Twistor Networks and Covariant Twisted Geometries

TL;DR

This paper explores the phase space of twistors and their relation to Lorentz group elements, introducing twistor networks as a classical analogue of spin foam boundary states and providing explicit formulas for their structure.

Contribution

It presents a detailed symplectic reduction of twistor phase space, defines twistor networks, and connects them to spin foam models with explicit measure expressions.

Findings

Derived Lorentz generators from twistors.

Defined twistor networks on graphs.

Expressed Haar measure in spinor variables.

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.