Categorical Operator Algebraic Foundations of Relational Quantum Theory

TL;DR

The work addresses reconstructing space-time from quantum correlations within a relational quantum framework by proposing a higher-C*-categorical, algebraic structure for systems as $C^*$-algebras and correlations as bimodules. It develops a spectral conjecture that quantum geometry can be captured by families of Fell bundles over involutive categories, connecting non-commutativity to categorical relations among points, and introduces hyper-C*-algebras and quantum higher $*$-categories to model multi-level observer interactions. By integrating Tomita-Takesaki modular theory, it defines modular spectral geometries associated with pairs of algebras and states and sketches a program to reconstruct a relational space-time from these modular data. Overall, the framework aims to derive geometry as a spectral consequence of quantum correlations within a higher categorical, modular setting, highlighting a path toward a relational quantum gravity theory.

Abstract

We provide an algebraic formulation of C.Rovelli's relational quantum theory that is based on suitable notions of "non-commutative" higher operator categories, originally developed in the study of categorical non-commutative geometry. As a way to implement C.Rovelli's original intuition on the relational origin of space-time, in the context of our proposed algebraic approach to quantum gravity via Tomita-Takesaki modular theory, we tentatively suggest to use this categorical formalism in order to spectrally reconstruct non-commutative relational space-time geometries from categories of correlation bimodules between operator algebras of observables. Parts of this work are joint collaborations with: Dr.Roberto Conti (Sapienza Universita' di Roma), Assoc.Prof.Wicharn Lewkeeratiyutkul (Chulalongkorn University, Bangkok), Dr.Rachel Dawe Martins (Istituto Superior Tecnico, Lisboa), Dr.Matti Raasakka (Paris 13 University), Dr.Noppakhun Suthichitranont.