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

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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.

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Notes

  1. At the time of writing there is a general consensus that the famous Poincaré conjecture which states that every orientable, simply connected, compact 3-manifold, is the 3-sphere, has now been proved by G. Perelman to be true. In other words, any 3-manifold that is topologically like the 3-sphere, is the 3-sphere.

References

  • Alfsen EM, Schultz FW (2003) Geometry of state spaces of operator algebras. Birkäuser, Boston–Basel–Berlin

    Google Scholar 

  • Andersen PB, Emmeche C, Finnemann NO, Christiansen PV (2000) Downward causation. minds, bodies and matter. Aarhus University Press, Aarhus DK

    Google Scholar 

  • Baez J (2001) Higher dimensional algebra and Planck scale physics. In: Callender C, Hugget N (eds) Physics Meets Philosophy at the Planck scale. Cambridge University Press, pp 177–195

  • Baez J, Dolan J (1995) Higher dimensional algebra and topological quantum field theory. J Math Phys 36:6073–6105

    Article  Google Scholar 

  • Baianu IC, Marinescu M (1968) Organismic supercategories: towards a unified theory of systems. Bull Math Biophys 30:148

    Article  Google Scholar 

  • Baianu IC (1970) Organismic supercategories: II. On multistable systems. Bull Math Biophys 32:539–561

    Article  Google Scholar 

  • Baianu IC (1971a) Organismic supercategories and qualitative dynamics of systems. Bull Math Biophys 33(3):339–354

    Article  Google Scholar 

  • Baianu IC (1971b) Categories, functors and quantum algebraic computations. In: Suppes P (ed) Proceed Fourth Intl Congress Logic-Mathematics-Philosophy of Science. September 1–4, 1971, Buch

  • Baianu IC, Scripcariu D (1973) On adjoint dynamical systems. Bull Math Biophys 35(4):475–486

    Article  Google Scholar 

  • Baianu IC (1973) Some algebraic properties of (M,R) – systems. Bull Math Biophys 35:213–217

    Google Scholar 

  • Baianu IC, Marinescu M (1974) A Functorial Construction of (M,R) – Systems. Revue Roumaine de Mathematiques Pures et Appliquees 19:388–391

    Google Scholar 

  • Baianu IC (1977) A logical model of genetic activities in Łukasiewicz algebras: the non-linear theory. Bull Math Biol 39:249–258

    Google Scholar 

  • Baianu IC (1980) Natural transformations of organismic structures. Bull Math Biol 42:431–446

    Google Scholar 

  • Baianu IC (1983) Natural transformation models in molecular biology. In: Proceedings of the SIAM Natl. Meet. Denver, CO. http://cogprints.org/3675/; http://cogprints.org/3675/0l/Naturaltransfmolbionu6.pdf

  • Baianu IC (1984) A molecular-set-variable model of structural and regulatory activities in metabolic and genetic networks. FASEB Proc 43:917

    Google Scholar 

  • Baianu IC (1987a) Computer models and automata theory in biology and medicine. In: Witten M (ed) Mathematical models in medicine, vol. 7. Pergamon Press, New York, 1513–1577; CERN Preprint No. EXT-2004-072: http://doe.cern.ch//archive/electronic/other/ext/ext-2004-072.pdf

  • Baianu IC (1987b) Molecular Models of Genetic and Organismic Structures. In: Proceedings of relational biology symposium Argentina; CERN Preprint No. EXT-2004-067: http://doc.cern.ch/archive/electronic/other/ext/ext2004067/Molecular Models ICB3.doc

  • Baianu IC (2004a) Quantum Nano-Automata (QNA): Microphysical Measurements with Microphysical QNA Instruments. CERN Preprint EXT–2004–125

  • Baianu IC (2004b) Quantum Interactomics and Cancer Mechanisms, Preprint No. 00001978: http://bioline.utsc.utoronto.ca/archive/00001978/01/QuantumInteractomics In Cancer–Sept13k4E– cuteprt.pdf. http://bioline.utsc.utoronto.ca/archive/00001978/

  • Baianu IC, Glazebrook JF, Georgescu G (2004) Categories of Quantum Automata and N-Valued Łukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R)–Systems and Their Higher Dimensional Algebra, Abstract and Preprint of Report: http://www.ag.uiuc.edu/fs401/QAuto.pdf and http://www.medicalupapers.com/quantum+automata+math+categories+baianu/

  • Baianu IC (2006) Robert Rosen’s Work and Complex Systems Biology. Axiomathes 16(1–2):25–34

    Article  Google Scholar 

  • Baianu IC, Brown R, Glazebrook JF (2006a) Quantum Algebraic Topology and Field Theories, (manuscript in preparation) http://www.ag.uiuc.edu/fs40l/QAT.pdf

  • Baianu IC, Brown R, Georgescu G, Glazebrook JF (2006b) Complex nonlinear biodynamics in categories, higher dimensional algebra and Łukasiewicz–Moisil topos: transformations of neuronal, genetic and neoplastic networks. Axiomathes 16(1–2):65–122

    Article  Google Scholar 

  • Baianu IC, Glazebrook JF, Georgescu G, Brown R (2007) Non-abelian Algebraic Topology Representations of Quantum Space–Time in a Generalized ‘Topos’ with a Quantum N–Valued Logic Classifier. (in submission)

  • Baianu IC, Poli R (2008) From simple to super- and ultra- complex systems: a paradigm shift towards a Non-Abelian dynamics. In: Poli R et al (eds) Theory and applications of ontology, vol 1. Springer, Berlin, p 30

    Google Scholar 

  • Bennett M, Hacker P (2003) Philosophical foundations of neuroscience. Blackwell Publishing, London

    Google Scholar 

  • Birkhoff G, von Neumann J (1936) The logic of quantum mechanics. Ann Math 37:823–843

    Article  Google Scholar 

  • Brown R, Glazebrook JF, Baianu IC (2007) A conceptual construction for complexity levels theory in spacetime categorical ontology: Non-Abelian algebraic topology, many-valued logics and dynamic systems. Axiomathes 17:(in this issue)

  • Brown R (2004) Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems. In: Proceedings of the Fields Institute Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories (September 23–28, 2002), Fields Institute Communications 43, 101–130

  • Brown R, Sivera R (2004) Non-Abelian Algebraic Topology: Part I, http://www.bangor.ac.uk/mas010/nonab-a-t.html; http://www.bangor.ac.uk/mas010/nonab-t/partI010604.pdf

  • Brown R, Janelidze (2004) Galois theory and a new homotopy double groupoid of a map of spaces. Appl Categ Struct 12:63–80

    Article  Google Scholar 

  • Brown R, Higgins PJ, Sivera R (2008) Noncommutative algebraic topology, to appear

  • Brown R, Paton R, Porter T (2004) Categorical language and hierarchical models for cell systems. In: Paton R, Bolouri H, Holcombe M, Parish JH, Tateson R (eds) Computation in Cells and Tissues - Perspectives and Tools of Thought. Natural Computing Series, Springer Verlag, Berlin, pp. 289–303

    Google Scholar 

  • Brown R, Porter T (2003) Category theory and higher dimensional algebra: potential descriptive tools in neuroscience. Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, edited by Nandini Singh, National Brain Research Centre, Conference Proceedings 1:80–92

  • Brown R, Hardie K, Kamps H, Porter T (2002) The homotopy double groupoid of a Hausdorff space. Theor. Appl. Categories 10:71–93.

    Google Scholar 

  • Brown R (2006) Topology and groupoids. BookSurge, LLC.

  • Butterfield J, Isham CJ (2001) Spacetime and the philosophical challenges of quantum gravity. In: Callender C, Hugget N (eds) Physics meets philosophy at the Planck scale. Cambridge University Press, pp 33–89

  • Butterfield J, Isham CJ 1998, 1999, 2000–2002, A topos perspective on the Kochen–Specker theorem I–IV, Int J Theor Phys, 37, No 11.: 2669–2733 38, No 3.: 827–859, 39, No 6.: 1413–1436, 41, No 4.: 613–639

  • Chalmers DJ (1996) The Conscious Mind–In search of a fundamental theory. Oxford University Press, Oxford, UK

    Google Scholar 

  • Chevalley C (1946) The theory of Lie groups. Princeton Univ Press, Princeton, NJ

    Google Scholar 

  • Chodos A, Detweiler S (1980) Where has the fifth diemsion gone? Phys Rev D 21:2167–2170

    Article  Google Scholar 

  • Connes A (1994) Noncommutative geometry. Academic Press, New York and London

    Google Scholar 

  • Cramer JG (1980) The transactional interpretation of quantum mechanics. Phys Rev D 22:362

    Article  Google Scholar 

  • Dalla Chiara M, Giuntini R, Greechie R (2004) Reasoning in quantum theory, trends in logic–studia logica library, vol 22. Kluwer, Dordrecht

    Google Scholar 

  • Ehresmann C (1965) Catégories et structures. Dunod, Paris

    Google Scholar 

  • Ehresmann C (1966) Trends toward unity in mathematics. Cahiers de Topologie et Geometrie Differentielle 8:1–7

    Google Scholar 

  • Ehresmann AC, Vanbremeersch J-P (1987) Hierarchical evolutive systems: a mathematical model for complex systems. Bull of Math Biol 49(1):13–50

    Google Scholar 

  • Ehresmann AC, Vanbremeersch J-P (2006) The memory evolutive systems as a model of Rosen’s organisms. Axiomathes 16(1–2):13–50

    Google Scholar 

  • Eilenberg S, Mac Lane S (1945) The general theory of natural equivalences. Trans Am Math Soc 58:231–294

    Article  Google Scholar 

  • Elsasser MW (1981) A form of logic suited for biology. In: Robert R (ed) Progress in theoretical biology, vol 6. Academic Press, New York and London, pp 23–62

    Google Scholar 

  • Gabor D (1946) Theory of communication. J IEE (London) 93(III):429–457

    Google Scholar 

  • Georgescu G, Popescu D (1968) On algebraic categories. Revue Roumaine de Mathematiques Pures et Appliquées 13:337–342

    Google Scholar 

  • Georgescu G, Vraciu C. (1970) On the characterization of Łukasiewicz-Moisil algebras. J. Algebra 16(4):486–495.

    Article  Google Scholar 

  • Georgescu G (2006) N-valued logics and Łukasiewicz–Moisil algebras. Axiomathes 16(1–2):123–136

    Article  Google Scholar 

  • Grothendieck A (1971) Revêtements Étales et Groupe Fondamental (SGA1), chapter VI: Catégories fibrées et descente. Lecture Notes in Math. 224, Springer–Verlag, Berlin

  • Grothendieck A (1957) Sur quelque point d-algébre homologique. Tohoku Math J 9:119–121

    Google Scholar 

  • Grothendieck A, Dieudonné J (1960–1967) Éléments de géométrie algébrique. Publ Inst des Hautes Etudes de Science Publ Math. : 4 (1960), 8(1961), 11 (1961), 17 (1963), 20 (1964), 24 (1965),28 (1966) and 32(1967)

  • Hawking SW, Ellis GFR (1973) The Large Scale Structure of Space–Time. Cambridge University Press

  • Heller A (1958) Homological algebra in Abelian categories. Ann of Math 68:484–525

    Article  Google Scholar 

  • Heller A, Rowe KA (1962) On the category of sheaves. Am J Math 84:205–216

    Article  Google Scholar 

  • Higgins PJ, Mackenzie KCH (1990) Fibrations and quotients of differentiable groupoids. J London Math Soc 42(1):101–110

    Article  Google Scholar 

  • Hurewicz E (1955) On the concept of fiber spaces. Proc Nat Acad Sci USA 41:956–961

    Article  Google Scholar 

  • van Kampen EH (1933) On the connection between the fundamental groups of some related spaces. Am J Math 55:261–267

    Google Scholar 

  • Kleisli H (1962) Homotopy theory in Abelian categories. Can J Math 14:139–169

    Google Scholar 

  • Knight JT (1970) On epimorphisms of non-commutative rings. Proc Cambridge Phil Soc 68:589–601

    Article  Google Scholar 

  • Krips H (1999) Measurement in quantum theory. In: Zalta EN (ed) The Stanford Encyclopedia of Philosophy (on line) (Winter 1999 Edition)

  • Landsman NP (1998) Mathematical topics between classical and quantum mechanics. Springer Verlag, New York

    Google Scholar 

  • Lawvere FW (1966) The category of categories as a foundation for mathematics. In: Eilenberg S et al (eds) Proc conf categorical algebra- La Jolla. Springer–Verlag, Berlin, Heidelberg and New York, pp 1–20

    Google Scholar 

  • Lawvere FW (1963) Functorial semantics of algebraic theories. Proc Natl Acad Sci USA Math 50:869–872

    Article  Google Scholar 

  • Lawvere FW (1969) Closed Cartesian Categories. Lecture held as a guest of the Romanian Academy of Sciences, Bucharest

    Google Scholar 

  • Lôfgren L (1968) An axiomatic explanation of complete self-reproduction. Bull Math Biophys 30:317–348

    Article  Google Scholar 

  • Lubkin S (1960) Imbedding of Abelian categories. Trans Am Math Soc 97:410–417

    Article  Google Scholar 

  • Mac Lane S (1963) Homology. Springer, Berlin

    Google Scholar 

  • Mac Lane S (2000) Categories for the working mathematician. Springer, New York–Berlin–Heidelberg

    Google Scholar 

  • Mac Lane S, Moerdijk I (1992) Sheaves in Geometry and Logic. A first introduction in topos theory. Springer-Verlag, New York

    Google Scholar 

  • Majid S (1995) Foundations of quantum group theory. Cambridge Univ Press

  • Majid S (2002) A quantum groups primer. Cambridge Univ Press, Cambridge, UK

    Google Scholar 

  • Mallios A, Raptis I (2003) Finitary, causal and quantal vacuum Einstein gravity. Int J Theor Phys 42:1479

    Article  Google Scholar 

  • Manders KL (1982) On the space-time ontology of physical theories. Phil Sci 49(4):575–590

    Article  Google Scholar 

  • McGinn C (1995) Consciousness and space. In: Metzinger T (ed) Conscious experience. Imprint Academic, Thorverton

  • Mitchell B (1965) Theory of categories. Academic Press, London

    Google Scholar 

  • Mitchell B (1964) The full imbedding theorem. Am J Math 86:619–637

    Article  Google Scholar 

  • Oberst U (1969) Duality theory for Grothendieck categories. Bull Am Math Soc 75:1401–1408

    Article  Google Scholar 

  • Oort F (1970) On the definition of an Abelian category. Proc Roy Neth Acad Sci 70:13–02

    Google Scholar 

  • Ore O (1931) Linear equations in non-commutative fields. Ann Math 32:463–477

    Article  Google Scholar 

  • Penrose R (1994) Shadows of the mind. Oxford University Press, Oxford, UK

    Google Scholar 

  • Plymen RJ, Robinson PL (1994) Spinors in Hilbert Space, Cambridge Tracts in Math. 114, Cambridge Univ Press, Cambridge, UK

  • Poli, Roberto (2006) Steps towards a synthetic methodology. In: Proc conf continuity and change: perspectives on science and religion. Metanexus Institute, June 2006, Philadelphia, PA. http://www.metanexus.net/conferences/pdf/conference2006/Poli.pdf

  • Poli R (2008) Ontology: the categorical stance. In: Poli R et al (eds) Theory and applications of ontology, vol 1. Springer, Berlin (in press)

  • Popescu N (1973) Abelian categories with applications to rings and modules, 2nd edn, 1975. Academic Press, New York 1975 (English translation by I.C. Baianu)

  • Popescu N (1966 and 1967) Elements of Sheaf theory. St. Cerc. Mat. V/VI, 18–19:205–240; 945–991

  • Popescu N (1967) La théorie générale de la décomposition. Rev Roum Math Pures et Appl 7:1365–1371

    Google Scholar 

  • Pribram KH (1991) Brain and Perception: Holonomy and Structure in Figural processing. Lawrence Erlbaum Assoc, Hillsdale

    Google Scholar 

  • Pribram KH (2000) Proposal for a quantum physical basis for selective learning. In: Farre (ed) Proceedings ECHO IV, pp 1–4

  • Raptis I, Zapatrin RR (2000) Quantisation of discretized spacetimes and the correspondence principle. Intl Jour Theor Phys 39:1

    Article  Google Scholar 

  • Raptis I (2003) Algebraic quantisation of causal sets. Int Jour Theor Phys 39:1233

    Article  Google Scholar 

  • Rashevsky N (1965) The representation of organisms in terms of predicates. Bull Math Biophys 27:477–491

    Article  Google Scholar 

  • Rashevsky N (1969) Outline of a unified approach to physics, biology and sociology. Bull Math Biophys 31:159–198

    Article  Google Scholar 

  • Roos JE (1964) Sur la condition Ab6 et ses variantes dans les catégories abéliennes. CRAS Paris 257:2368–2371

    Google Scholar 

  • Roux A (1964) Sur une équivalence de catégories abéliennes. CRAS Paris 258:5566–5569

    Google Scholar 

  • Roberts JE (2004) More lectures on algebraic quantum field theory. In: Connes A et al Noncommutative Geometry. Springer, Berlin

  • Rosen R (1985) Anticipatory systems. Pergamon Press, New York

    Google Scholar 

  • Rosen R (1958a) A relational theory of biological systems. Bull Math Biophys 20:245–260

    Article  Google Scholar 

  • Rosen R (1958b) The representation of biological systems from the standpoint of the theory of categories. Bull Math Biophys 20:317–341

    Article  Google Scholar 

  • Rovelli C (1998) Loop Quantum Gravity. In: Dadhich N et al (eds) Living reviews in relativity (refereed electronic journal) http://www.livingreviews.org/Articles/Volume1/1998 1 rovelli

  • Russell B, Whitehead AN (1925) Principia mathematica. Cambridge Univ Press, Cambridge

    Google Scholar 

  • Ryle G (1949) The concept of mind. Hutchinson, London

    Google Scholar 

  • Silver L (1967) Non-commutative localization and applications. J Algebra 7:44–76

    Article  Google Scholar 

  • Sorkin RD (1991) Finitary substitute for continuous topology. Int J Theor Phys 30(7):923–947

    Article  Google Scholar 

  • Smolin L (2001) Three roads to quantum gravity. Basic Books, New York

    Google Scholar 

  • Spanier EH (1966) Algebraic Topology. McGraw Hill, New York

    Google Scholar 

  • Spencer-Brown G (1969) Laws of form. George Allen and Unwin, London

    Google Scholar 

  • Stenström B (1968) Direct sum decomposition in Grothendieck categories. Arkiv Math 7:427–432

    Article  Google Scholar 

  • Szabo RJ (2003) Quantum field theory on non-commutative spaces. Phys Rep 378:207–209

    Article  Google Scholar 

  • Takahashi H (1963) Adjoint pair of functors on Abelian categories. J Fac Soc Univ Tokyo 13:175–181

    Google Scholar 

  • Unruh WG (2001) Black holes, dumb holes, and entropy. In: Callender C, Hugget N (eds) Physics meets philosophy at the Planck scale. Cambridge University Press, Cambridge, UK, pp 152–173

    Google Scholar 

  • Várilly JC (1997) An introduction to noncommutative geometry arXiv:physics/9709045

  • Velmans M (2000) Understanding consciousness. Routledge, London

    Google Scholar 

  • von Neumann J (1932) Mathematische Grundlagen der Quantenmechanik. Springer, Berlin

    Google Scholar 

  • Wess J, Bagger J (1983) Supersymmetry and supergravity. Princeton University Press, Princeton, NJ

    Google Scholar 

  • Wheeler J, Zurek W (1983) Quantum theory and measurement. Princeton University Press, Princeton NJ

    Google Scholar 

  • Weinberg S (1995–2000) The quantum theory of fields, vols I–III. Cambridge Univ Press, Cambridge, UK

    Google Scholar 

  • Weinstein A (1996) Groupoids: unifying internal and external symmetry. Notices Am Math Soc 43:744–752

    Google Scholar 

  • Whitehead JHC (1941) On adding relations to homotopy groups. Ann Math 42(2):409–428

    Article  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge fruitful suggestions from Dr. Roberto Poli. I.C.B gratefully acknowledges the support in part by a grant from Renessen Biotechnology Co.

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Authors and Affiliations

  1. FSHN and NPRE Departments, AFC-NMR and NIR Microspectroscopy Facility, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    I. C. Baianu

  2. School of Informatics, University of Wales, Dean St. Bangor, Gwynedd, LL57 1UT, UK

    R. Brown

  3. Department of Mathematics and Computer Science, Eastern Illinois University, 600 Lincoln Ave, Charleston, IL, 61920-3099, USA

    J. F. Glazebrook

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  1. I. C. Baianu
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Baianu, I.C., Brown, R. & Glazebrook, J.F. A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity. Axiomathes 17, 353–408 (2007). https://doi.org/10.1007/s10516-007-9012-1

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