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 $(2n)$ and odd ($2n+1$) representations, ($n=1,2,3\dots$), spanning the complete Hilbert space $\mathcal{H}={\mathcal{H}}_{\mathrm{odd}}\oplus {\mathcal{H}}_{\mathrm{even}}$. 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 $\alpha =0$ (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.

Figure 1

Oscillation between the chiral and antichiral part of the bispinor $\varrho (t)$, or Zitterbewegung, for suitable values of the group parameters. This oscillation reflects the underlying chiral and antichiral quantum structure of the spacetime.

Figure 2

The ${\mathcal{H}}_{1/4}$ discrete (number representation) states occupy the even sector of the full Hilbert space $\mathcal{H}$. This irreducible representation of the Mp (2) group is not dense (in a topological sense) but it contains the ground state $|0⟩$.

Figure 3

The ${\mathcal{H}}_{3/4}$ discrete (number representation) states occupy the sector odd of the full Hilbert space $\mathcal{H}$. This Irreducible representation of Mp (2) is not dense (in a topological sense) but its lower or fundamental state is the first excited state $|1⟩$.

Figure 4

The squeezed quasi-probability Wigner function $W(X,T)$ of quantum space-time. $W(X,T)$ clearly shows the hyperbolic light-cone space-time structure and with symmetric form. For the coherent states, $W_{\alpha }(X,T)$ endows the hyperbolic structure but with a linear $\alpha$ tail deformation in $X$ or $T$.