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Register a new userCategorical Operator Algebraic Foundations of Relational Quantum Theory
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Authors
Paolo Bertozzini
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We provide an algebraic formulation of C.Rovellis 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.Rovellis 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.
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When working with distant collaborators on the same documents, one often uses a version control system, which is a program tracking the history of files and helping importing modifications brought by others as patches. The implementation of such a system requires to handle lots of situations depending on the operations performed by users on files, and it is thus difficult to ensure that all the corner cases have been correctly addressed. Here, instead of verifying the implementation of such a system, we adopt a complementary approach: we introduce a theoretical model, which is defined abstractly by the universal property that it should satisfy, and work out a concrete description of it. We begin by defining a category of files and patches, where the operation of merging the effect of two coinitial patches is defined by pushout. Since two patches can be incompatible, such a pushout does not necessarily exist in the category, which raises the question of which is the correct category to represent and manipulate files in conflicting state. We provide an answer by investigating the free completion of the category of files under finite colimits, and give an explicit description of this category: its objects are finite sets labeled by lines equipped with a transitive relation and morphisms are partial functions respecting labeling and relations.
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