Original Paper

Axiomathes

, Volume 17, Issue 3, pp 353-408

First online:

A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity

  • I. C. BaianuAffiliated withFSHN and NPRE Departments, AFC-NMR and NIR Microspectroscopy Facility, University of Illinois at Urbana-Champaign Email author 
  • , R. BrownAffiliated withSchool of Informatics, University of Wales
  • , J. F. GlazebrookAffiliated withDepartment of Mathematics and Computer Science, Eastern Illinois University

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Abstract

A non-Abelian, Universal SpaceTime Ontology is introduced in terms of Categories, Functors, Natural Transformations, Higher Dimensional Algebra and the Theory of Levels. A Paradigm shift towards Non-Commutative Spacetime structures with remarkable asymmetries or broken symmetries, such as the CPT-symmetry violation, is proposed. This has the potential for novel applications of Higher Dimensional Algebra to SpaceTime structure determination in terms of universal, topological invariants of ‘hidden’ symmetry. Fundamental concepts of Quantum Algebra and Quantum Algebraic Topology, such as Quantum Groups, von Neumann and Hopf Algebras are first considered with a view to their possible extensions and future applications in Quantum Field theories. New, non-Abelian results may be obtained through Higher Homotopy, General van Kampen Theorems, Lie Groupoids/Algebroids and Groupoid Atlases, possibly with novel applications to Quantum Dynamics and Local-to-Global Problems, Quantum Logics and Logic Algebras. Many-valued Logics, Łukasiewicz–Moisil Logics lead to Generalized LM-Toposes as global representations of SpaceTime Structures in the presence of intense Quantum Gravitational Fields. Such novel representations have the potential to develop a Quantum/General Relativity Theory in the context of Supersymmetry, Supergravity, Supersymmetry Algebras and the Metric Superfield in the Planck limit of spacetime. Quantum Gravity and Physical Cosmology issues are also considered here from the perspective of multiverses, thus leading also to novel types of Generalized, non-Abelian, Topological, Higher Homotopy Quantum Field Theories (HHQFT) and Non-Abelian Quantum Algebraic Topology (NA-QAT) theories.

Keywords

Universal SpaceTime Ontology, Axioms of Abelian Categories and Categorical Ontology/the Theory of Levels The Paradigm Shift towards Non-Abelian Ontology, Relations and Abstract Structures Commutativity vs. Non-Commutativity, Symmetry vs. Asymmetry, CPT Symmetry violation Quantum Field Theory and Quantum Dynamics Quantum Logics and Logic Algebras, Many-valued Logics, Łukasiewicz–Moisil Logics and Generalized LM-Toposes (GLM) Quantum Fields, General Relativity, Supersymmetry, Supergravity, and the Metric Superfield Supersymmetry Algebras, Symmetry Breaking, and Quantum General Relativity Quantum Gravity and Physical Cosmology, Non-Abelian Gauge Theories and Higgs bosons, Chronotopoids and Multiverses Higher Dimensional Algebra (HDA) in SpaceTime Ontology, Higher Homotopy and General van Kampen Theorem (HHvKT) Local-to-Global Problems and Combinations of Local Procedures (COLP), Lie Groupoids, Groupoid Atlases and Graded Lie Algebroids in Non-Linear Quantum Gravity Fundamental Quantum Double Groupoids and Spacetime Topological Invariants