Area spectrum in Lorentz covariant loop gravity

TL;DR

This work develops a manifestly Lorentz covariant canonical formulation of gravity and computes the area spectrum using Wilson lines built from a true Lorentz connection. By constructing a unique shifted connection ${\cal A}$, the authors diagonalize the area operator and express its eigenvalues in terms of the quadratic Casimirs $C_2(so(3,1))$ and $C_2(so(3))$, showing no dependence on the Immirzi parameter $\beta$. The result, $\mathcal{S} \sim \hbar \sqrt{-C_2(so(3,1)) + C_2(so(3))}$, preserves Lorentz covariance and provides a potentially robust alternative to non-covariant loop formulations, with implications for spin-foam approaches and the role of the Immirzi parameter in quantum gravity. The analysis also discusses vacuum choices and the ongoing question of the spectral character (discrete vs continuous) due to the non-compact Lorentz group.

Abstract

We use the manifestly Lorentz covariant canonical formalism to evaluate eigenvalues of the area operator acting on Wilson lines. To this end we modify the standard definition of the loop states to make it applicable to the present case of non-commutative connections. The area operator is diagonalized by using the usual shift ambiguity in definition of the connection. The eigenvalues are then expressed through quadratic Casimir operators. No dependence on the Immirzi parameter appears.