Flipped spinfoam vertex and loop gravity

Abstract

We introduce a vertex amplitude for 4d loop quantum gravity. We derive it from a conventional quantization of a Regge discretization of euclidean general relativity. This yields a spinfoam sum that corrects some difficulties of the Barrett-Crane theory. The second class simplicity constraints are imposed weakly, and not strongly as in Barrett-Crane theory. Thanks to a flip in the quantum algebra, the boundary states turn out to match those of SO(3) loop quantum gravity -- the two can be identified as eigenstates of the same physical quantities -- providing a solution to the problem of connecting the covariant SO(4) spinfoam formalism with the canonical SO(3) spin-network one. The vertex amplitude is SO(3) and SO(4)-covariant. It rectifies the triviality of the intertwiner dependence of the Barrett-Crane vertex, which is responsible for its failure to yield the correct propagator tensorial structure. The construction provides also an independent derivation of the kinematics of loop quantum gravity and of the result that geometry is quantized.

37 pages, 4 figures

References in corpus (7)

• LQG vertex with finite Immirzi parameter
• A new spinfoam vertex for quantum gravity
• The loop-quantum-gravity vertex-amplitude
• A "general boundary" formulation for quantum mechanics and quantum gravity
• The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
• Linearized dynamics from the 4-simplex Regge action
• Spin foam model from canonical quantization

Cited by in corpus (10)

• LQG vertex with finite Immirzi parameter
• Area-angle variables for general relativity
• The complete LQG propagator: II. Asymptotic behavior of the vertex
• Simplicity and closure constraints in spin foam models of gravity
• The perturbative Regge-calculus regime of Loop Quantum Gravity
• Coherent states, constraint classes, and area operators in the new spin-foam models
• A New Class of Group Field Theories for 1st Order Discrete Quantum Gravity
• Spin-Foam Models and the Physical Scalar Product
• Numerical indications on the semiclassical limit of the flipped vertex
• Tensorial Structure of the LQG graviton propagator