QUANTIZATION OF CONSTANTLY CURVED TETRAHEDRON

Quantum tetrahedron is a key building block in the theory of Loop Quantum Gravity (LQG) and plays a crucial role in the boundary states of the spinfoam amplitude of LQG. In LQG with vanishing cosmological constant, the physical Hilbert space of the quantum at tetrahedron is the 4-valent SU(2) intertwiner space labeled by irreducible representation, each assigned to a face of the quantum at tetrahedron. Furthermore, the space is the solution space of the quantum at closure condition. The area spectrum of each face of the quantum at tetrahedron is discrete and is characterized by a spin label. Classically, the correspondence between a set of solutions of at closure condition and at tetrahedron is guaranteed by the Minkowski theorem. This theorem has been generalized to the curved case, where a curved closure condition applies. The curved Minkowski theorem allows us to reconstruct homogeneously curved tetrahedra (spherical or hyperbolic tetrahedra) from a family of four SU(2) holonomies that satisfy the curved closure condition Although the quantization of the closure condition for a at tetrahedron has been extensively studied in LQG, the quantization of the curved closure condition and curved tetrahedron has not been explored yet. The homogeneously curved tetrahedron has played an important role in the recent construction of the spinfoam model with cosmological constant in 3+1 dimensional LQG. It is anticipated that the quantization of a curved tetrahedron should deFIne the building block for the boundary Hilbert space of the spinfoam model.