Holomorphic Lorentzian Simplicity Constraints

TL;DR

This paper introduces a Hamiltonian framework for sl(2,C) algebra using twistors, develops holomorphic simplicity constraints that commute, and links them to geometric 3d polyhedra in Minkowski space.

Contribution

It presents a novel holomorphic form of simplicity constraints that Poisson-commute and are equivalent to traditional constraints, advancing the mathematical structure of Lorentzian spin foam models.

Findings

Holomorphic simplicity constraints Poisson-commute

Complete set of invariants generates a closed algebra including gl(N,C)

Reduction to 3d spacelike polyhedra in Minkowski space

Abstract

We develop an Hamiltonian representation of the sl(2,C) algebra on a phase space consisting of N copies of twistors, or bi-spinors. We identify a complete set of global invariants, and show that they generate a closed algebra including gl(N,C) as a subalgebra. Then, we define the linear and quadratic simplicity constraints which reduce the spinor variables to (framed) 3d spacelike polyhedra embedded in Minkowski spacetime. Finally, we introduce a new version of the simplicity constraints which (i) are holomorphic and (ii) Poisson-commute with each other, and show their equivalence to the linear and quadratic constraints.