Categorical Operator Algebraic Foundations of Relational Quantum Theory

TL;DR

This paper develops an algebraic framework for relational quantum theory using non-commutative higher operator categories, aiming to reconstruct non-commutative space-time geometries from categorical correlations in quantum gravity.

Contribution

It introduces a novel algebraic and categorical formalism for relational quantum theory based on non-commutative operator categories, connecting quantum gravity and non-commutative geometry.

Findings

Proposes a categorical algebraic approach to relational quantum space-time.

Suggests spectral reconstruction of non-commutative geometries from operator bimodules.

Links relational quantum theory with Tomita-Takesaki modular 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.