$SU(\infty)$ Quantum Gravity: Emergence of Gravity in an Infinitely Divisible Quantum Universe

Abstract: $SU(\infty)-QGR$ is a foundationally quantum approach to cosmology and gravity. It assumes that the Hilbert space of the Universe as a whole represents the symmetry group $SU(\infty)$, and demonstrates this symmetry for Hilbert spaces of infinite number of subsystems, which randomly emerge and represent arbitrary finite rank {\it internal} symmetries. The aim of present work is in depth study of the foundation and properties of this model. We show that the global $SU(\infty)$ symmetry manifests itself through the entanglement of each subsystem with the rest of the Universe. We demonstrate that the states of subsystems depend on a dimensionful parameter arising due to the breaking of a global $U(1)$ symmetry. A relative dynamics arises when an arbitrary subsystem is selected as a quantum clock with a time parameter. Thus, states of subsystems are characterized by 4 continuous parameters related to their $SU(\infty)$ symmetry and dynamics, plus discrete parameters characterizing their internal symmetries. We demonstrate the irrelevance of the geometry of this parameter space for observables. On the other hand, we use quantum speed limits to show that the perceived classical spacetime and its Lorentzian geometry emerge as an average path of subsystems in their Hilbert space. In this respect, $SU(\infty)-QGR$ fundamentally deviates from gauge-gravity duality models. Invariance under reparameterization restricts the action of subsystems dynamics to a Yang-Mills quantum field theory defined on the (3+1)-dimensional parameter space for both $SU(\infty)$ - gravity - and internal symmetries. Consequently, $SU(\infty)-QGR$ is renormalizable, but predicts a spin-1 mediator for quantum gravity. Nonetheless, it is proved that when quantum gravity effects are not detectable, the dynamics is perceived as the Einstein-Hilbert action. We briefly discuss $SU(\infty)-QGR$ specific models for dark energy.