Holomorphic Lorentzian Simplicity Constraints

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.

20 pages. v2: explicit counting of the holomorphic constraints added, and minor amendments

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• Deformation Operators of Spin Networks and Coarse-Graining
• Pachner moves in a 4d Riemannian holomorphic Spin Foam model
• Linking covariant and canonical LQG II: Spin foam projector
• Deformations of Polyhedra and Polygons by the Unitary Group
• On the Implementation of the Canonical Quantum Simplicity Constraint
• Holonomy Operator and Quantization Ambiguities on Spinor Space
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• Null twisted geometries
• Bubble Networks: Framed Discrete Geometry for Quantum Gravity
• New action for simplicial gravity in four dimensions
• Twisted geometries, twistors and conformal transformations
• A note on the secondary simplicity constraints in loop quantum gravity
• Twisted Geometries Coherent States for Loop Quantum Gravity
• Bulk amplitude and degree of divergence in 4d spin foams
• Hamiltonian spinfoam gravity
• Quantum Surface and Intertwiner Dynamics in Loop Quantum Gravity
• 3d Lorentzian loop quantum gravity and the spinor approach
• Spinfoams: Foundations
• Schwinger Pair Production in SL$(2,\mathbb{C})$ Topologically Non-Trivial Fields via Non-Abelian Worldline Instantons
• Loop-Quantum-Gravity Simplicity Constraint as Surface Defect in Complex Chern-Simons Theory
• Classical dynamics for Loop Gravity: The 2-vertex model
• Spin foams
• A simpler way of imposing simplicity constraints
• Deformations of Lorentzian Polyhedra: Kapovich-Millson phase space and SU(1,1) Intertwiners
• An octahedron of complex null rays, and conformal symmetry breaking
• Commuting Simplicity and Closure Constraints for 4D Spin Foam Models
• Polyhedra in spacetime from null vectors