Twistor Networks and Covariant Twisted Geometries

TL;DR

This work develops a covariant twistor formulation for loop quantum gravity by doubling twistors on each graph edge and imposing a complex area-matching constraint, which reduces the edge phase space to $T^*\mathrm{SL}(2,\mathbb{C})$. It provides explicit spinor-based parameterizations of Lorentz generators $\vec J,\vec K$ and the holonomy $G$, derives the associated Poisson structure, and interprets the reduced space in terms of covariant twisted geometries via a BV-like bivector decomposition and a para-ell decomposition of holonomies. The authors introduce twistor networks as a Lorentz-covariant generalization of spinor networks, with a first-order action linking spinors and edge holonomies, and identify simple twistor networks as the classical analogue of simple projected spin networks relevant to the EPRL/FK boundary states. They also present a spinor-based expression for the Haar measure on $\mathrm{SL}(2,\mathbb{C})$, enabling practical calculations in this formalism. Overall, the framework provides new tools to study covariant properties of loop quantum gravity and spin foams, with potential applications to coherent states and spin-foam amplitudes.

Abstract

We study the symplectic reduction of the phase space of two twistors to the cotangent bundle of the Lorentz group. We provide expressions for the Lorentz generators and group elements in terms of the spinors defining the twistors. We use this to define twistor networks as a graph carrying the phase space of two twistors on each edge. We also introduce simple twistor networks, which provide a classical version of the simple projected spin networks living on the boundary Hilbert space of EPRL/FK spin foam models. Finally, we give an expression for the Haar measure in terms of spinors.

Theorems & Definitions (1)

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