Polyhedra in spacetime from null vectors

TL;DR

This paper introduces a null vector parametrization of convex spacelike polyhedra in Minkowski space, linking classical geometry with quantum gravity models and simplifying the analysis of spin foam intertwiners.

Contribution

It provides a dimension-independent null vector parametrization of polyhedra, connecting classical shapes with quantum gravity structures and deriving a 4-simplex action consistent with Barrett-Crane amplitudes.

Findings

Null face normals parametrization with no constraints

Closure constraint satisfied via SL(2,C) rotation in 3+1d

Derived a 4-simplex action matching large-spin Barrett-Crane behavior

Abstract

We consider convex spacelike polyhedra oriented in Minkowski space. These are the classical analogues of spinfoam intertwiners. We point out a parametrization of these shapes using null face normals, with no constraints or redundancies. Our construction is dimension-independent. In 3+1d, it provides the spacetime picture behind a well-known property of the loop quantum gravity intertwiner space in spinor form, namely that the closure constraint is always satisfied after some SL(2,C) rotation. As a simple application of our variables, we incorporate them in a 4-simplex action that reproduces the large-spin behavior of the Barrett-Crane vertex amplitude.