Towards a Covariant Loop Quantum Gravity

TL;DR

The paper addresses the tension between Lorentz covariance and the SU(2) gauge used in standard LQG, along with the Immirzi ambiguity. It develops Covariant Loop Quantum Gravity by performing a Lorentz-covariant canonical analysis of the Palatini-Holst action without time gauge, leading to a Lorentz-spin-network kinematics and a covariant, potentially noncommutative, connection. A key result is that the area spectrum can be continuous and independent of the Immirzi parameter in the covariant framework, while a commutative SU(2) extension recovers the familiar LQG spectrum in the time gauge; projected and simple spin networks play central roles in making the Lorentz structure compatible with the area operator. Dynamics are addressed via spin foams, notably the Barrett-Crane model, which provides a covariant evolution framework linking Covariant LQG to a path-integral formulation. Together, these developments offer a path to unify canonical and covariant approaches and to resolve the Immirzi ambiguity, albeit with open challenges in defining a rigorous Hilbert space and a complete Hamiltonian constraint.

Abstract

We review the canonical analysis of the Palatini action without going to the time gauge as in the standard derivation of Loop Quantum Gravity. This allows to keep track of the Lorentz gauge symmetry and leads to a theory of Covariant Loop Quantum Gravity. This new formulation does not suffer from the Immirzi ambiguity, it has a continuous area spectrum and uses spin networks for the Lorentz group. Finally, its dynamics can easily be related to Barrett-Crane like spin foam models.