Quantum state of fields in $SU(\infty)$ Quantum Gravity ($SU(\infty)$-QGR)
Abstract
Our Universe is ruled by quantum mechanics and should be treated as a quantum system. $SU(\infty)$-QGR is a recently proposed quantum model for the Universe, in which gravity is associated to $SU(\infty)$ symmetry of its Hilbert space. Fragmentation of its infinite dimensional state due to random quantum fluctuations divides the Universe to approximately isolated subsystems. In addition to parameters of their {\it internal} finite rank symmetries, states and dynamics of subsystems are characterized by 4 continuous parameters and the perceived classical spacetime is their effective representation, reflecting quantum states of subsystems and their relative evolution. At lowest order the effective Lagrangian of $SU(\infty)$-QGR~has the form of Yang-Mills gauge theories for both $SU(\infty)$ - gravity - and internal symmetries defined on the aforementioned 4D parameter space. In the present work we study more thoroughly some of the fundamental aspects of $SU(\infty)$-QGR. Specifically, we clarify the impact of the degeneracies of $\mathcal{SU}(\infty)$ algebra on the construction of the model; describe mixed states of subsystems and their purification; calculate measures of their entanglement to the rest of the Universe; and discuss their role in the emergence of local gauge symmetries. We also describe the relationship between what is called {\it internal space} of $SU(\infty)$ Yang-Mills with the 4D parameter space, and analytically demonstrate irrelevance of the geometry of…
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