On the Constant that Fixes the Area Spectrum in Canonical Quantum Gravity

TL;DR

This paper determines the precise proportionality constant for the area spectrum in loop quantum gravity, showing it is 8*pi*gamma times the square root of j(j+1) times the Planck length squared.

Contribution

It calculates and fixes the value of the proportionality coefficient in the area spectrum formula within loop quantum gravity.

Findings

The proportionality factor is 8*pi*gamma.

Each spin network edge contributes an area proportional to 8*pi*gamma*sqrt{j(j+1)}.

Clarifies the coefficient used in the area spectrum formula.

Abstract

The formula for the area eigenvalues that was obtained by many authors within the approach known as loop quantum gravity states that each edge of a spin network contributes an area proportional to sqrt{j(j+1)} times Planck length squared to any surface it transversely intersects. However, some confusion exists in the literature as to a value of the proportionality coefficient. The purpose of this rather technical note is to fix this coefficient. We present a calculation which shows that in a sector of quantum theory based on the connection A=Gamma-gamma*K, where Gamma is the spin connection compatible with the triad field, K is the extrinsic curvature and gamma is Immirzi parameter, the value of the multiplicative factor is 8*pi*gamma. In other words, each edge of a spin network contributes an area 8*pi*gamma*l_p^2*sqrt{j(j+1)} to any surface it transversely intersects.