Degenerate Plebanski Sector and Spin Foam Quantization

Abstract

We show that the degenerate sector of Spin(4) Plebanski formulation of four-dimensional gravity is exactly solvable and describes covariantly embedded SU(2) BF theory. This fact ensures that its spin foam quantization is given by the SU(2) Crane-Yetter model and allows to test various approaches of imposing the simplicity constraints. Our analysis strongly suggests that restricting representations and intertwiners in the state sum for Spin(4) BF theory is not sufficient to get the correct vertex amplitude. Instead, for a general theory of Plebanski type, we propose a quantization procedure which is by construction equivalent to the canonical path integral quantization and, being applied to our model, reproduces the SU(2) Crane-Yetter state sum. A characteristic feature of this procedure is the use of secondary second class constraints on an equal footing with the primary simplicity constraints, which leads to a new formula for the vertex amplitude.

34 pages; changes in the abstract and introduction, a few references added

References in corpus (16)

• LQG vertex with finite Immirzi parameter
• A New Spin Foam Model for 4d Gravity
• Flipped spinfoam vertex and loop gravity
• Group field theory with non-commutative metric variables
• Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity
• Consistently Solving the Simplicity Constraints for Spinfoam Quantum Gravity
• Lorentz covariance of loop quantum gravity
• Spin Foams and Canonical Quantization
• Spin foam model from canonical quantization
• Simplicity and closure constraints in spin foam models of gravity
• Hamiltonian Analysis of non-chiral Plebanski Theory and its Generalizations
• Plebanski Theory and Covariant Canonical Formulation
• Covariant canonical formalism for four-dimensional BF theory
• Gauge symmetries in spinfoam gravity: the case for "cellular quantization"
• Testing the imposition of the Spin Foam Simplicity Constraints
• Spontaneously broken Lorentz symmetry for Hamiltonian gravity

Cited by in corpus (6)

• The Spin Foam Approach to Quantum Gravity
• Discrete Gravity Models and Loop Quantum Gravity: a Short Review
• Chiral description of ghost-free massive gravity
• First order gravity on the light front
• A note on the secondary simplicity constraints in loop quantum gravity
• Spinfoams: Foundations