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.