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Holomorphic Lorentzian Simplicity Constraints

Maité Dupuis, Laurent Freidel, Etera R. Livine, Simone Speziale

TL;DR

The paper develops a covariant, Lorentzian spinor phase space based on ${N}$ twistors to address simplicity constraints in spin foam models. It identifies a complete set of ${\mathrm{SL}}(2,\mathbb{C})}$ invariants that realize a closed ${\mathfrak{gl}}(N,\mathbb{C})$ structure and introduces holomorphic simplicity constraints that commute and are equivalent to the standard quadratic and linear forms when the SL(2,C) closure is imposed. This holomorphic set enables a coherent, Gupta–Bleuler–style imposition on quantum states and yields a covariant lift of SU(2) intertwiners via a Y-map, connecting Lorentzian spin foams to the SU(2) boundary Hilbert space. The work paves the way for quantizing the holomorphic constraints and analyzing quantum corrections while preserving the correct semiclassical limit, with follow-up work SL2Cquant planned.

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.

Holomorphic Lorentzian Simplicity Constraints

TL;DR

The paper develops a covariant, Lorentzian spinor phase space based on twistors to address simplicity constraints in spin foam models. It identifies a complete set of invariants that realize a closed structure and introduces holomorphic simplicity constraints that commute and are equivalent to the standard quadratic and linear forms when the SL(2,C) closure is imposed. This holomorphic set enables a coherent, Gupta–Bleuler–style imposition on quantum states and yields a covariant lift of SU(2) intertwiners via a Y-map, connecting Lorentzian spin foams to the SU(2) boundary Hilbert space. The work paves the way for quantizing the holomorphic constraints and analyzing quantum corrections while preserving the correct semiclassical limit, with follow-up work SL2Cquant planned.

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.

Paper Structure

This paper contains 16 sections, 3 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Spinor Phase Space and $\mathrm{U}(N)$-Formalism for $\mathrm{SU}(2)$-Intertwiners
  3. Spinor Phase Space and ${\mathfrak{su}}(2)$-Algebra
  4. Classical Geometry of Spinors and Polyhedra
  5. Probing the $\mathrm{U}(N)$ Structure
  6. Generalizing to $\mathrm{SL}(2,{\mathbb C})$: the Basic Idea
  7. The Double-Spinor Phase Space for $\mathrm{SL}(2,{\mathbb C})$
  8. Representing $\sl(2,{\mathbb C})$ with a Pair of Spinors
  9. $\mathrm{SL}(2,{\mathbb C})$-Invariants and the ${\mathfrak{gl}}(N,{\mathbb C})$-Structure
  10. The $\mathrm{GL}(N,{\mathbb C})$ Action and Squashed Spinor Configurations
  11. Holomorphic Simplicity Constraints
  12. Holomorphic Simplicity Constraints
  13. Reduction from $\mathrm{SL}(2,{\mathbb C})$ to $\mathrm{SU}(2)$-Intertwiners and the $Y$-map
  14. Conclusions and Outlook
  15. ${\mathfrak{so}}(3,2)$ Algebra from a single Spinor
  16. ...and 1 more sections

Key Result

Proposition 1

If the holomorphic simplicity constraints ${\mathcal{C}}_{ij}=0$ are satisfied for all pairs $i,j$, and the $\sl(2,{\mathbb C})$-closure constraints hold, $\sum_i \vec{J}_i=\sum_i\vec{K}_i=0$, then the quadratic simplicity constraints are satisfied, with

Theorems & Definitions (5)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Proposition 2