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Categorical Physics

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This is a relatively new area in mathematical and theoretical physics that is concerned with category theory applications to physics, especially non-Abelian categories and non-Abelian algebraic topology concepts and results in mathematical physics and physical mathematics. Applications range from QFT, AQFT, non-Abelian gauge theories and quantum gravity to complex systems, categorical dynamics, complex categorical dynamics, mathematical biophysics and relational biology. Other applications are related to graph theory approaches to Quantum Chemistry.

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References

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  1. ↑ Baez, J. and Dolan, J., 1998b, "Categorification", Higher Category Theory, Contemporary Mathematics , 230, Providence: AMS, 1--36.
  2. ↑ Baianu, I. and M. Marinescu: 1968, Organismic Supercategories: Towards a Unitary Theory of Systems. Bulletin of Mathematical Biophysics 30 , 148-159.
  3. ↑ Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. Bulletin of Mathematical Biophysics , 32 : 539-561.
  4. ↑ Baianu,I.C.: 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. Bulletin of Mathematical Biophysics , 33 (3), 339--354.
  5. ↑ Baianu, I.C.: 1971b, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), In Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science , 14 pages, September 1--4, 1971.
  6. ↑ Baianu, I.C. : \L ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. Health Physics and Radiation Effects (June 29, 2004).
  7. ↑ Baianu, I.C. and D. Scripcariu: 1973, On Adjoint Dynamical Systems. The Bulletin of Mathematical Biophysics , 35 (4), 475--486.
  8. ↑ Baez, J. \& Dolan, J., 2001, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited -- 2001 and Beyond , Berlin: Springer, 29--50.
  9. ↑ Baez, J., 1997, An Introduction to n-Categories, in Category Theory and Computer Science, Lecture Notes in Computer Science , 1290, Berlin: Springer-Verlag, 1--33.
  10. ↑ Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and \L{}ukasiewicz-Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., Axiomathes , 16 Nos. 1-2: 65-122.
  11. ↑ Baianu, I.C. and D. Scripcariu: 1973, On Adjoint Dynamical Systems. Bulletin of Mathematical Biophysics , 35 (4): 475-486.
  12. ↑ Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued \L ukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R) -Systems and Their Higher Dimensional Algebra, Preprint of Report .
  13. ↑ Brown, R., Higgins, P. J. and R. Sivera,: 2007, Non-Abelian Algebraic Topology , vol.I pdf doc.; Review of Part I and full contents PDF doc.
  14. ↑ R. Brown. 2008. Higher Dimensional Algebra Preprint as pdf and ps docs. at arXiv:math/0212274v6 [math.AT]
  15. ↑ Brown R. and T. Porter: 2003, Category theory and higher dimensional algebra: potential descriptive tools in neuroscience, In: Proceedings of the International Conference on Theoretical Neurobiology , Delhi, February 2003, edited by Nandini Singh, National Brain Research Centre, Conference Proceedings 1 : 80-92.
  16. ↑ Chaician, M. and A. Demichev. 1996. Introduction to Quantum Groups, World Scientific .
  17. ↑ Connes, A. 1994. Noncommutative geometry . Academic Press: New York.
  18. ↑ Crane, Louis. 1993. Categorical Physics., 9 pages, with free download at ("A new mathematical form for the quantum theory of gravity coupled to matter. The motivation is from the connection between CSW TQFT and the Ashtekar variables").
  19. ↑ Croisot, R. and Lesieur, L. 1963. Alg\`ebre noeth\'erienne non-commutative. , Gauthier-Villard: Paris.
  20. ↑ Dieudonn\'e, J. and Grothendieck, A., 1960, [1971], \'El\'ements de G\'eom\'etrie Alg\'ebrique , Berlin: Springer-Verlag.
  21. ↑ Dixmier, J., 1981, Von Neumann Algebras, Amsterdam: North-Holland Publishing Company. [First published in French in 1957: Les Algebres d'Operateurs dans l'Espace Hilbertien, Paris: Gauthier--Villars.]
  22. ↑ M. Durdevich : Geometry of quantum principal bundles I, Commun. Math. Phys. 175 (3) (1996), 457--521.
  23. ↑ M. Durdevich : Geometry of quantum principal bundles II, Rev.Math. Phys. 9 (5) (1997), 531-607.
  24. ↑ Ehresmann, C.: 1952, Structures locales et structures infinit\'esimales, C.R.A.S. Paris 274 : 587-589.
  25. ↑ Ehresmann, C.: 1959, Cat\'egories topologiques et cat\'egories diff\'erentiables, Coll. G\'eom. Diff. Glob. Bruxelles, pp.137-150.
  26. ↑ Ehresmann, C.: 1966, Trends Toward Unity in Mathematics., Cahiers de Topologie et Geometrie Differentielle 8 : 1-7.
  27. ↑ Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., American Mathematical Society 43 : 757-831.
  28. ↑ Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58 : 231-294.
  29. ↑ Ezawa,Z.F., G. Tsitsishvilli and K. Hasebe : Noncommutative geometry, extended algebra and Grassmannian solitons in multicomponent Hall systems, (at arXiv:hep--th/0209198).
  30. ↑ Gabriel, P. and N. Popescu: 1964, Caract\'{e}risation des cat\'egories ab\'eliennes avec g\'{e}n\'{e}rateurs et limites inductives. , CRAS Paris 258 : 4188-4191.
  31. ↑ Galli, A. \& Reyes, G. \& Sagastume, M., 2000, Completeness Theorems via the Double Dual Functor, Studia Logica , 64 , no. 1: 61--81.
  32. ↑ Grothendieck, A. and J. Dieudon\'{e}.: 1960, El\'{e}ments de geometrie alg\'{e}brique., Publ. Inst. des Hautes Etudes de Science , 4 .
  33. ↑ Kan, D. M., 1958, Adjoint Functors, Transactions of the American Mathematical Society 87, 294-329.
  34. ↑ H. Krips : Measurement in Quantum Theory, The Stanford Encyclopedia of Philosophy (Winter 1999 Edition), Edward N. Zalta (ed.)
  35. ↑ Landsman, N. P. : Compact quantum groupoids, (at arXiv:math\^a~@~Tph/9912006).
  36. ↑ Lawvere, F. W., 1964, An Elementary Theory of the Category of Sets, Proceedings of the National Academy of Sciences U.S.A. , 52, 1506--1511.
  37. ↑ Lawvere, F. W., 1965, Algebraic Theories, Algebraic Categories, and Algebraic Functors, Theory of Models , Amsterdam: North Holland, 413--418.
  38. ↑ Lawvere, F. W.: 1966, The Category of Categories as a Foundation for Mathematics., in Proc. Conf. Categorical Algebra- La Jolla ., Eilenberg, S. et al., eds. Springer--Verlag: Berlin, Heidelberg and New York., pp. 1-20.
  39. ↑ Lawvere, F. W., 1969b, Adjointness in Foundations, Dialectica , 23 : 281--295.
  40. ↑ Lawvere, F. W., 1992, Categories of Space and of Quantity, The Space of Mathematics, Foundations of Communication and Cognition , Berlin: De Gruyter, 14-30.
  41. ↑ Lawvere, F. W., 2002, Categorical Algebra for Continuum Micro-Physics, Journal of Pure and Applied Algebra , 175, no. 1--3, 267--287.
  42. ↑ Li, M. and P. Vitanyi: 1997, An introduction to Kolmogorov Complexity and its Applications , Springer Verlag: New York.
  43. ↑ L\"{o}fgren, L.: 1968, An Axiomatic Explanation of Complete Self-Reproduction, Bulletin of Mathematical Biophysics , 30 : 317-348.
  44. ↑ K. C. H. Mackenzie : Lie Groupoids and Lie Algebroids in Differential Geometry , LMS Lect. Notes 124 , Cambridge University Press, 1987
  45. ↑ MacLane, S.: 1948. Groups, categories, and duality., Proc. Natl. Acad. Sci.U.S.A , 34 : 263-267.
  46. ↑ MacLane, S., 1969, Foundations for Categories and Sets, in Category Theory, Homology Theory and their Applications II , Berlin: Springer, 146--164.
  47. ↑ MacLane, S., 1971, Categorical algebra and Set-Theoretic Foundations, in Axiomatic Set Theory , Providence: AMS, 231--240.
  48. ↑ MacLane, S., 1950, Dualities for Groups, Bulletin of the American Mathematical Society , 56, 485-516.
  49. ↑ MacLane, S., 1996, Structure in Mathematics. Mathematical Structuralism., Philosophia Mathematica , 4, 2, 174-183.
  50. ↑ MacLane, S., 1997, Categories for the Working Mathematician, 2nd edition, New York: Springer-Verlag.
  51. ↑ Majid, S.: 1995, Foundations of Quantum Group Theory , Cambridge Univ. Press: Cambridge, UK.
  52. ↑ Majid, S.: 2002, A Quantum Groups Primer , Cambridge Univ.Press: Cambridge, UK.
  53. ↑ May, J.P. 1999, A Concise Course in Algebraic Topology , The University of Chicago Press: Chicago.
  54. ↑ Mc Larty, C., 1994, Category Theory in Real Time, Philosophia Mathematica , 2 , no. 1, 36-44.
  55. ↑ Mc Larty, C., 1991, Axiomatizing a Category of Categories, Journal of Symbolic Logic , 56, no. 4, 1243-1260.
  56. ↑ Mitchell, B.: 1965, Theory of Categories , Academic Press:London.
  57. ↑ Ore, O., 1931, Linear equations on non-commutative fields, Ann. Math. 32 : 463-477.
  58. ↑ Plymen, R.J. and P. L. Robinson: 1994, Spinors in Hilbert Space , Cambridge Tracts in Math. 114 , Cambridge Univ. Press, Cambridge.
  59. ↑ Popescu, N.: 1973, Abelian Categories with Applications to Rings and Modules. New York and London: Academic Press., 2nd edn. 1975. (English translation by I.C. Baianu) .
  60. ↑ Pareigis, B., 1970, Categories and Functors, New York: Academic Press.
  61. ↑ Pedicchio, M. C. and Tholen, W., 2004, Categorical Foundations, Cambridge: Cambridge University Press.
  62. ↑ Peirce, B., 1991, Basic Category Theory for Computer Scientists, Cambridge: MIT Press.
  63. ↑ Pradines, J.: 1966, Th\'eorie de Lie pour les groupoides diff\'erentiable, relation entre propri\'etes locales et globales, C. R. Acad Sci. Paris S\'er. A 268 : 907-910.
  64. ↑ Raptis, I.: 2003, Algebraic quantisation of causal sets, Int. Jour. Theor. Phys. 39 : 1233.
  65. ↑ M. A. Rieffel : Group C*--algebras as compact quantum metric spaces, Documenta Math. 7 (2002), 605-651.
  66. ↑ Roberts, J. E.: 2004, More lectures on algebraic quantum field theory, in A. Connes, et al. Noncommutative Geometry , Springer: Berlin and New York.
  67. ↑ Rosen, R.: 1958b, The Representation of Biological Systems from the Standpoint of the Theory of Categories., Bulletin of Mathematical Biophysics 20 : 317-341.
  68. ↑ Szabo, R. J.: 2003, Quantum field theory on non-commutative spaces, Phys. Rep. 378 : 207--209.
  69. ↑ V\'arilly, J. C.: 1997, An introduction to noncommutative geometry. (at arXiv:physics/9709045)
  70. ↑ Weinstein, A.: 1996, Groupoids : unifying internal and external symmetry, Notices of the Amer. Math. Soc. 43 : 744--752.
  71. ↑ Wess J. and J. Bagger: 1983, Supersymmetry and Supergravity , Princeton University Press: Princeton, NJ.
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