Complex Ashtekar variables, the Kodama state and spinfoam gravity

Abstract

Starting from a Hamiltonian description of four dimensional general relativity in presence of a cosmological constant we perform the program of canonical quantisation. This is done using complex Ashtekar variables while keeping the Barbero--Immirzi parameter real. Introducing the SL(2,C) Kodama state formally solving all first class constraints we propose a spinfoam vertex amplitude. We construct SL(2,C) boundary spinnetwork functions coloured by finite dimensional representations of the group, and derive the skein relations needed to calculate the amplitude. The space of boundary states is shown to carry a representation of the holonomy flux algebra and can naturally be equipped with a non-degenerate inner product. It fails to be positive definite, but cylindrical consistency is perfectly satisfied.

30 pages, 3 figures

References in corpus (10)

• LQG vertex with finite Immirzi parameter
• Flipped spinfoam vertex and loop gravity
• Quantum gravity with a positive cosmological constant
• Lorentz covariance of loop quantum gravity
• Cosmological Constant in LQG Vertex Amplitude
• A Note On The Chern-Simons And Kodama Wavefunctions
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• On the q-quantum gravity loop algebra

Cited by in corpus (5)

• SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry
• A note on the geometrical interpretation of quantum groups and non-commutative spaces in gravity
• Quantum cosmology of a dynamical Lambda
• Chern-Simons Theory, Vassiliev Invariants, Loop Quantum Gravity and Functional Integration Without Integration
• Revisiting EPRL: All Finite-Dimensional Solutions by Naimark's Fundamental Theorem