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A curvature operator for a regular tetrahedron shape in LQG. (English) Zbl 1421.83042

Summary: An alternative approach introducing a 3-dimensional (3D) Ricci scalar curvature quantum operator given in terms of volume and area as well as new edge length operators is proposed. An example of monochromatic 4-valent node intertwiner state (equilateral tetrahedra) is studied and the scalar curvature measure for a regular tetrahedron shape is constructed. It is shown that all regular tetrahedron states are in the negative scalar curvature regime and for the semi-classical limit the spectrum is very close to the Euclidean regime.

MSC:

83C45 Quantization of the gravitational field
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
57N16 Geometric structures on manifolds of high or arbitrary dimension
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