Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Euclidean Theory

Details Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Euclidean Theory Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Euclidean Theory

2011-09-02

arxiv.org/abs/1109.0500v2

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

27 pages, 5 figures, references added

Scientific paper

We study the large-j asymptotics of the Euclidean EPRL/FK spin foam amplitude on a 4d simplicial complex with arbitrary number of simplices. We show that for a critical configuration (j_f, g_{ve}, n_{ef}) in general, there exists a partition of the simplicial complex into three regions: Non-degenerate region, Type-A degenerate region and Type-B degenerate region. On both the non-degenerate and Type-A degenerate regions, the critical configuration implies a non-degenerate Euclidean geometry, while on the Type-B degenerate region, the critical configuration implies a vector geometry. Furthermore we can split the Non-degenerate and Type-A regions into sub-complexes according to the sign of Euclidean oriented 4-simplex volume. On each sub-complex, the spin foam amplitude at critical configuration gives a Regge action that contains a sign factor sgn(V_4(v)) of the oriented 4-simplices volume. Therefore the Regge action reproduced here can be viewed as a discretized Palatini action with on-shell connection. The asymptotic formula of the spin foam amplitude is given by a sum of the amplitudes evaluated at all possible critical configurations, which are the products of the amplitudes associated to different type of geometries.

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