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On Nonlinear Quantum Mechanics, Noncommutative Phase Spaces, Fractal-Scale Calculus and Vacuum Energy

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Abstract

A (to our knowledge) novel Generalized Nonlinear Schrödinger equation based on the modifications of Nottale-Cresson’s fractal-scale calculus and resulting from the noncommutativity of the phase space coordinates is explicitly derived. The modifications to the ground state energy of a harmonic oscillator yields the observed value of the vacuum energy density. In the concluding remarks we discuss how nonlinear and nonlocal QM wave equations arise naturally from this fractal-scale calculus formalism which may have a key role in the final formulation of Quantum Gravity.

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References

  1. Abrams, D.S., Lloyd, S.: Phys. Rev. Lett. 81, 3992 (1998)

    Article  ADS  Google Scholar 

  2. Acatrinei, C.: J. High Energy Phys. 0109, 007 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  3. Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  4. Ambjorn, J., Jurkiewicz, J., Loll, R.: Phys. Rev. Lett. 93, 131301 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  5. Ambjorn, J., Jurkiewicz, J., Loll, R.: Phys. Lett. B 607, 205 (2005)

    ADS  Google Scholar 

  6. Banerjee, R.: Mod. Phys. Lett. A 17, 631 (2002)

    Article  MATH  ADS  Google Scholar 

  7. Bekenstein, J.: Alternatives to dark matter: modified gravity as an alternative to dark matter. arXiv:1001.3876

  8. Bellucci, S., Nersessian, A., Sochichiu, C.: Phys. Lett. B 522, 345 (2001)

    MATH  MathSciNet  ADS  Google Scholar 

  9. Bender, C.: Introduction to PT-Symmetric Quantum Theory. Contemp. Phys. 46, 277 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  10. Bender, C., Cavero-Pelaez, I., Milton, K.A., Shajesh, K.V.: Phys. Lett. B 613, 97 (2005)

    MathSciNet  ADS  Google Scholar 

  11. Bernevig, B., Hu, J., Toumbas, N., Zhang, S.C.: Phys. Rev. Lett. 91, 236803 (2003)

    Article  ADS  Google Scholar 

  12. Białynicki-Birula, I., Mycielsky, J.: Ann. Phys. 100, 62 (1976)

    Article  ADS  Google Scholar 

  13. Bohm, D., Vigier, J.: Phys. Rev. 96, 208 (1954)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Born, M.: Proc. R. Soc. A 165, 291 (1938)

    Article  ADS  Google Scholar 

  15. Born, M.: Rev. Mod. Phys. 21, 463 (1949)

    Article  MATH  ADS  Google Scholar 

  16. Borsten, L., Dahanayake, D., Duff, M.J., Ebrahim, H., Rubens, W.: Black holes, qubits and octonions. arXiv:0809.4685

  17. Castro, C.: J. Math. Phys. 31(11), 2633 (1990)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Castro, C.: Found. Phys. Lett. 4, 81 (1991)

    Article  Google Scholar 

  19. Castro, C.: Found. Phys. 22, 569 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  20. Castro, C.: Found. Phys. 8, 1301 (2000)

    Article  Google Scholar 

  21. Castro, C.: Chaos Solitons Fractals 11(11), 1663 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Castro, C.: Chaos Solitons Fractals 12, 101 (2001)

    Article  MATH  Google Scholar 

  23. Castro, C.: Prog. Phys. 1, 20 (2005)

    MathSciNet  Google Scholar 

  24. Castro, C.: Found. Phys. 35(6), 971 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. Castro, C.: Prog. Phys. 2, 86 (2006)

    MathSciNet  Google Scholar 

  26. Castro, C.: J. Math. Phys. 48(7), 073517 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  27. Castro, C.: Adv. Stud. Theor. Phys. 1(12), 603 (2007)

    MATH  MathSciNet  Google Scholar 

  28. Castro, C.: Phys. Lett. B 668, 442 (2008)

    MathSciNet  ADS  Google Scholar 

  29. Castro, C.: Phys. Lett. B 675, 226 (2009)

    MathSciNet  ADS  Google Scholar 

  30. Castro, C.: On n-ary algebras, branes and polyvector gauge theories in noncommutative Clifford spaces. J. Phys. A, Math. Theor. 43, 365201 (2010)

    Article  ADS  Google Scholar 

  31. Castro, C., Mahecha, J.: Prog. Phys. 1, 38 (2006)

    Google Scholar 

  32. Castro, C., Pavsic, M.: Prog. Phys. 1, 31 (2005)

    MathSciNet  Google Scholar 

  33. Celerier, M., Nottale, L.: Dirac equation in scale relativity. arXiv:hep-th/0112213

  34. Chang, M.C., Niu, Q.: Phys. Rev. Lett. 75, 1348 (1995)

    Article  ADS  Google Scholar 

  35. Cresson, J.: J. Math. Phys. 44, 4907 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  36. Cresson, J.: Fractional embedding of differential operators and Lagrangian systems. arXiv:math/0605752

  37. Cresson, J., Frederico, G., Torres, D.: Topol. Methods Nonlinear Anal. 33(2), 217 (2009)

    MATH  MathSciNet  Google Scholar 

  38. Dito, G.: Lett. Math. Phys. 48, 307 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  39. Djemai, A.E.F., Smail, H.: Commun. Theor. Phys. 41, 837 (2004)

    MATH  MathSciNet  Google Scholar 

  40. Duval, C., Horvathy, P.: Phys. Lett. B 479, 284 (2000)

    MATH  MathSciNet  ADS  Google Scholar 

  41. Duval, C., Horvathy, P.: J. Phys. A 34, 10097 (2001)

    MATH  MathSciNet  ADS  Google Scholar 

  42. Duval, C., Horvathy, P.A.: Phys. Lett. B 457, 306 (2002)

    MathSciNet  ADS  Google Scholar 

  43. Feher, L.: Ph.D. Thesis (1987). Unpublished

  44. Gamboa, J., Loewe, M., Mendez, F., Rojas, J.C.: Phys. Rev. D 64, 06701 (2001)

    Google Scholar 

  45. Gutt, S.: Lett. Math. Phys. 7, 249 (1983)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  46. Horvarthy, P.: Ann. Phys. 299, 128 (2002)

    Article  ADS  Google Scholar 

  47. Horvathy, P.A., Plyuschchay, M.: Phys. Lett. B 595, 547 (2004)

    MathSciNet  ADS  Google Scholar 

  48. Horvathy, P.A., Plyushchay, M.S.: J. High Energy Phys. 06, 033 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  49. Horvathy, P.A., Martina, L., Stichel, P.C.: Exotic Galilean symmetry and non-commutative mechanics. arXiv:1002.4772

  50. Jackiw, R., Nair, V.P.: Phys. Rev. D 43, 1991 (1933)

    MathSciNet  Google Scholar 

  51. Jackiw, R., Nair, V.P.: Phys. Lett. B 480, 237 (2000)

    MATH  MathSciNet  ADS  Google Scholar 

  52. Kontsevich, M.: Lett. Math. Phys. 66, 157 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  53. Low, S.: J. Math. Phys. 38, 2197 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  54. Low, S.: J. Phys A, Math. Gen. 35, 5711 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  55. Milgrom, M.: New physics at low accelerations (MOND): an alternative to dark matter. In: Alimi, J.M. et al. (eds.) Proceedings of The Invisible Universe International Conference, Paris (June 2009, to appear). arXiv:0912.2678

  56. Moiseyev, N.: Phys. Rep. 302, 211 (1998)

    Article  ADS  Google Scholar 

  57. Nair, V.P., Polychronakos, A.P.: Phys. Lett. B 505, 267 (2001)

    MATH  MathSciNet  ADS  Google Scholar 

  58. Nottale, L.: Int. J. Mod. Phys. A 4, 5047 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  59. Nottale, L.: Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity. World Scientific, Singapore (1993)

    MATH  Google Scholar 

  60. Nottale, L.: Chaos Solitons Fractals 4(3), 361 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  61. Nottale, L.: Chaos Solitons Fractals 7, 877 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  62. Nottale, L.: Astron. Astrophys. 377, 867 (1997)

    ADS  Google Scholar 

  63. Nottale, L.: J. Phys. A, Math. Theor. 42, 275306 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  64. Nottale, L., Celerier, M.N., Lehner, T.: J. Math. Phys. 47, 032303 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  65. Ord, G.N.: J. Phys. A, Math. Gen. 16, 1869 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  66. Petrosky, T., Prigogine, I.: Chaos Solitons Fractals 4(3), 311 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  67. Pissondes, J.: Chaos Solitons Fractals 10(2–3), 513 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  68. Plyushchay, M.S.: Phys. Lett. B 248, 107 (1990)

    MathSciNet  ADS  Google Scholar 

  69. Puszkarz, W.: On the Staruszkiewicz modification of the Schrödinger equation. arXiv:quant-ph/9912006

  70. Reuter, M.: Phys. Rev. D 57, 971 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  71. Reuter, M., Saueressig, F.: arXiv:0708.1317

  72. Santamato, E.: Phys. Rev D 29, 216 (1984)

    MathSciNet  ADS  Google Scholar 

  73. Santamato, E.: J. Math. Phys. 25, 2477 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  74. Santamato, E.: Phys. Rev. D 32, 2615 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  75. Skagerstam, B.-S., Stern, A.: Int. J. Mod. Phys. A 5, 1575 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  76. Sochichiu, C.: Appl. Sci. 3, 48 (2001)

    MATH  MathSciNet  Google Scholar 

  77. Staruszkiewicz, A.: Acta Phys. Pol. 14, 907 (1983)

    MathSciNet  Google Scholar 

  78. Szabo, R.J.: Phys. Rep. 378, 207 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  79. Valentini, A.: Beyond the quantum. arXiv:1001.2758

  80. Weinberg, S.: In: Hawking, S.W., Israel, W. (eds.) General Relativity, an Einstein Centenary Survey. Cambridge University Press, Cambridge (1979)

    Google Scholar 

  81. Weinberg, S.: Ann. Phys. 194, 336 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  82. Wetterich, C.: Warping with dilation symmetry and self tuning of the cosmological constant. arXiv:1003.3809

  83. Yang, C.N.: Phys. Rev. 72, 874 (1947)

    Article  ADS  Google Scholar 

  84. Yang, C.N.: In: Proceedings of the International Conference on Elementary Particles, Kyoto, pp. 322–323 (1965)

  85. Yaris, R., Winkler, P.: J. Phys. B, At. Mol. Phys. 11, 1481 (1978)

    ADS  Google Scholar 

  86. Yaris, R., Winkler, P.: J. Phys. B, At. Mol. Phys. 11, 1475 (1978)

    ADS  Google Scholar 

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  1. Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, GA, 30314, USA

    Carlos Castro

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Castro, C. On Nonlinear Quantum Mechanics, Noncommutative Phase Spaces, Fractal-Scale Calculus and Vacuum Energy. Found Phys 40, 1712–1730 (2010). https://doi.org/10.1007/s10701-010-9476-5

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