Abstract
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.
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References
C.J. Isham, Topological and global aspects of quantum theory, in Relativity, groups and topology II — Lectures 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].
N.M.J. Woodhouse, Geometric quantization, Oxford mathematical monographs, Clarendon, U.K. (1992).
J.-S. Park, Topological open p-branes, hep-th/0012141 [INSPIRE].
C. Klimčík and T. Strobl, WZW-Poisson manifolds, J. Geom. Phys. 43 (2002) 341 [math/0104189] [INSPIRE].
P. Ševera and A. WEinstein, Poisson geometry with a 3 form background, Prog. Theor. Phys. Suppl. 144 (2001) 145 [math/0107133] [INSPIRE].
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].
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].
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].
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].
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].
D. Lüst, T-duality and closed string non-commutative (doubled) geometry, JHEP 12 (2010) 084 [arXiv:1010.1361] [INSPIRE].
R. Blumenhagen and E. Plauschinn, Nonassociative gravity in string theory?, J. Phys. A 44 (2011) 015401 [arXiv:1010.1263] [INSPIRE].
D. Lüst, Twisted Poisson structures and non-commutative/non-associative closed string geometry, PoS(CORFU2011)086 [arXiv:1205.0100] [INSPIRE].
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].
H.J. Lipkin, W.I. Weisberger and M. Peshkin, Magnetic charge quantization and angular momentum, Annals Phys. 53 (1969) 203 [INSPIRE].
M.J.P. Gingras, Observing monopoles in a magnetic analog of ice, Science 326 (2009) 375 [arXiv:1005.3557].
K. Johnson and F.E. Low, Current algebras in a simple model, Prog. Theor. Phys. Suppl. 37 (1966) 74 [INSPIRE].
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.
S.G. Jo, Commutators in an anomalous non-Abelian chiral gauge theory, Phys. Lett. B 163 (1985) 353 [INSPIRE].
M. Günaydin and B. Zumino, Magnetic charge and non-associative algebras, in Symposium to honor G.C. Wick, Pisa Italy (1984) [INSPIRE].
M. Günaydin and D. Minic, Nonassociativity, Malcev algebras and string theory, Fortsch. Phys. 61 (2013) 873 [arXiv:1304.0410] [INSPIRE].
R. Moufang, Alternativekörper und der Satz vom vollständigen Vierseit (in German), Abh. Math. Sem. Univ. Hamburg 9 (1933) 207.
M. Günaydin, C. Piron and H. Ruegg, Moufang plane and octonionic quantum mechanics, Commun. Math. Phys. 61 (1978) 69 [INSPIRE].
R. Haag, Local quantum physics, Springer-Verlag, Berlin, Heidelberg Germany and New York U.S.A. (1992).
W. Thirring, Quantum mathematical physics, Springer, New York U.S.A. (2002).
M. Bojowald and A. Skirzewski, Effective equations of motion for quantum systems, Rev. Math. Phys. 18 (2006) 713 [math-ph/0511043] [INSPIRE].
M. Bojowald and A. Skirzewski, Quantum gravity and higher curvature actions, in Proceedings of “Current 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].
M. Bojowald and A. Tsobanjan, Effective Casimir conditions and group coherent states, Class. Quant. Grav. 31 (2014) 115006 [arXiv:1401.5352] [INSPIRE].
D. Brizuela, Statistical moments for classical and quantum dynamics: formalism and generalized uncertainty relations, Phys. Rev. D 90 (2014) 085027 [arXiv:1410.5776] [INSPIRE].
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].
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Bojowald, M., Brahma, S., Büyükçam, U. et al. States in non-associative quantum mechanics: uncertainty relations and semiclassical evolution. J. High Energ. Phys. 2015, 93 (2015). https://doi.org/10.1007/JHEP03(2015)093
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DOI: https://doi.org/10.1007/JHEP03(2015)093