Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 051, 70 pages      arXiv:0904.3644      https://doi.org/10.3842/SIGMA.2009.051

Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review

Ion C. Baianu a, James F. Glazebrook b and Ronald Brown c
a) FSHN and NPRE Departments, University of Illinois at Urbana-Champaign, AFC-NMR & FT-NIR Microspectroscopy Facility, Urbana IL 61801 USA
b) Department of Mathematics and Computer Science, Eastern Illinois University, 600 Lincoln Avenue, Charleston IL 61920 USA
    Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana IL 61801 USA
c) School of Computer Science, University of Bangor, Dean Street, Bangor Gwynedd LL57 1UT UK

Received November 25, 2008, in final form April 09, 2009; Published online April 23, 2009

Abstract
A novel algebraic topology approach to supersymmetry (SUSY) and symmetry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non-Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier-Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin) models with the extended quantum symmetry of entangled, 'string-net condensed' (ground) states.

Key words: extended quantum symmetries; groupoids and algebroids; quantum algebraic topology (QAT); algebraic topology of quantum systems; symmetry breaking, paracrystals, superfluids, spin networks and spin glasses; convolution algebras and quantum algebroids; nuclear Fréchet spaces and GNS representations of quantum state spaces (QSS); groupoid and functor representations in relation to extended quantum symmetries in QAT; quantization procedures; quantum algebras: von Neumann algebra factors; paragroups and Kac algebras; quantum groups and ring structures; Lie algebras; Lie algebroids; Grassmann-Hopf, weak C*-Hopf and graded Lie algebras; weak C*-Hopf algebroids; compact quantum groupoids; quantum groupoid C*-algebras; relativistic quantum gravity (RQG); supergravity and supersymmetry theories; fluctuating quantum spacetimes; intense gravitational fields; Hamiltonian algebroids in quantum gravity; Poisson-Lie manifolds and quantum gravity theories; quantum fundamental groupoids; tensor products of algebroids and categories; quantum double groupoids and algebroids; higher dimensional quantum symmetries; applications of generalized van Kampen theorem (GvKT) to quantum spacetime invariants.

pdf (826 kb)   ps (428 kb)   tex (96 kb)

References

  1. Abragam A., Bleaney B., Electron paramagnetic resonance of transition ions, Clarendon Press, Oxford, 1970.
  2. Aguiar M., Andruskiewitsch N., Representations of matched pairs of groupoids and applications to weak Hopf algebras, Contemp. Math. 376 (2005), 127-173, math.QA/0402118.
  3. Aguiar M.C.O., Dobrosavljevic V., Abrahams E., Kotliar G., Critical behavior at Mott-Anderson transition: a TMT-DMFT perspective, Phys. Rev. Lett. 102 (2009), 156402, 4 pages, arXiv:0811.4612.
    Aguiar M.C.O., Dobrosavljevic V., Abrahams E., Kotliar G., Scaling behavior of an Anderson impurity close to the Mott-Anderson transition, Phys. Rev. B 73 (2006), 115117, 7 pages, cond-mat/0509706.
  4. Alfsen E.M., Schultz F.W., Geometry of state spaces of operator algebras, Birkhäuser, Boston - Basel - Berlin, 2003.
  5. Altintash A.A., Arika M., The inhomogeneous invariance quantum supergroup of supersymmetry algebra, Phys. Lett. A 372 (2008), 5955-5958.
  6. Anderson P.W., Absence of diffusion in certain random lattices, Phys. Rev. 109 (1958), 1492-1505.
  7. Anderson P.W., Topology of glasses and mictomagnets, Lecture presented at The Cavendish Laboratory, Cambridge, UK, 1977.
  8. Baez J., This week's finds in mathematical physics, 1999, Week 137, available at http://math.ucr.edu/home/baez/week137.html.
  9. Baez J.C., Langford L., Higher-dimensional algebra. IV. 2-tangles, Adv. Math. 180 (2003), 705-764, math.QA/9811139.
  10. Baianu I.C., Categories, functors and automata theory: a novel approach to quantum automata through algebraic-topological quantum computation, in Proceedings of the 4th Intl. Congress of Logic, Methodology and Philosophy of Science (Bucharest, August - September, 1971), University of Bucharest, Bucharest, 1971, 256-257.
  11. Baianu I.C., Structural studies of erythrocyte and bacterial cytoplasmic membranes by X-ray diffraction and electron microscopy, PhD Thesis, Queen Elizabeth College, University of London, 1974.
  12. Baianu I.C., X-ray scattering by partially disordered membrane systems, Acta Cryst. A 34 (1978), 751-753.
  13. Baianu I.C., Boden N., Levine Y.K., Lightowlers D., Dipolar coupling between groups of three spin-1/2 undergoing hindered reorientation in solids, Solid State Comm. 27 (1978), 474-478.
  14. Baianu I.C., Boden N., Levine Y.K., Ross S.M., Quasi-quadrupolar NMR spin-Echoes in solids containing dipolar coupled methyl groups, J. Phys. C: Solid State Phys. 11 (1978), L37-L41.
  15. Baianu I.C., Boden N., Mortimer M., Lightowlers D., A new approach to the structure of concentrated aqueous electrolyte solutions by pulsed N.M.R., Chem. Phys. Lett. 54 (1978), 169-175.
  16. Baianu I.C., Rubinson K.A., Patterson J., The observation of structural relaxation in a FeNiPB class by X-ray scattering and ferromagnetic resonance, Phys. Status Solidi A 53 (1979), K133-K135.
  17. Baianu I.C., Rubinson K.A., Patterson J., Ferromagnetic resonance and spin-wave excitations in metallic glasses, J. Phys. Chem. Solids 40 (1979), 940-951.
  18. Baianu I.C., Structural order and partial disorder in biological systems, Bull. Math. Biology 42 (1980), 464-468.
  19. Baianu I.C., Boden N., Lightowlers D., NMR spin-echo responses of dipolar-coupled spin-1/2 triads in solids, J. Magnetic Resonance 43 (1981), 101-111.
  20. Baianu I.C., Lukasiewicz-Topos models of neural networks, cell genome and interactome nonlinear dynamic models, Preprint, 2004, http://cogprints.org/3701/.
  21. Baianu I.C., Les Categories des Connexions Cellulaire dans le Topos sur les Algebres des Logiques Lukasiewicz pour les Neuronnes et le Genome Cellulaire compris dans un Interactome Dynamique, CERN Preprint EXT-2004-059, http://cdsweb.cern.ch/record/746663.
  22. Baianu I.C., Glazebrook J.F., Brown R., Georgescu G., Complex nonlinear biodynamics in categories, higher dimensional algebra and Lukasiewicz-Moisil topos: transformation of neural, genetic and neoplastic networks, Axiomathes 16 (2006), 65-122.
  23. Baianu I.C., Glazebrook J.F., Brown R., A non-Abelian, categorical ontology of spacetimes and quantum gravity, Axiomathes 17 (2007), 353-408.
  24. Baianu I.C., Glazebrook J.F., Brown R., Georgescu,G., Towards quantum non-Abelian algebraic topology, in preparation.
  25. Baianu I.C., Glazebrook J.F., Brown R., On physical applications of non-Abelian algebraic topology in quantum field theories, in preparation.
  26. Bais F.A., Schroers B.J., Slingerland J.K., Broken quantum symmetry and confinement phases in planar physics, Phys. Rev. Lett. 89 (2002), 181601, 4 pages, hep-th/0205117.
  27. Ball R.C., Fermions without fermion fields, Phys. Rev. Lett. 95 (2005), 176407, 4 pages, cond-mat/0409485.
  28. Barrett J.W., Geometrical measurements in three-dimensional quantum gravity, in Proceedings of the Tenth Oporto Meeting on Geometry, Topology and Physics (2001), Internat. J. Modern Phys. A 18 (2003), October, suppl., 97-113, gr-qc/0203018.
  29. Barrett J.W., Mackaay M., Categorical representation of categorical groups, Theory Appl. Categ. 16 (2006), 529-557, math.CT/0407463.
  30. Baues H.J., Conduché D., On the tensor algebra of a nonabelian group and applications, K-Theory 5 (1991/92), 531-554.
  31. Blaom A.D., Lie algebroids and Cartan's method of equivalence, math.DG/0509071.
  32. Blok B., Wen X.-G., Many-body systems with non-Abelian statistics, Nuclear Phys. B 374 (1992), 615-619.
  33. Böckenhauer J., Evans D., Kawahigashi Y., Chiral structure of modular invariants for subfactors, Comm. Math. Phys. 210 (2000), 733-784, math.OA/9907149.
  34. Böhm G., Nill F., Szlachányi K., Weak Hopf algebras. I. Integral theory and C*-structure, J. Algebra 221 (1999), 385-438, math.OA/9805116.
  35. Bohm A., Gadella M., Dirac kets, Gamow vectors and Gelfan'd triples. The rigged Hilbert space formulation of quantum mechanics, Lecture Notes in Physics, Vol. 348, Springer-Verlag, Berlin, 1989.
  36. Borceux F., Janelidze G., Galois Theories, Cambridge Studies in Advanced Mathematics, Vol. 72, Cambridge University Press, Cambridge, 2001.
  37. Bos R., Continuous representations of groupoids, math.RT/0612639.
  38. Bos R., Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids, Int. J. Geom. Methods Mod. Phys. 4 (2007), 389-436, math.SG/0604027.
  39. Bos R., Groupoids in geometric quantization, PhD thesis, Radboud University Nijmegen, 2007.
  40. Bos R., Continuous representations of groupoids, Houston J. Math., to appear, available at http://www.math.ru.nl/~rdbos/ContinReps.pdf.
  41. Brown R., Hardie K., Kamps H., Porter T., The homotopy double groupoid of a Hausdorff space, Theory Appl. Categ. 10 (2002), 71-93.
  42. Brown R., Topology and groupoids, BookSurge, LLC, Charleston, SC, 2006.
  43. Brown R., Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups, J. Homotopy Relat. Struct. 3 (2008), 331-342.
  44. Brown R., Glazebrook J.F., Baianu I.C., A categorical and higher dimensional algebra framework for complex systems and spacetime structures, Axiomathes 17 (2007), 409-493.
  45. Brown R., Higgins P. J., Tensor products and homotopies for w-groupoids and crossed complexes, J. Pure Appl. Algebra 47 (1987), 1-33.
  46. Brown R., Higgins P.J., Sivera R., Nonabelian algebraic topology, Vols. 1 and 2, in preparation, available at http://www.bangor.ac.uk/r.brown/nonab-a-t.html.
  47. Brown R., Janelidze G., Galois theory and a new homotopy double groupoid of a map of spaces, Appl. Categ. Structures 12 (2004), 63-80, math.AT/0208211.
  48. Brown R., Janelidze G., Van Kampen theorems for categories of covering morphisms in lextensive categories, J. Pure Appl. Algebra 119 (1997), 255-263.
  49. Brown R., Loday J.-L., Homotopical excision, and Hurewicz theorems for n-cubes of spaces, Proc. London Math. Soc. (3) 54 (1987) 176-192.
  50. Brown R., Loday J.-L., Van Kampen theorems for diagrams of spaces, Topology 26 (1987), 311-335.
  51. Brown R., Mosa G.H., Double algebroids and crossed modules of algebroids, Preprint University of Wales-Bangor, 1986.
  52. Brown R., Spencer C.B., Double groupoids and crossed modules, Cahiers Top. Géom. Différentielle 17 (1976), 343-362.
  53. Brown R., Spencer C.B., G-groupoids, crossed modules and the fundamental groupoid of a topological group, Indag. Math. 38 (1976), 296-302.
  54. Buneci M.R., Groupoid representations, Ed. Mirton, Timishoara, Romania, 2003.
  55. Buneci M.R., The structure of the Haar systems on locally compact groupoids, Math. Phys. Electron. J. 8 (2002), Paper 4, 11 pages.
  56. Buneci M.R., Consequences of the Hahn structure theorem for Haar measure, Math. Rep. (Bucur.) 4(54) (2002), 321-334.
  57. Buneci M.R., Amenable unitary representations of measured groupoids, Rev. Roumaine Math. Pures Appl. 48 (2003), 129-133.
  58. Buneci M.R., Invariant means on groupoids, Rev. Roumaine Math. Pures Appl. 48 (2003), 13-29.
  59. Buneci M.R., Groupoid algebras for transitive groupoids, Math. Rep. (Bucur.) 5(55) (2003), 9-26.
  60. Buneci M.R., Haar systems and homomorphism on groupoids, in Operator Algebras and Mathematical Physics (Constantza, 2001), Editors J.-M. Combes, J. Cuntz, G. Elliott, Gh. Nenciu, H. Siedentop and S. Stratila, Theta, Bucharest, 2003, 35-49.
  61. Buneci M.R., Necessary and sufficient conditions for the continuity of a pre-Haar system at a unit with singleton orbit, Quasigroups and Related Systems 12 (2004), 29-38.
  62. Buneci M.R., Isomorphic groupoid C*-algebras associated with different Haar systems, New York J. Math. 11 (2005), 225-245, math.OA/0504024.
  63. Buneci M.R., The equality of the reduced and the full C*-algebras and the amenability of a topological groupoid, in Recent Advances in Operator Theory, Operator Algebras, and Their Applications, Oper. Theory Adv. Appl., Vol. 153, Birkhäuser, Basel, 2005, 61-78.
  64. Buneci M.R., Groupoid C*-algebras, Surveys in Math. and its Appl. 1 (2006), 71-98.
  65. Butterfield J., Isham C.J., A topos perspective on the Kochen-Specker theorem. I. Quantum states as generalized valuations, Internat. J. Theoret. Phys. 37 (1998), 2669-2733, quant-ph/9803055.
    Butterfield J., Isham C.J., A topos perspective on the Kochen-Specker theorem. II. Conceptual aspects and classical analogues, Internat. J. Theoret. Phys. 38 (1999), 827-859, quant-ph/9808067.
    Butterfield J., Isham C.J., A topos perspective on the Kochen-Specker theorem. III. Von Neumann algebras as the base category, Internat. J. Theoret. Phys. 39 (2000), 1413-1436, quant-ph/9911020.
    Butterfield J., Isham C.J., A topos perspective on the Kochen-Specker theorem. IV. Interval valuations, Internat. J. Theoret. Phys. 41 (2002), 613-639, quant-ph/0107123.
  66. Byczuk K., Hofstetter W., Vollhardt D., Mott-Hubbard transition vs. Anderson localization of correlated, disordered electrons, Phys. Rev. Lett. 94 (2005), 401-404, cond-mat/0403765.
  67. Carter J.S., Saito M., Knotted surfaces, braid moves, and beyond, in Knots and Quantum Gravity (Riverside, CA, 1993), Oxford Lecture Ser. Math. Appl., Vol. 1, Oxford Univ. Press, New York, 1994, 191-229.
  68. Chaician M., Demichev A., Introduction to quantum groups, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
  69. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
  70. Chamon C., Quantum glassiness in strongly correlated clean systems: an example of topological overprotection, Phys. Rev. Lett. 94 (2005), 040402, 4 pages, cond-mat/0404182.
  71. Coleman P., De Luccia F., Gravitational effects on and of vacuum decay, Phys. Rev. D 21 (1980), 3305-3309.
  72. Connes A., Sur la théorie non-commutative de l'intégration, in Algèbres d'opérateurs (Sém., Les Plans-sur-Bex, 1978), Lecture Notes in Math., Vol. 725, Springer, Berlin, 1979, 19-143.
  73. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  74. Crane L., Frenkel I.B., Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. Topology and physics, J. Math. Phys. 35 (1994), 5136-5154, hep-th/9405183.
  75. Crane L., Kauffman L.H., Yetter D.N., State-sum invariants of 4-manifolds, J. Knot Theory Ramifications 6 (1997), 177-234, hep-th/9409167.
  76. Day B.J., Street R., Monoidal bicategories and Hopf algebroids, Adv. Math. 129 (1997), 99-157.
  77. Deligne P., Catégories tensorielles, Moscow Math. J. 2 (2002), 227-248.
  78. Deligne P., Morgan J.W., Notes on supersymmetry (following Joseph Berenstein), in Quantum Fields and Strings: a Course for Mathematicians (Princeton, NJ, 1996/1997), Editors J.W. Morgan et al., Vols. 1, 2, Amer. Math. Soc., Providence RI, 1999, 41-97.
  79. Dennis E., Kitaev A., Landahl A., Preskill J., Topological quantum memory, J. Math. Phys. 43 (2002), 4452-4459, quant-ph/0110143.
  80. Dixmier J., Les C*-algèbras et leurs représentations, Deuxième èdition, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969.
  81. Doebner H.-D., Hennig J.-D. (Editors), Quantum groups, Proceedings of the Eighth International Workshop on Mathematical Physics (Technical University of Clausthal, Clausthal, July 19-26, 1989), Lecture Notes in Physics, Vol. 370, Springer-Verlag, Berlin, 1990.
  82. Drechsler W., Tuckey P.A., On quantum and parallel transport in a Hilbert bundle over spacetime, Classical Quantum Gravity 13 (1996), 611-632, gr-qc/9506039.
  83. Drinfel'd V.G., Quantum groups, in Proceedings Intl. Cong. of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Editor A. Gleason, Amer. Math. Soc., Providence, RI, 1987, 798-820.
  84. Drinfel'd V.G., Structure of quasi-triangular quasi-Hopf algebras, Funct. Anal. Appl. 26 (1992), 63-65.
  85. Ellis G.J., Higher dimensional crossed modules of algebras, J. Pure Appl. Algebra 52 (1988), 277-282.
  86. Etingof P.I., Ostrik V., Finite tensor categories, math.QA/0301027.
  87. Etingof P.I., Varchenko A.N., Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups, Comm. Math. Phys. 196 (1998), 591-640, q-alg/9708015.
  88. Etingof P.I., Varchenko A.N., Exchange dynamical quantum groups, Comm. Math. Phys. 205 (1999), 19-52, math.QA/9801135.
  89. Etingof P.I., Schiffmann O., Lectures on the dynamical Yang-Baxter equations, in Quantum Groups and Lie Theory (Durham, 1999), London Math. Soc. Lecture Note Ser., Vol. 290, Cambridge Univ. Press, Cambridge, 2001, 89-129, math.QA/9908064.
  90. Eymard P., L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236.
  91. Fauser B., A treatise on quantum Clifford algebras, Konstanz, Habilitationsschrift, math.QA/0202059.
  92. Fauser B., Grade free product formulae from Grassman-Hopf gebras, in Clifford Algebras: Applications to Mathematics, Physics and Engineering, Editor R. Ablamowicz, Prog. Math. Phys., Vol. 34, Birkhäuser Boston, Boston, MA, 2004, 279-303, math-ph/0208018.
  93. Fell J.M.G., The dual spaces of C*-algebras, Trans. Amer. Math. Soc. 94 (1960), 365-403.
  94. Fernandez F.M., Castro E.A., Lie algebraic methods in quantum chemistry and physics, Mathematical Chemistry Series, CRC Press, Boca Raton, FL, 1996.
  95. Feynman R.P., Space-time approach to non-relativistic quantum mechanics, Rev. Modern Phys. 20 (1948), 367-387.
  96. Freedman M.H., Kitaev A., Larsen M.J., Wang Z., Topological quantum computation, Bull. Amer. Math. Soc. (N.S.) 40 (2003), 31-38, quant-ph/0101025.
  97. Fröhlich A., Non-Abelian homological algebra. I. Derived functors and satellites, Proc. London Math. Soc. (3) 11 (1961), 239-275.
  98. Gel'fand I., Naimark M., On the imbedding of normed rings into the ring of operators in Hilbert space, Rec. Math. [Mat. Sbornik] N.S. 12(54) (1943), 197-213.
  99. Georgescu G., N-valued logics and Lukasiewicz-Moisil algebras, Axiomathes 16 (2006), 123-136.
  100. Georgescu G., Popescu D., On algebraic categories, Rev. Roumaine Math. Pures Appl. 13 (1968), 337-342.
  101. Georgescu G., Vraciu C., On the characterization of Lukasiewicz algebras, J. Algebra 16 (1970), 486-495.
  102. Gilmore R., Lie groups, Lie algebras and some of their applications, Dover Publs. Inc., Mineola - New York, 2005.
  103. Grabowski J., Marmo G., Generalized Lie bialgebroids and Jacobi structures, J. Geom Phys. 40 (2001), 176-199.
  104. Grandis M., Mauri L., Cubical sets and their site, Theory Appl. Categ. 11 (2003), no. 8, 185-211.
  105. Grisaru M.T., Pendleton H.N., Some properties of scattering amplitudes in supersymmetric theories, Nuclear Phys. B 124 (1977), 81-92.
  106. Grothendieck A., Sur quelque point d-algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119-121.
  107. Grothendieck A., Technique de descente et théorèmes d'existence en géométrie algébrique. II, Séminaire Bourbaki 12 (1959/1960), exp. 195, Secrétariat Mathématique, Paris, 1961.
  108. Grothendieck A., Technique de construction en géométrie analytique. IV. Formalisme général des foncteurs représentables, Séminaire H. Cartan 13 (1960/1961), exp. 11, Secrétariat Mathématique, Paris, 1962.
  109. Grothendieck A., Revêtements Étales et Groupe Fondamental (SGA1), chapter VI: Catégories fibrées et descente, Lecture Notes in Math., Vol. 224, Springer-Verlag, Berlin, 1971.
  110. Grothendieck A., Dieudonné J., Eléments de geometrie algèbrique, Publ. Inst. des Hautes Études de Science 4 (1960), 5-365.
    Grothendieck A., Dieudonné J., Étude cohomologique des faisceaux coherents, Publ. Inst. des Hautes Études de Science 11 (1961), 5-167.
  111. Gu Z.-G., Wen X.-G., A lattice bosonic model as a quantum theory of gravity, gr-qc/0606100.
  112. Hahn P., Haar measure for measure groupoids, Trans. Amer. Math. Soc. 242 (1978), 1-33.
  113. Hahn P., The regular representations of measure groupoids, Trans. Amer. Math. Soc. 242 (1978), 34-72.
  114. Harrison B.K., The differential form method for finding symmetries, SIGMA 1 (2005), 001, 12 pages, math-ph/0510068.
  115. Hazewinkel A. (Editor), Handbook of algebra, Vol. 4, Elsevier, St. Louis, Chatswood - Singapore, 2006.
  116. Heynman R., Lifschitz S., Lie groups and Lie algebras, Nelson Press, New York - London, 1958.
  117. Heunen C., Landsman N.P., Spitters B., A topos for algebraic quantum theory, Comm. Math. Phys., to appear, arXiv:0709.4364.
  118. Higgins P.J., Presentations of groupoids, with applications to groups, Proc. Cambridge Philos. Soc. 60 (1964), 7-20.
  119. Higgins P.J., Notes on categories and groupoids, Van Nostrand Rienhold Mathematical Studies, no. 32, Van Nostrand Reinhold Co., London - New York - Melbourne, 1971 (reprinted in Theory Appl. Categ. 7 (2005)).
  120. Hindeleh A.M., Hosemann R., Paracrystals representing the physical state of matter, Solid State Phys. 21 (1988), 4155-4170.
  121. Hosemann R., Bagchi R.N., Direct analysis of diffraction by matter, North-Holland Publs., Amsterdam - New York, 1962.
  122. Hosemann R., Vogel W., Weick D., Balta-Calleja F.J., Novel aspects of the real paracrystal, Acta Cryst. A 37 (1981), 85-91.
  123. Isham C.J., Butterfield J., Some possible roles for topos theory in quantum theory and quantum gravity, Found. Phys. 30 (2000), 1707-1735, gr-qc/9910005.
  124. Janelidze G., Precategories and Galois theory, in Category theory (Como, 1990), Springer Lecture Notes in Math., Vol. 1488, Springer, Berlin, 1991, 157-173.
  125. Janelidze G., Pure Galois theory in categories, J. Algebra 132 (1990), 270-286.
  126. Joyal A., Street R., Braided tensor categories, Adv. Math. 102 (1993), 20-78.
  127. Kac V., Lie superalgebras, Adv. Math. 26 (1977), 8-96.
  128. Kamps K.H., Porter T., A homotopy 2-groupoid from a fibration, Homology Homotopy Appl. 1 (1999), 79-93.
  129. Kapitsa P.L., Plasma and the controlled thermonuclear reaction, The Nobel Prize lecture in Physics 1978, in Nobel Lectures, Physics 1971-1980, Editor S. Lundqvist, World Scientific Publishing Co., Singapore, 1992.
  130. Karaali G., On Hopf algebras and their generalizations, Comm. Algebra 36 (2008), 4341-4367, math.QA/0703441.
  131. Kirillov A. Jr., Ostrik V., On q-analog of McKay correspondence and ADE classification of sl(2) conformal field theories, math.QA/0101219.
  132. Konno H., Elliptic quantum group Uq,p(^sl2), Hopf algebroid structure and elliptic hypergeometric series, arXiv:0803.2292.
    Konno H., Dynamical R matrices of elliptic quantum groups and connection matrices for the q-KZ equations, SIGMA 2 (2006), 091, 25 pages, math.QA/0612558.
  133. Khoroshkin S.M., Tolstoy V.N., Universal R-matrix for quantum supergroups, in Group Theoretical Methods in Physics (Moscow, 1990), Lecture Notes in Phys., Vol. 382, Springer, Berlin, 1991, 229-232.
  134. Kitaev A.Y., Fault-tolerant quantum computation by anyons, Ann. Physics 303 (2003), 2-30, quant-ph/9707021.
  135. Kustermans J., Vaes S., The operator algebra approach to quantum groups, Proc. Natl. Acad. Sci. USA 97 (2000), 547-552.
  136. Lambe L.A., Radford D.E., Introduction to the quantum Yang-Baxter equation and quantum groups: an algebraic approach, Mathematics and its Applications, Vol. 423, Kluwer Academic Publishers, Dordrecht, 1997.
  137. Lance E.C., Hilbert C*-modules. A toolkit for operator algebraists, London Mathematical Society Lecture Note Series, Vol. 210, Cambridge University Press, Cambridge, 1995.
  138. Landsman N.P., Mathematical topics between classical and quantum mechanics, Springer Monographs in Mathematics, Springer-Verlag, New York, 1998.
  139. Landsman N.P., Compact quantum groupoids, in Quantum Theory and Symmetries (Goslar, July 18-22, 1999), Editors H.-D. Doebner et al., World Sci. Publ., River Edge, NJ, 2000, 421-431, math-ph/9912006.
  140. Landsman N.P., Ramazan B., Quantization of Poisson algebras associated to Lie algebroids, in Proc. Conf. on Groupoids in Physics, Analysis and Geometry(Boulder CO, 1999), Editors J. Kaminker et al., Contemp. Math. 282 (2001), 159-192, math-ph/0001005.
  141. Lee P.A., Nagaosa N., Wen X.-G., Doping a Mott insulator: physics of high temperature superconductivity, cond-mat/0410445.
  142. Levin A., Olshanetsky M., Hamiltonian algebroids and deformations of complex structures on Riemann curves, hep-th/0301078.
  143. Levin M., Wen X.-G., Fermions, strings, and gauge fields in lattice spin models, Phys. Rev. B 67 (2003), 245316, 4 pages, cond-mat/0302460.
  144. Levin M., Wen X.-G., Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96 (2006), 110405, 4 pages, cond-mat/0510613.
  145. Levin M., Wen X.-G., Colloquium: photons and electrons as emergent phenomena, Rev. Modern Phys. 77 (2005), 871-879, cond-mat/0407140.
    Levin M., Wen X.-G., Quantum ether: photons and electrons from a rotor model, Phys. Rev. B 73 (2006), 035122, 10 pages, hep-th/0507118.
  146. Loday J.L., Spaces with finitely many homotopy groups, J. Pure Appl. Algebra 24 (1982), 179-201.
  147. Lu J.-H., Hopf algebroids and quantum groupoids, Internat. J. Math. 7 (1996), 47-70, q-alg/9505024.
  148. Mack G., Schomerus V., Quasi Hopf quantum symmetry in quantum theory, Nuclear Phys. B 370 (1992), 185-230.
  149. Mackaay M., Spherical 2-categories and 4-manifold invariants, Adv. Math. 143 (1999), 288-348, math.QA/9805030.
  150. Mackenzie K.C.H., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge University Press, Cambridge, 2005.
  151. Mackey G.W., The scope and history of commutative and noncommutative harmonic analysis, History of Mathematics, Vol. 5, American Mathematical Society, Providence, RI, London Mathematical Society, London, 1992.
  152. Mac Lane S., Categorical algebra, Bull. Amer. Math. Soc. 71 (1965), 40-106.
  153. Mac Lane S., Moerdijk I., Sheaves in geometry and logic. A first introduction to topos theory, Springer Verlag, New York, 1994.
  154. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  155. Maltsiniotis G., Groupoïdes quantiques, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), 249-252.
  156. Martin J.F., Porter T., On Yetter's invariant and an extension of the Dijkgraaf-Witten invariant to categorical groups, Theory Appl. Categ., 18 (2007), no. 4, 118-150.
  157. Miranda E., Introduction to bosonization, Brazil. J. Phys. 33 (2003), 1-33.
  158. Mitchell B., The theory of categories, Pure and Applied Mathematics, Vol. 17, Academic Press, New York - London 1965.
  159. Mitchell B., Rings with several objects, Adv. Math. 8 (1972), 1-161.
  160. Mitchell B., Separable algebroids, Mem. Amer. Math. Soc. 57 (1985), no. 333.
  161. Moerdijk I., Svensson J.A., Algebraic classification of equivariant 2-types, J. Pure Appl. Algebra 89 (1993), 187-216.
  162. Mosa G.H., Higher dimensional algebroids and Crossed complexes, PhD Thesis, University of Wales, Bangor, 1986.
  163. Mott N.F., Electrons in glass, Nobel Lecture, 11 pages, December 8, 1977.
  164. Mott N.F., Electrons in solids with partial disorder, Lecture presented at The Cavendish Laboratory, Cambridge, UK, 1978.
  165. Mott N.F., Davis E.A., Electronic processes in non-crystalline materials, Oxford University Press, 1971, 2nd ed., 1978.
  166. Mott N.F. et al., The Anderson transition, Proc. Royal Soc. London Ser. A 345 (1975), 169-178.
  167. Moultaka G., Rausch M. de Traubenberg, Tanasa A., Cubic supersymmetry and abelian gauge invariance, Internat. J. Modern Phys. A 20 (2005), 5779-5806, hep-th/0411198.
  168. Mrcun J., On spectral representation of coalgebras and Hopf algebroids, math.QA/0208199.
  169. Mrcun, J., On the duality between étale groupoids and Hopf algebroids, J. Pure Appl. Algebra 210 (2007), 267-282.
  170. Muhly P., Renault J., Williams D., Equivalence and isomorphism for groupoid C*-algebras, J. Operator Theory 17 (1987), 3-22.
  171. Nayak C., Simon S.H., Stern A., Freedman M., Das Sarma S., Non-Abelian anyons and topological quantum computation, Rev. Modern Phys. 80 (2008), 1083, 77 pages, arXiv:0707.1889.
  172. Nikshych D.A., Vainerman L., A characterization of depth 2 subfactors of II1 factors, J. Funct. Anal. 171 (2000), 278-307, math.QA/9810028.
  173. Nishimura H., Logical quantization of topos theory, Internat. J. Theoret. Phys. 35 (1996), 2555-2596.
  174. Neuchl M., Representation theory of Hopf categories, PhD Thesis, University of Münich, 1997.
  175. Ocneanu A., Quantized groups, string algebras and Galois theory for algebras, in Operator Algebras and Applications, London Math. Soc. Lecture Note Ser., Vol. 136, Cambridge University Press, Cambridge, 1988, Vol. 2, 119-172.
  176. Ocneanu A., Operator algebras, topology and subgroups of quantum symmetry - construction of subgroups of quantum groups, in Taniguchi Conference on Mathematics Nara '98, Adv. Stud. Pure Math., Vol. 31, Math. Soc. Japan, Tokyo, 2001, 235-263.
  177. Ostrik V., Module categories over representations of SLq(2) in the non-simple case, Geom. Funct. Anal. 17 (2008), 2005-2017, math.QA/0509530.
  178. Patera J., Orbit functions of compact semisimple, however, as special functions, in Proceedings of Fifth International Conference "Symmetry in Nonlinear Mathematical Physics" (June 23-29, 2003, Kyiev), Editors A.G. Nikitin, V.M. Boyko, R.O. Popovych and I.A. Yehorchenko, Proceedings of the Institute of Mathematics, Kyiv 50 (2004), Part 1, 1152-1160.
  179. Paterson A.L.T., Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics, Vol. 170, Birkhäuser Boston, Inc., Boston, MA, 1999.
  180. Paterson A.L.T., The Fourier-Stieltjes and Fourier algebras for locally compact groupoids, in Trends in Banach Spaces and Operator Theory (Memphis, TN, 2001), Contemp. Math. 321 (2003), 223-237, math.OA/0310138.
  181. Paterson A.L.T., Graph inverse semigroups, groupoids and their C*-algebras, J. Operator Theory 48 (2002), 645-662, math.OA/0304355.
  182. Popescu N., The theory of Abelian categories with applications to rings and modules, London Mathematical Society Monographs, no. 3, Academic Press, London - New York, 1973.
  183. Prigogine I., From being to becoming time and complexity in the physical sciences, W.H. Freeman and Co., San Francisco, 1980.
  184. Ramsay A., Walter M.E., Fourier-Stieltjes algebras of locally compact groupoids, J. Funct. Anal. 148 (1997), 314-367, math.OA/9602219.
  185. Ran Y., Wen X.-G., Detecting topological order through a continuous quantum phase transition, Phys. Rev Lett. 96 (2006), 026802, 4 pages, cond-mat/0509155.
  186. Raptis I., Zapatrin R.R., Quantization of discretized spacetimes and the correspondence principle, Internat. J. Theoret. Phys. 39 (2000), 1-13, gr-qc/9904079.
  187. Regge T., General relativity without coordinates, Nuovo Cimento (10) 19 (1961), 558-571.
  188. Rehren H.-K., Weak C*-Hopf symmetry, in Proceedings of Quantum Group Symposium at Group 21 (Goslar, 1996), Heron Press, Sofia, 1997, 62-69, q-alg/9611007.
  189. Renault J., A groupoid approach to C*-algebras, Lecture Notes in Mathematics, Vol. 793, Springer, Berlin, 1980.
  190. Renault J., Représentations des produits croisés d'algèbres de groupoïdes, J. Operator Theory 18 (1987), 67-97.
  191. Renault J., The Fourier algebra of a measured groupoid and its multiplier, J. Funct. Anal. 145 (1997), 455-490.
  192. Ribeiro T.C., Wen X.-G., New mean field theory of the tt¢t¢¢J model applied to the high-Tc superconductors, Phys. Rev. Lett. 95 (2005), 057001, 4 pages, cond-mat/0410750.
  193. Roberts J.E., Skein theory and Turaev-Viro invariants, Topology 34 (1995), 771-787.
  194. Roberts J.E., Refined state-sum invariants of 3-and 4-manifolds, in Geometric Topology (Athens, GA, 1993), AMS/IP Stud. Adv. Math., Vol. 2.1, Amer. Math. Soc., Providence, RI, 1997, 217-234.
  195. Rovelli C., Loop quantum gravity, Living Rev. Relativ. 1 (1998), 1998-1, 68 pages, gr-qc/9710008.
  196. Sato N., Wakui M., (2+1)-dimensional topological quantum field theory with a Verlinde basis and Turaev-Viro-Ocneanu invariants of 3-manifolds, in Invariants of Knots and 3-Manifolds (Kyoto, 2001), Geom. Topol. Monogr., Vol. 4, Geom. Topol. Publ., Coventry, 2002, 281-294, math.QA/0210368.
  197. Schwartz L., Généralisation de la notion de fonction, de dérivation, de transformation de Fourier et applications mathématiques et physiques, Ann. Univ. Grenoble. Sect. Sci. Math. Phys. (N.S.) 21 (1945), 57-74.
  198. Schwartz L., Théorie des distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg, no. IX-X, Hermann, Paris, 1966.
  199. Seda A.K., Haar measures for groupoids, Proc. Roy. Irish Acad. Sect. A 76 (1976), no. 5, 25-36.
  200. Seda A.K., Banach bundles of continuous functions and an integral representation theorem, Trans. Amer. Math. Soc. 270 (1982), 327-332.
  201. Seda A.K., On the Continuity of Haar measures on topological groupoids, Proc. Amer. Math. Soc. 96 (1986), 115-120.
  202. Stern A., Halperin B.I., Proposed experiments to probe the non-Abelian n = 5/2 quantum hall state, Phys. Rev. Lett. 96 (2006), 016802, 4 pages, cond-mat/0508447.
  203. Stradling R.A., Quantum transport, Phys. Bulletin 29 (1978), 559-562.
  204. Szlachányi K., The double algebraic view of finite quantum groupoids, J. Algebra 280 (2004), 249-294.
  205. Sweedler M.E., Hopf algebras, Mathematics Lecture Note Series, W.A. Benjamin, Inc., New York, 1969.
  206. Tanasa A., Extension of the Poincaré symmetry and its field theoretical interpretation, SIGMA 2 (2006), 056, 23 pages, hep-th/0510268.
  207. Tsui D.C., Allen S.J. Jr., Mott-Anderson localization in the two-dimensional band tail of Si inversion layers, Phys. Rev. Lett. 32 (1974), 1200-1203.
  208. Turaev V.G., Viro O.Ya., State sum invariants of 3-manifolds and quantum 6j-symbols, Topology 31 (1992), 865-902.
  209. van Kampen E.H., On the connection between the fundamental groups of some related spaces, Amer. J. Math. 55 (1933), 261-267.
  210. Várilly J.C., An introduction to noncommutative geometry, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2006.
  211. Weinberg S., The quantum theory of fields. Vol. I. Foundations, Cambridge University Press, Cambridge, 2005.
    Weinberg S., The quantum theory of fields. Vol. II. Modern applications, Cambridge University Press, Cambridge, 2005.
    Weinberg S., The quantum theory of fields. Vol. III. Supersymmetry, Cambridge University Press, Cambridge, 2005.
  212. Weinstein A., Groupoids: unifying internal and external symmetry. A tour through some examples, Notices Amer. Math. Soc. 43 (1996), 744-752, math.RT/9602220.
  213. Wen X.-G., Non-Abelian statistics in the fractional quantum Hall states, Phys. Rev. Lett. 66 (1991), 802-805.
  214. Wen X.-G., Projective construction of non-Abelian quantum hall liquids, Phys. Rev. B 60 (1999), 8827, 4 pages, cond-mat/9811111.
  215. Wen X.-G., Quantum order from string-net condensations and origin of light and massless fermions, Phys. Rev. D 68 (2003), 024501, 25 pages, hep-th/0302201.
  216. Wen X.-G., Quantum field theory of many-body systems - from the origin of sound to an origin of light and electrons, Oxford University Press, Oxford, 2004.
  217. Wess J., Bagger J., Supersymmetry and supergravity, Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1983.
  218. Wickramasekara S., Bohm A., Symmetry representations in the rigged Hilbert space formulation of quantum mechanics, J. Phys. A: Math. Gen. 35 (2002), 807-829, math-ph/0302018.
  219. Wightman A.S., Quantum field theory in terms of vacuum expectation values, Phys. Rev. 101 (1956), 860-866.
  220. Wightman A.S., Hilbert's sixth problem: mathematical treatment of the axioms of physics, in Mathematical Developments Arising from Hilbert Problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974), Amer. Math. Soc., Providence, R.I., 1976, 147-240.
  221. Wightman A.S., Gärding L., Fields as operator-valued distributions in relativistic quantum theory, Ark. Fys. 28 (1964), 129-184.
  222. Witten E., Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351-355.
  223. Witten E., Anti de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998), 253-291, hep-th/9802150.
  224. Woronowicz S.L., Twisted SU(2) group. An example of a non-commutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), 117-181.
  225. Xu P., Quantum groupoids and deformation quantization, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 289-294, q-alg/9708020.
  226. Yang C.N., Mills R.L., Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96 (1954), 191-195.
  227. Yetter D.N., TQFTs from homotopy 2-types, J. Knot Theory Ramifications 2 (1993), 113-123.
  228. Zhang R.B., Invariants of the quantum supergroup Uq(gl(m/1)), J. Phys. A: Math. Gen. 24 (1991), L1327-L1332.
  229. Zhang Y.-Z., Gould M.D., Quasi-Hopf superalgebras and elliptic quantum supergroups, J. Math. Phys. 40 (1999), 5264-5282, math.QA/9809156.

Previous article   Next article   Contents of Volume 5 (2009)