A new look at Lorentz-Covariant Loop Quantum Gravity

TL;DR

This paper investigates the classical and quantum aspects of a Lorentz-covariant connection in loop quantum gravity, revealing its relation to SU(2) structures and confirming the consistency of the time-gauge approach at the quantum level.

Contribution

It introduces a unique Lorentz-covariant connection that simplifies to SU(2) spin-network states, preserving the quantum spectrum and confirming the gauge's consistency.

Findings

The Lorentz-covariant connection lies in the conjugacy class of an SU(2) connection.

The Lorentz-covariant electric field is unique up to trivial equivalence.

The area operator's algebraic structure is related to Casimir operators.

Abstract

In this work, we study the classical and quantum properties of the unique commutative Lorentz-covariant connection for loop quantum gravity. This connection has been found after solving the second-class constraints inherited from the canonical analysis of the Holst action without the time-gauge. We show that it has the property of lying in the conjugacy class of a pure $\su(2)$ connection, a result which enables one to construct the kinematical Hilbert space of the Lorentz-covariant theory in terms of the usual $\SU(2)$ spin-network states. Furthermore, we show that there is a unique Lorentz-covariant electric field, up to trivial and natural equivalence relations. The Lorentz-covariant electric field transforms under the adjoint action of the Lorentz group, and the associated Casimir operators are shown to be proportional to the area density. This gives a very interesting algebraic interpretation of the area. Finally, we show that the action of the surface operator on the Lorentz-covariant holonomies reproduces exactly the usual discrete $\SU(2)$ spectrum of time-gauge loop quantum gravity. In other words, the use of the time-gauge does not introduce anomalies in the quantum theory.