Comparing Quantum Gravity Models: String Theory, Loop Quantum Gravity, and Entanglement gravity versus $SU(\infty)$-QGR

In a previous work [arXiv:2009.03428] we proposed a new model for Quantum GRavity(QGR) and cosmology, dubbed $SU(\infty)$-QGR. One of the axioms of this model is that Hilbert spaces of the Universe and its subsystems represent $SU(\infty)$ symmetry group. In this framework, the classical spacetime is interpreted as being the parameter space characterizing states of the $SU(\infty)$ representing Hilbert spaces. Using quantum uncertainty relations, it is shown that the parameter space - the spacetime - has a 3+1 dimensional Lorentzian geometry. Here after a review of $SU(\infty)$-QGR, including the demonstration that its classical limit is Einstein gravity, we compare it with several QGR proposals, including: string and M-theories, loop quantum gravity and related models, and QGR proposals inspired by holographic principle and quantum entanglement. The purpose is to find their common and analogous features, even if they apparently seem to have different roles and interpretations. The hope is that such exercise gives a better understanding of gravity as a universal quantum force and clarifies the physical nature of the spacetime. We identify several common features among the studied models: importance of 2D structures; algebraic decomposition to tensor products; special role of $SU(2)$ group in their formulation; necessity of a quantum time as a relational observable. We discuss how these features can be considered as analogous in different models. We also show that they arise in $SU(\infty)$-QGR without fine-tuning, additional assumptions, or restrictions.