Lorentz covariance of loop quantum gravity

TL;DR

This paper presents a Lorentz-covariant formulation of loop quantum gravity by mapping SU(2) spin-network states to SL(2,C) functions, clarifying the Lorentz invariance of spinfoam dynamics and boundary states.

Contribution

It introduces a manifestly Lorentz-covariant framework for loop quantum gravity using SL(2,C) functions, connecting boundary states with bulk dynamics and clarifying the role of Lorentz invariance.

Findings

SU(2) spin-network states can be represented by SL(2,C) functions.

Spinfoam dynamics are locally SL(2,C)-invariant in the bulk.

Boundary states are precisely in the space K of SL(2,C) functions.

Abstract

The kinematics of loop gravity can be given a manifestly Lorentz-covariant formulation: the conventional SU(2)-spin-network Hilbert space can be mapped to a space K of SL(2,C) functions, where Lorentz covariance is manifest. K can be described in terms of a certain subset of the "projected" spin networks studied by Livine, Alexandrov and Dupuis. It is formed by SL(2,C) functions completely determined by their restriction on SU(2). These are square-integrable in the SU(2) scalar product, but not in the SL(2,C) one. Thus, SU(2)-spin-network states can be represented by Lorentz-covariant SL(2,C) functions, as two-component photons can be described in the Lorentz-covariant Gupta-Bleuler formalism. As shown by Wolfgang Wieland in a related paper, this manifestly Lorentz-covariant formulation can also be directly obtained from canonical quantization. We show that the spinfoam dynamics of loop quantum gravity is locally SL(2,C)-invariant in the bulk, and yields states that are preciseley in K on the boundary. This clarifies how the SL(2,C) spinfoam formalism yields an SU(2) theory on the boundary. These structures define a tidy Lorentz-covariant formalism for loop gravity.