations for General Relativity (GR) can now be written as commutation relations in this GLA framework including relativistic Quantum Theory (QT) and the Standard Model (SM) with novel predictions. We begin with an Abelian Lie algebra of n “position” operators, X, whose simultaneous eigenvalues, y, define a real n-dimensional space R(n) with a Hilbert space representation. Then with n new operators defined as independent functions, X^′(X), we define contravariant and covariant tensors in terms of their eigenvalues, y and y^′ with Dirac notation. We then define n additional operators, D, whose exponential map is, by definition, to translate X in a noncommutative algebra of operators (observables) where the “structure constants” are shown to be the metric functions of the X operators to allow for spatial curvature. The D operators then have a Hilbert space position-diagonal representation as a generalized differential operator plus a Christoffel symbol, Γ^µ (y), an arbitrary vector function A^µ (y), and the derivative of a scalar function g^µn∂ϕ(y)/∂yⁿ. One can then express the Christoffel symbols, and the Riemann, Ricci, and other tensors as commutators in this representation thereby framing RG as a GLA. We then show that this GLA provides a more general framework for RG to support GR, QT, the SM with novel predictions.