Coherent states of quantum spacetimes for black holes and de Sitter spacetime

We provide a group theory approach to coherent states describing quantum space-time and its properties. This provides a relativistic framework for the metric of a Riemmanian space with bosonic and fermionic coordinates, its continuum and discrete states, and a kind of "quantum optics" for the space-time. New results of this paper are: (i) The space-time is described as a physical coherent state of the complete covering of the SL(2C) group, e.g., the metaplectic group Mp(n). (ii) (The discrete structure arises from its two irreducible even (2 n ) and odd (2 n +1 ) representations, (n =1 ,2 ,3 … ), spanning the complete Hilbert space H =H_odd⊕H_even. Such a global or complete covering guarantees the CPT symmetry and unitarity. Large n yields the classical and continuum manifold, as it must be. (iii) The coherent and squeezed states and Wigner functions of quantum-space-time for black holes and de Sitter, and (iv) for the quantum space-imaginary time (instantons), black holes in particular. They encompass the semiclassical space-time behavior plus high quantum phase oscillations, and notably account for the classical-quantum gravity duality and trans-Planckian domain. The Planck scale consistently corresponds to the coherent state eigenvalue α =0 (and to the n =0 level in the discrete representation). It is remarkable the power of coherent states in describing both continuum and discrete space-time. The quantum space-time description is regular, there is no any space-time singularity here, as it must be.