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arXiv:2008.13514 · International Journal of Geometric Methods in Modern Physics

Contextual extensions of quantum gravity models

Xiao-Kan Guo

Published Jan 27, 2021• 0 citations
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Abstract

We present a simple way of incorporating the structure of contextual extensions into quantum gravity models. The contextual extensions of [Formula: see text]-algebras, originally proposed for contextual hidden variables, are generalized to the cones indexed by the contexts and their limit in a category. By abstracting the quantum gravity models as functors, we study the contextual extensions as the categorical limits of these functors in several quantum gravity models. Such contextual extensions of quantum gravity models are useful for building topos-theoretic models of quantum gravity.

DOI
10.1142/s0219887821500857

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We present a simple way of incorporating the structure of contextual extensions into quantum gravity models. The contextual extensions of [Formula: see text]-algebras, originally proposed for contextual hidden variables, are generalized to the cones indexed by the contexts and their limit in a category. By abstracting the quantum gravity models as functors, we study the contextual extensions as the categorical limits of these functors in several quantum gravity models. Such contextual extensions of quantum gravity models are useful for building topos-theoretic models of quantum gravity.
We present a simple way of incorporating the structure of contextual extensions into quantum gravity models. The contextual extensions of C∗C^*C∗-algebras, originally proposed for contextual hidden variables, are generalized to the cones indexed by the contexts and their limit in a category. By abstracting the quantum gravity models as functors, we study the contextual extensions as the categorical limits of these functors in several quantum gravity models. Such contextual extensions of quantum gravity models are useful for building topos-theoretic models of quantum gravity.
Contextuality has become a cornerstone of modern foundational approaches to quantum physics. This paper introduces contextual mathematics (CM), a categorical framework for context-indexed observations and agent-dependent evaluation. Contexts are organized into a category Ctx, and observation is modeled as a contravariant bifunctor into an epistemic category E; context refinements induce restriction maps, and a Grothendieck construction organizes evaluations over agent–context pairs. We show how the established sheaf-theoretic account of Kochen–Specker contextuality (Abramsky–Brandenburger) can be represented inside this framework, recovering non-existence of a global section as the relevant obstruction. We then formulate, as comparison problems rather than established equivalences, several further candidate obstruction theories that a fully developed CM might connect to this sheaf-theoretic core: homological extension classes (Ext1 ), Kan-extension reconstruction, deformation-quantization cocycles, and device-dependent restriction in algebraic quantum field theory (AQFT) over globally hyperbolic spacetimes. We are explicit about which of these connections are proved, which are illustrative analogies, and which remain open.
The paper is the contribution to quantum toposophy focusing on the abstract orthomodular structures (following Dunn-Moss-Wang terminology). Early quantum toposophical approach to "abstract quantum logic" was proposed based on the topos of functors [E,Sets]\mathsf{[E,Sets]}[E,Sets] where E\mathsf{E}E is a so-called orthomodular preorder category – a modification of categorically rewritten orthomodular lattice (taking into account that like any lattice it will be a finite co-complete preorder category). In the paper another kind of categorical semantics of quantum logic is discussed which is based on the modification of the topos construction itself – so called quantosquantosquantos – which would be evaluated as a non-classical modification of topos with some extra structure allowing to take into consideration the peculiarity of negation in orthomodular quantum logic. The algebra of subobjects of quantos is not the Heyting algebra but an orthomodular lattice. Quantoses might be apprehended as an abstract reflection of Landsman's proposal of "Bohrification", i.e., the mathematical interpretation of Bohr's classical concepts by commutative C∗C^*C∗-algebras, which in turn are studied in their quantum habitat of noncommutative C∗C^*C∗-algebras – more fundamental structures than commutative C∗C^*C∗-algebras. The Bohrification suggests that topos-theoretic approach also should be modified. Since topos by its nature is an intuitionistic construction then Bohrification in abstract case should be transformed in an application of categorical structure based on an orthomodular lattice which is more general construction than Heyting algebra – orthomodular lattices are non-distributive while Heyting algebras are distributive ones. Toposes thus should be studied in their quantum habitat of "orthomodular" categories i.e. of quntoses. Also an interpretation of some well-known systems of orthomodular quantum logic in quantos of functors [E,QSets]\mathsf{[E,QSets]}[E,QSets] is constructed where QSets\mathsf{QSets}QSets is a quantos (not a topos) of quantum sets. The completeness of those systems in respect to the semantics proposed is proved.
This paper introduces the Quantum Contextual Topos (QCT), a novel framework that extends traditional quantum logic by embedding contextual elements within a topos-theoretic structure. This framework seeks to provide a classically-obedient tool for exploring the logical foundations of quantum mechanics. The QCT framework aims to address the limitations of classical quantum logic, particularly its challenges in capturing the dynamic and contextual nature of quantum phenomena. By integrating modal operators and classical propositional logic within a topos structure, the QCT offers a unified approach to modeling quantum systems. The main result of this work is demonstrating that the internal logic of QCT corresponds to a form of classical propositional polymodal logic. We do this by generalizing Stone's Representation Theorem for a specific case of polymodal algebras and their underlying Stone Spaces.
The endeavor to refute hidden variable theories underlying quantum theory has yielded the discipline of contextual sets. A plethora of various kinds of sets of arbitrary structure in any dimension have been developed, alongside extensive experimental validation. These advancements incited us to investigate to what extent we might move beyond hidden variables, engineer contextual sets, and find their applications within quantum theory itself, without any reference to hidden variable models. To this end, we consider possible applications of contextual sets in quantum computation, cryptography, pseudo-telepathy, and nonlocality, as well as generating them from error-correction protocols, complex Hadamard gates, and simple quantum gates. We found that the results in this field are still scarce, and we therefore investigated the directions in which future research might be carried out and the potential obstacles to realizing such undertakings in the past.