#
APS April Meeting 2020

## Volume 65, Number 2

##
Saturday–Tuesday, April 18–21, 2020;
Washington D.C.

### Session C15: Quantum Theory of Gravity

1:30 PM–3:18 PM,
Saturday, April 18, 2020

Room: Virginia B

Sponsoring
Unit:
DGRAV

Chair: David Craig, Oregon State University

### Abstract: C15.00007 : Riemannian Geometry and General Relativity Reframed as a Generalized Lie Algebra

2:42 PM–2:54 PM
On Demand

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Abstract

####
Author:

Joseph Johnson

(Univ of South Carolina)

Quantum Theory (QT) and the Standard Model (SM) are expressible in Lie
algebra frameworks while General Relativity (GR) is framed in the non-linear
differential equations of Riemannian Geometry (RG), a very different
framework that makes their union difficult. We show that RG can be reframed
as a NonCommutative Algebra (NCA) that is a generalization of a Lie algebra
(LA) where ``structure functions'' of position (X) generalize the LA
structure constants. Such a NCA becomes an (approximate) LA in small regions
of space-time. We begin with an Abelian algebra of n Hermitian operators
X$^{\mathrm{\mu }}$ ($\mu \quad =$ 0, 1, .. n-1) with representations on a
Hilbert space whose eigenvalues represent independent variables such as
space-time. We define operators D$^{\mathrm{\mu }}$ that by definition
translate the corresponding eigenvalues of X$^{\mathrm{\mu }}$ each by a
distance ds as dX$^{\mathrm{\lambda }}$(ds) $=$ exp(a ds $\eta
_{\mathrm{\mu \thinspace }}$D$^{\mathrm{\mu }})$ X$^{\mathrm{\lambda
\thinspace }}$exp(-a ds $\eta_{\mathrm{\nu \thinspace }}$D$^{\mathrm{\nu
}}$ ) - X$^{\mathrm{\lambda }} \quad =$ ds $\eta_{\mathrm{\mu \thinspace }}$[
D$^{\mathrm{\mu }}$, X$^{\mathrm{\thinspace \lambda }}$]/a $+$ ho where a is
a constant and $\eta_{\mathrm{\mu }}$ is a unit vector for the
translation. We define the functions g$^{\mathrm{\mu \nu }}$(X) $=$
[D$^{\mathrm{\mu }}$, X$^{\mathrm{\nu }}$]/a and show that ds$^{\mathrm{2}}$
$=$ g$_{\mathrm{\mu \nu }}$(X) dX$^{\mathrm{\mu }}_{\mathrm{\thinspace
}}$dX$^{\mathrm{\nu }}$ proving that g$_{\mathrm{\mu \nu }}$(X) is the
metric for the space taken in the position diagonal representation where
D$^{\mathrm{\thinspace \mu }} \quad =$ a g$^{\mathrm{\mu \upsilon }}$(y)
($\partial $/$\partial $y$^{\mathrm{\nu }}) \quad +$ A$^{\mathrm{\mu }}$ (y)
thereby defining [D$^{\mathrm{\thinspace \mu }}$, D$^{\mathrm{\thinspace \nu
}}$]. Integration with QT gives a $=$ iž. Details and predictions are
discussed.