Entanglement and Generalized Berry Geometrical Phases in Quantum Gravity

TL;DR

This paper introduces a new formalism based on the Minimum Group Representation Principle to analyze quantum spacetime, revealing novel entanglement and geometrical phases in cosmological and black hole contexts, with implications for quantum gravity dualities.

Contribution

It develops a formalism using the Metaplectic group to describe quantum spacetime, uncovering new entanglement structures and Berry phases relevant to cosmology and black holes.

Findings

Berry phases in inflationary cosmology linked to observable indices

Pure entangled states characterized in the Metaplectic group framework

Entanglement between different spacetime regions and quantum gravity duality

Abstract

A new formalism is introduced describe the physical and geometric content of quantum spacetime. It is based in the Minimum Group Representation Principle. New results for entanglement and geometrical/topological phases are found and implemented in cosmological and black hole space-times. Our main results here are: (i) The Berry phases for inflation, for the cosmological perturbations, and its expression in terms of observables, as the spectral scalar and tensor indices, $n_S$ an $n_T$, and their ratio $r$. The Berry phase for de Sitter inflation is imaginary, its sign describing the exponential acceleration. (ii) The pure entangled states in the minimum group (metaplectic) $Mp(n)$ representation for quantum de Sitter space-time and black holes are found. (iii) For entanglement, the relation between the Schmidt type representation and the physical states of the $Mp(n)$ group is found: This is a new non-diagonal coherent state representation complementary to the known Sudarshan diagonal one. (iv) The mean $Mp(2)$ generator values are related to the space-time topological charge. (v) The basic even and odd $n$ -sectors of the Hilbert space are intrinsic to the quantum spacetime and its discrete levels (continuum for $n \rightarrow \infty$) and are it entangled. (vi) The gravity or cosmological domains on one side and another of the Planck scale are entangled. Examples: The primordial quantum trans-Planckian de Sitter vacuum and the late classical gravity de Sitter vacuum today; the central quantum reqion and the external classical region of black holes. The classical and quantum dual gravity regions of the space-time are entangled. (vii) The general classical-quantum gravity duality is associated to the Metaplectic $Mp(n)$ group symmetry which provides the complete full covering of the phase space and of the quantum space-time mapped from it.