States in non-associative quantum mechanics: uncertainty relations and semiclassical evolution

  • Martin Bojowald
  • Suddhasattwa Brahma
  • Umut Büyükçam
  • Thomas Strobl
Open Access
Regular Article - Theoretical Physics


A non-associative algebra of observables cannot be represented as operators on a Hilbert space, but it may appear in certain physical situations. This article employs algebraic methods in order to derive uncertainty relations and semiclassical equations, based on general properties of quantum moments.


Differential and Algebraic Geometry Flux compactifications 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    C.J. Isham, Topological and global aspects of quantum theory, in Relativity, groups and topology IILectures given at the 1983 Les Houches Summer School on Relativity, Groups and Topology, B.S. DeWitt and R. Stora eds., North-Holland, Amsterdam The Netherlands (1983) [INSPIRE].
  2. [2]
    N.M.J. Woodhouse, Geometric quantization, Oxford mathematical monographs, Clarendon, U.K. (1992).Google Scholar
  3. [3]
    J.-S. Park, Topological open p-branes, hep-th/0012141 [INSPIRE].
  4. [4]
    C. Klimčík and T. Strobl, WZW-Poisson manifolds, J. Geom. Phys. 43 (2002) 341 [math/0104189] [INSPIRE].CrossRefADSzbMATHMathSciNetGoogle Scholar
  5. [5]
    P. Ševera and A. WEinstein, Poisson geometry with a 3 form background, Prog. Theor. Phys. Suppl. 144 (2001) 145 [math/0107133] [INSPIRE].ADSGoogle Scholar
  6. [6]
    R. Blumenhagen, A. Deser, D. Lüst, E. Plauschinn and F. Rennecke, Non-geometric fluxes, asymmetric strings and nonassociative geometry, J. Phys. A 44 (2011) 385401 [arXiv:1106.0316] [INSPIRE].ADSGoogle Scholar
  7. [7]
    D. Mylonas, P. Schupp and R.J. Szabo, Membrane σ-models and quantization of non-geometric flux backgrounds, JHEP 09 (2012) 012 [arXiv:1207.0926] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  8. [8]
    I. Bakas and D. Lüst, 3-cocycles, non-associative star-products and the magnetic paradigm of R-flux string vacua, JHEP 01 (2014) 171 [arXiv:1309.3172] [INSPIRE].CrossRefADSGoogle Scholar
  9. [9]
    D. Mylonas, P. Schupp and R.J. Szabo, Non-geometric fluxes, quasi-Hopf twist deformations and nonassociative quantum mechanics, J. Math. Phys. 55 (2014) 122301 [arXiv:1312.1621] [INSPIRE].CrossRefADSGoogle Scholar
  10. [10]
    D. Mylonas, P. Schupp and R.J. Szabo, Nonassociative geometry and twist deformations in non-geometric string theory, PoS(ICMP2013)007 [arXiv:1402.7306] [INSPIRE].
  11. [11]
    D. Lüst, T-duality and closed string non-commutative (doubled) geometry, JHEP 12 (2010) 084 [arXiv:1010.1361] [INSPIRE].CrossRefGoogle Scholar
  12. [12]
    R. Blumenhagen and E. Plauschinn, Nonassociative gravity in string theory?, J. Phys. A 44 (2011) 015401 [arXiv:1010.1263] [INSPIRE].ADSMathSciNetGoogle Scholar
  13. [13]
    D. Lüst, Twisted Poisson structures and non-commutative/non-associative closed string geometry, PoS(CORFU2011)086 [arXiv:1205.0100] [INSPIRE].
  14. [14]
    R. Blumenhagen, M. Fuchs, F. Haßler, D. Lüst and R. Sun, Non-associative deformations of geometry in double field theory, JHEP 04 (2014) 141 [arXiv:1312.0719] [INSPIRE].CrossRefADSGoogle Scholar
  15. [15]
    H.J. Lipkin, W.I. Weisberger and M. Peshkin, Magnetic charge quantization and angular momentum, Annals Phys. 53 (1969) 203 [INSPIRE].CrossRefADSGoogle Scholar
  16. [16]
    M.J.P. Gingras, Observing monopoles in a magnetic analog of ice, Science 326 (2009) 375 [arXiv:1005.3557].CrossRefMathSciNetGoogle Scholar
  17. [17]
    K. Johnson and F.E. Low, Current algebras in a simple model, Prog. Theor. Phys. Suppl. 37 (1966) 74 [INSPIRE].CrossRefADSGoogle Scholar
  18. [18]
    F. Buccella, G. Veneziano, R. Gatto and S. Okubo, Necessity of additional unitary-antisymmetric q-number terms in the commutators of spatial current components, Phys. Rev. 149 (1966) 1268.CrossRefADSGoogle Scholar
  19. [19]
    S.G. Jo, Commutators in an anomalous non-Abelian chiral gauge theory, Phys. Lett. B 163 (1985) 353 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  20. [20]
    M. Günaydin and B. Zumino, Magnetic charge and non-associative algebras, in Symposium to honor G.C. Wick, Pisa Italy (1984) [INSPIRE].
  21. [21]
    M. Günaydin and D. Minic, Nonassociativity, Malcev algebras and string theory, Fortsch. Phys. 61 (2013) 873 [arXiv:1304.0410] [INSPIRE].Google Scholar
  22. [22]
    R. Moufang, Alternativekörper und der Satz vom vollständigen Vierseit (in German), Abh. Math. Sem. Univ. Hamburg 9 (1933) 207.CrossRefMathSciNetGoogle Scholar
  23. [23]
    M. Günaydin, C. Piron and H. Ruegg, Moufang plane and octonionic quantum mechanics, Commun. Math. Phys. 61 (1978) 69 [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  24. [24]
    R. Haag, Local quantum physics, Springer-Verlag, Berlin, Heidelberg Germany and New York U.S.A. (1992).CrossRefzbMATHGoogle Scholar
  25. [25]
    W. Thirring, Quantum mathematical physics, Springer, New York U.S.A. (2002).CrossRefGoogle Scholar
  26. [26]
    M. Bojowald and A. Skirzewski, Effective equations of motion for quantum systems, Rev. Math. Phys. 18 (2006) 713 [math-ph/0511043] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  27. [27]
    M. Bojowald and A. Skirzewski, Quantum gravity and higher curvature actions, in Proceedings ofCurrent Mathematical Topics in Gravitation and Cosmology(42nd Karpacz Winter School of Theoretical Physics), A. Borowiec and M. Francaviglia eds., eConf C 0602061 (2006) 03 [Int. J. Geom. Meth. Mod. Phys. 4 (2007) 25] [hep-th/0606232] [INSPIRE].
  28. [28]
    M. Bojowald and A. Tsobanjan, Effective Casimir conditions and group coherent states, Class. Quant. Grav. 31 (2014) 115006 [arXiv:1401.5352] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  29. [29]
    D. Brizuela, Statistical moments for classical and quantum dynamics: formalism and generalized uncertainty relations, Phys. Rev. D 90 (2014) 085027 [arXiv:1410.5776] [INSPIRE].ADSGoogle Scholar
  30. [30]
    M. Bojowald and A. Kempf, Generalized uncertainty principles and localization of a particle in discrete space, Phys. Rev. D 86 (2012) 085017 [arXiv:1112.0994] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Martin Bojowald
    • 1
  • Suddhasattwa Brahma
    • 1
  • Umut Büyükçam
    • 1
  • Thomas Strobl
    • 2
  1. 1.Institute for Gravitation and the CosmosThe Pennsylvania State UniversityUniversity ParkU.S.A.
  2. 2.Institut Camille JordanUniversité Claude Bernard Lyon 1Villeurbanne cedexFrance

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