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Quaternion Algebra on 4D Superfluid Quantum Space-Time: Gravitomagnetism

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Abstract

Gravitomagnetic equations result from applying quaternionic differential operators to the energy–momentum tensor. These equations are similar to the Maxwell’s EM equations. Both sets of the equations are isomorphic after changing orientation of either the gravitomagnetic orbital force or the magnetic induction. The gravitomagnetic equations turn out to be parent equations generating the following set of equations: (a) the vorticity equation giving solutions of vortices with nonzero vortex cores and with infinite lifetime; (b) the Hamilton–Jacobi equation loaded by the quantum potential. This equation in pair with the continuity equation leads to getting the Schrödinger equation describing a state of the superfluid quantum medium (a modern version of the old ether); (c) gravitomagnetic wave equations loaded by forces acting on the outer space. These waves obey to the Planck’s law of radiation.

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Notes

  1. In a Clifford bundle setting \(\mathcal D\) would identify as a Dirac–Kahler operator [83].

  2. Here we distinguish two components of the gravitomagnetic field—gravitoelectric and gyromagnetic fields. It is due to that masses attract each other like opposite electric charges, while gyroscopes create the vorticity like magnets creating the magnetic induction.

  3. It should be noted that the Navier–Stokes equation is the case of a trace–torsion geometry given by the velocity \({\varvec{v}}\) [72, 73, 75].

  4. Observe that \(\varPhi \), having dimension [length\(^2\)/time], is equal to S / m, where S is the action function and m is mass, and \({\mathbf {A}}\) is proportional to the angular momentum divided by mass.

  5. According to Shipov [102], a simplest geometry with torsion, built on a variety of orientable points (points with spin) is the geometry of absolute parallelism.

  6. The quantum potential is a keystone of the quantum mechanics [14, 15, 50]. Its computations [25, 33, 60, 74, 77, 78, 93] under a variety of physical circumstances shed light on the nature of the ether underlying the quantum realm.

  7. More detail consideration of the random generalized Brownian motion has been performed by Rapoport in [75, 76].

  8. In a general case we should define the viscous stress tensor [24, 84, 85, 93]. At the transition to the nonviscous superfluid we come to the noise tensor [75] fluctuating about zero.

References

  1. Ade, P.A.R.: 257 co-authors of Planck collaboration: Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys. 594(A13), 1–63 (2016). https://doi.org/10.1051/0004-6361/201525830

    Article  Google Scholar 

  2. Agamalyan, M.M., Drabkin, G.M., Sbitnev, V.I.: Spatial spin resonance of polarized neutrons. A tunable slow neutron filter. Phys. Rep. 168(5), 265–303 (1988). https://doi.org/10.1016/0370-1573(88)90081-6

    Article  ADS  Google Scholar 

  3. Albareti, F.D., Cembranos, J.A.R., Maroto, A.L.: The large-scale structure of vacuum. Int. J. Mod. Phys. D 23(12), 1442,019 (7 pp) (2014). https://doi.org/10.1142/S021827181442019X

    Article  MATH  Google Scholar 

  4. Albareti, F.D., Cembranos, J.A.R., Maroto, A.L.: Vacuum energy as dark matter. Phys. Rev. D 90, 123509 (2014). https://doi.org/10.1103/PhysRevD.90.123509

    Article  ADS  Google Scholar 

  5. Anderson, J.D., Laing, P.A., Lau, E.L., Liu, A.S., Nieto, M.M., Turyshev, S.G.: Indication, from Pioneer 10/11, Galileo, and Ulysses data, of an apparent anomalous, weak long-range acceleration. Phys. Rev. Lett. 81, 2858–2861 (1998). https://doi.org/10.1103/PhysRevLett.81.2858

    Article  ADS  Google Scholar 

  6. Atiyah, M., Dunajski, M., Mason, L.J.: Twistor theory at fifty: from contour integrals to twistor strings. Proc. R. Soc. A 473, 20170530 (2017). https://doi.org/10.1098/rspa.2017.0530

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Bars, I., Terning, J.: Extra Dimensions in Space and Time. Springer, New York (2010). https://doi.org/10.1007/978-0-387-77638-5

    Book  MATH  Google Scholar 

  8. Baylis, W.B. (ed.): Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering. Burkhäuser, Boston (1996)

    MATH  Google Scholar 

  9. Behera, H.: Comments on gravitoelectromagnetism of Ummarino and Gallerati in “Superconductor in a weak static gravitational field” vs other versions. Eur. Phys. J. C 77, 822 (2017). https://doi.org/10.1140/epjc/s10052-017-5386-4

    Article  ADS  Google Scholar 

  10. Behera, H., Naik, P.C.: Gravitomagnetic moments and dynamics of dirac’s (spin 1/2) fermions in flat space-time maxwellian gravity. Int. J. Mod. Phys. A 19, 4207–4230 (2004). https://doi.org/10.1142/S0217751X04017768

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Berezhiani, L., Khoury, J.: Theory of dark matter superfluidity. Phys. Rev. D 92, 103,510 (2015). https://doi.org/10.1103/PhysRevD.92.103510. arXiv:1507.01019 (hep-th)

    Article  Google Scholar 

  12. Berry, M.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. A 392, 45–57 (1984). https://doi.org/10.1098/rspa.1984.0023

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Berry, M., Marzoli, I., Schleich, W.: Quantum carpets, carpets of light. Phys. World 14(6), 39–44 (2001). https://doi.org/10.1088/2058-7058/14/6/30

    Article  Google Scholar 

  14. Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables I. Phys. Rev. 85(2), 166–179 (1952). https://doi.org/10.1103/PhysRev.85.166

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Bohm, D.: A suggested interpretation of the quantun theory in terms of “hidden” variables II. Phys. Rev. 85(2), 180–193 (1952). https://doi.org/10.1103/PhysRev.85.180

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Bohm, D.: Wholeness and the Implicate Order. Taylor and Francis Group, London (2006)

    Google Scholar 

  17. Cashen, B., Aker, A., Kesden, M.: Gravitomagnetic dynamical friction. Phys. Rev. D 95, 064,014 (2017). https://doi.org/10.1103/PhysRevD.95.064014

    Article  MathSciNet  Google Scholar 

  18. Ciufolini, I., Wheeler, J.A.: Gravitation and Inertia. Princeton University Press, Princeton (1995). Page 8, Fig. 1.2. The directionalities of gravitomagnetism, magnetism, and fluid drag compared and contrasted

  19. Clark, S.J., Tucker, R.W.: Gauge symmetry and gravito-electromagnetism. Class. Quantum Gravity 17(19), 4125–4157 (2000). https://doi.org/10.1088/0264-9381/17/19/311

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Das, S., Bhaduri, R.K.: Dark matter and dark energy from Bose–Einstein condensate. Class. Quant. Grav. 32, 105,003 (6 pp) (2015). https://doi.org/10.1088/0264-9381/32/10/105003

    Article  MATH  Google Scholar 

  21. Davies, P.: The Goldilocks Enigma. A Mariner Book. Houghton Mifflin Co., Boston (2006)

    Google Scholar 

  22. de Blok, W.J.G., McGaugh, S.S., Rubin, V.C.: High-resolution rotation curves of low surface brightness galaxies II. Mass models. Astron. J. 122, 2396–2427 (2001). https://doi.org/10.1086/323450

    Article  ADS  Google Scholar 

  23. De Matos, C.J., Tajmar, M.: Gravitomagnetic fields in rotating superconductors to solve Tate’s Cooper pair mass anomaly. In: El-Genk, M.S. (ed.) AIP Conference Proceedings, vol. 813, p. 1415 (2006). https://doi.org/10.1063/1.2169327

  24. Disconzi, M.M., Kephart, T.W., Scherrer, R.J.: On a viable first order formulation of relativistic viscous fluids and its applications to cosmology (2017). arXiv:1510.07187

  25. Fiscaletti, D.: The geometrodynamic nature of the quantum potential. Ukr. J. Phys. 57(5), 560–573 (2012)

    Google Scholar 

  26. Freedman, W.L., Madore, B.F., Gibson, B.K., Ferrarese, L., Kelson, D.D., Sakai, S., Mould, J.R., Kennicutt Jr., R.C., Ford, H.C., Graham, J.A., Huchra, J.P., Hughes, S.M.G., Illingworth, G.D., Macri, L.M., Stetson, P.B.: Final results from the hubble space telescope key project to measure the hubble constant. Astrophys. J. 553, 47–72 (2001). https://doi.org/10.1086/320638

    Article  ADS  Google Scholar 

  27. Friedman, A.: Über die Krümming des Raumes. Z. Phys. 10(1), 377–386 (1922)

    Article  ADS  Google Scholar 

  28. Girard, P.R.: The quaternion group and modern physics. Eur. J. Phys. 5, 25–32 (1984). http://iopscience.iop.org/0143-0807/5/1/007

  29. Girard, P.R.: Quaternions, Clifford Algebras and Relativistic Physics. Birkhäuser Verlag AG, Basel (2007)

    MATH  Google Scholar 

  30. Goenner, H.F.M.: On the history of unified field theories. Living Rev. Relativ. 7(2), 1–153 (2004). http://www.livingreviews.org/lrr-2004-2

  31. Goenner, H.F.M.: On the history of unified field theories. Part II. (ca. 1930-ca. 1965). Living Rev. Relativ. 17(1), 5 (2014). https://doi.org/10.12942/lrr-2014-5

    Article  ADS  MATH  Google Scholar 

  32. Griffin, A., Nikuni, T., Zaremba, E.: Ch. 14: Landau’s theory of superfluidity. In: Bose-Condensed Gases at Finite Temperatures (pp. 309–321). Cambridge University Press, Cambridge (2009). https://doi.org/10.1017/CBO9780511575150

  33. Grössing, G.: On the thermodynamic origin of the quantum potential. Phys. A 388, 811–823 (2009)

    Article  MathSciNet  Google Scholar 

  34. Hamermesh, M.: Group Theory and Its Application to Physical Problems. Dover Publ., Inc., New York (1962)

    Book  MATH  Google Scholar 

  35. Heaviside, O.: A gravitational and electromagnetic analogy. Electrician 31, 281–282 (1893)

    Google Scholar 

  36. Hehl, F.W., Obukhov, Y.N.: Élie Cartan’s torsion in geometry and in field theory, an essay. Annales de la Fondation Louis de Broglie 32(2–3), 157–194 (2007)

    MATH  Google Scholar 

  37. Hestenes, D.: Space-Time Algebra. Springer, Basel (2015)

    Book  MATH  Google Scholar 

  38. Huang, K.: Dark energy and dark matter in a superfluid universe. Int. J. Mod. Phys. A 28, 1330,049 (2013). https://doi.org/10.1142/S021775X13300494. arXiv:1309.5707

    Article  MathSciNet  Google Scholar 

  39. Huang, K., Xiong, C., Zhao, X.: Scalar-field theory of dark matter. Int. J. Mod. Phys. A 29, 1450,074 (2014). https://doi.org/10.1142/S0217751X14500742

    Article  Google Scholar 

  40. Ioffe, A.I., Kirsanov, S.G., Sbitnev, V.I., Zabiyakin, V.S.: Geometric phase in neutron spin resonance. Phys. Lett. A 158(9), 433–435 (1991). https://doi.org/10.1016/0375-9601(91)90453-F

    Article  ADS  Google Scholar 

  41. Jackiw, R., Nair, V.P., Pi, S.Y., Polychronakos, A.P.: Perfect fluid theory and its extensions. J. Phys. A 37, R327–R432 (2004). https://doi.org/10.1088/0305-4470/37/42/R01

    Article  MathSciNet  MATH  Google Scholar 

  42. Jenke, T., Geltenbort, P., Lemmel, H., Abele, H.: Realization of a gravity-resonance-spectroscopy technique. Nat. Phys. 7, 468–472 (2011). https://www.nature.com/articles/nphys1970

  43. Jenke, T., Cronenberg, G., Burgdörfer, J., Chizhova, L.A., Geltenbort, P., Ivanov, A.N., Lauer, T., Lins, T., Rotter, S., Saul, H., Schmidt, U., Abele, H.: Gravity resonance spectroscopy constrains dark energy and dark matter scenarios. Phys. Rev. Lett. 112, 151105 (2014). https://doi.org/10.1103/PhysRevLett.112.151105

    Article  ADS  Google Scholar 

  44. Kambe, T.: A new formulation of equations of compressible fluids by analogy with Maxwell’s equations. Fluid Dyn. Res. 42, 055,502 (2010). https://doi.org/10.1088/0169-5983/42/5/055502

    Article  MathSciNet  MATH  Google Scholar 

  45. Kleineberg, K.K., Friedrich, R.: Gaussian vortex approximation to the instanton equations of two-dimensional turbulence. Phys. Rev. E 87, 033,007 (2013). https://doi.org/10.1103/PhysRevE.87.033007

    Article  Google Scholar 

  46. Kundu, P., Cohen, I.: Fluid Mechanics, p. 92101-4495. Academic Press, San Diego (2002)

    Google Scholar 

  47. Li, T.P., Wu, M.: Thermal gravitational radiation and condensed universe (2016). arXiv:1612.04199

  48. Lighthill, M.J.: An Informal Introduction to Theoretical Fluid Mechanics. Oxford University Press, Oxford (1986)

    MATH  Google Scholar 

  49. Lounesto, P.: Clifford Algebras and Spinors. London Mathematical Society Lecture Note. Series 286. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  50. Madelung, E.: Quantumtheorie in hydrodynamische form. Zts. f. Phys. 40, 322–326 (1926)

    Article  ADS  MATH  Google Scholar 

  51. Marcer, P., Rowlands, P.: Nilpotent quantum mechanics: analogs and applications. Front. Phys. 5, 28 (2017). https://doi.org/10.3389/fphy.2017.00028

    Article  Google Scholar 

  52. Marmanis, H.: Analogy between the Navier–Stokes equations and Maxwell’s equations: application to turbulence. Phys. Fluids 10, 1428 (1998). https://doi.org/10.1063/1.869762

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. Martins, A.A.: Fluidic electrodynamics: on parallels between electromagnetic and fluidic inertia (2012). arXiv:1202.4611

  54. Martins, A.A., Pinheiro, M.J.: Inertia, electromagnetism and fluid dynamics. In: El-Genk, M.S. (ed.) AIP Conference Proceedings-STAIF 2008, vol. 969, pp. 1154–1162 (2008). https://doi.org/10.1063/1.2844955

  55. Martins, A.A., Pinheiro, M.J.: Fluidic electrodynamics: approach to electromagnetic propulsion. Phys. Fluids 21, 097,103 (2009). https://doi.org/10.1063/1.3236802

    Article  MATH  Google Scholar 

  56. Mashhoon, B., Gronwald, F., Lichtenegger, H.I.M.: Gravitomagnetism and the clock effect. Lect. Notes Phys. 562, 83–108 (2001)

    Article  ADS  MATH  Google Scholar 

  57. Mezei, F.: Fundamentals of neutron spin echo spectroscopy. In: Mezei, F., Pappas, A.G.T. (eds.) Neutron Spin Echo Spectroscopy. Lecture Notes in Physics, vol. 601. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  58. Mielczarek, J., Stachowiak, T., Szydlowski, M.: Vortex in axion condensate as a dark matter halo. Int. J. Mod. Phys. D 19, 1843–1855 (2010). https://doi.org/10.1142/S0218271810018037

    Article  ADS  MATH  Google Scholar 

  59. Negretti, M.E., Billant, P.: Stability of a Gaussian pancake vortex in a stratified fluid. J. Fluid Mech. 718, 457–480 (2013). https://doi.org/10.1017/jfm.2012.624

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150, 1079–1085 (1966). https://doi.org/10.1103/PhysRev.150.1079

    Article  ADS  Google Scholar 

  61. Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)

    MATH  Google Scholar 

  62. Nelson, E.: Review of stochastic mechanics. J. Phys. 361, 012,011 (2012). https://doi.org/10.1088/1742-6596/361/1/012011

    Article  Google Scholar 

  63. Nesvizhevsky, V.V., Börner, H.G., Petukhov, A.K., Abele, H., Baessler, S., Ruess, F.J., Stöferle, T., Westphal, A., Gagarski, A.M., Petrov, G.A., Strelkov, A.V.: Quantum states of neutrons in the Earth’s gravitational field. Nature 415, 297–299 (2002). https://doi.org/10.1038/415297a

    Article  ADS  Google Scholar 

  64. Nozaria, K., Anvaria, S.F., Sefiedgara, A.S.: Black body radiation in a model universe with large extra dimensions and quantum gravity effects (2018). arXiv:1206.5631v1

  65. Penrose, R.: Twistor algebra. J. Math. Phys. 8, 345 (1967). https://doi.org/10.1063/1.1705200

    Article  ADS  MathSciNet  MATH  Google Scholar 

  66. Penrose, R.: Twistor quantization and curved space-time. Int. J. Theor. Phys. 1, 61–99 (1968). https://doi.org/10.1007/BF00668831

    Article  Google Scholar 

  67. Penrose, R.: Spinors and torsion in general relativity. Found. Phys. 13(3), 325–339 (1983). https://doi.org/10.1007/BF01906181

    Article  ADS  MathSciNet  Google Scholar 

  68. Penrose, R., Rindler, W.: Spinors and Space-Time Volume 1: Two-Spinor Calculus and Relativistic Fields. Cambridge University Press, Cambridge (1984)

    Book  MATH  Google Scholar 

  69. Penrose, R., Rindler, W.: Spinors and Space-Time. Volume 2: Spinor and Twistor Methods in Space-Time Geometry. Cambridge University Press, Cambridge (1986)

    Book  MATH  Google Scholar 

  70. Provenzale, A., Babiano, A., Bracco, A., Pasquero, C., Weiss, J.B.: Coherent vortices and tracer transport. Lect. Notes Phys. 744, 101–116 (2008). https://doi.org/10.1007/978-3-540-75215-85

    Article  ADS  MATH  Google Scholar 

  71. Rapoport, D.L.: Riemann–Cartan–Weyl quantum geometry. II Cartan stochastic copying method, Fokker–Planck operator and Maxwell–de Rham equations. Int. J. Theor. Phys. 36(10), 2115–2152 (1997). https://doi.org/10.1007/BF02435948

    Article  MathSciNet  MATH  Google Scholar 

  72. Rapoport, D.L.: Stochastic differential geometry and the random flows of viscous and magnetized fluids in smooth manifolds and Euclidean Space (2000). arXiv:math-ph/0012032

  73. Rapoport, D.L.: Martingale problem approach to the representations of the Navier–Stokes equations on smooth-boundary manifolds and semispace. Random Oper. Stoch. Equ. 11(2), 109–136 (2003). https://doi.org/10.1515/156939703322386887

    Article  MathSciNet  MATH  Google Scholar 

  74. Rapoport, D.L.: Cartan–Weyl dirac and laplacian operators, Brownian motions: the quantum potential and scalar curvature, Maxwell’s and Dirac–Hestenes equations, and supersymmetric systems. Found. Phys. 35(8), 1383–1431 (2005). https://doi.org/10.1007/s10701-005-6443-7

    Article  ADS  MathSciNet  MATH  Google Scholar 

  75. Rapoport, D.L.: On the unification of geometric and random structures through torsion fields: Brownian motions, viscous and magneto-fluid-dynamics. Found. Phys. 35(7), 1205–1244 (2005). https://doi.org/10.1007/s10701-005-6407-y

    Article  ADS  MathSciNet  MATH  Google Scholar 

  76. Rapoport, D.L.: Torsion fields, Cartan–Weyl space-time and state-space quantum geometries, their Brownian motions, and the time variables. Found. Phys. 37(4), 813–854 (2007). https://doi.org/10.1007/s10701-007-9126-8

    Article  ADS  MathSciNet  MATH  Google Scholar 

  77. Rapoport, D.L.: Torsion fields, the extended photon, quantum jumps, the eikonal equation, the twistor geometry of cognitive space and the laws of thought. In: Levy, J., Dutty, M.C. (eds.) Ether Space-Time and Cosmology: Physical Vacuum, Relativity and Quantum Physics, vol. 3, pp. 389–457. Apeiron, Montreal (2009)

    Google Scholar 

  78. Rapoport, D.L.: Torsion fields, propagating singularities, nilpotence, quantum jumps and the eikonal equations (2010). https://doi.org/10.1063/1.3527144. https://www.researchgate.net/publication/234943262

  79. Rapoport, D.L.: Klein bottle logophysics, self-reference, heterarchies, genomic topologies, harmonics and evolution. Part I: morphomechanics, space and time in biology & physics, cognition, non-linearity and the structure of uncertainty. Quantum Biosyst. 7(1), 1–73 (2016)

    MathSciNet  Google Scholar 

  80. Rapoport, D.L.: Klein bottle logophysics, self-reference, heterarchies, genomic topologies, harmonics and evolution. Part II: non-orientability,cognition, chemical topology and eversions in nature. Quantum Biosyst. 7(1), 74–106 (2016)

    MathSciNet  Google Scholar 

  81. Rapoport, D.L., Sternberg, S.: Classical mechanics without Lagrangians and Hamiltonians. Nuovo Cimento A 80(3), 371–383 (1984). https://doi.org/10.1007/BF02785808

    Article  ADS  MathSciNet  Google Scholar 

  82. Rapoport, D.L., Sternberg, S.: On the interaction of spin and torsion. Ann. Phys. 158(2), 447–475 (1984). https://doi.org/10.1016/0003-4916(84)90128-3

    Article  ADS  MathSciNet  MATH  Google Scholar 

  83. Rodrigues, W.A., Capelas de Oliveira, E.: The many faces of Maxwell, Dirac and Einstein equations. A Clifford Bundle Approach. LNP 722. Springer, Berlin (2007). https://doi.org/10.1007/978-3-540-71293-0

  84. Romatschke, P.: New developments in relativistic viscous hydrodynamics. Int. J. Mod. Phys. E 19, 1–53 (2010). https://doi.org/10.1142/S0218301310014613. arXiv:0902.3663

    Article  ADS  Google Scholar 

  85. Romatschke, P.: Relativistic viscous fluid dynamics and non-equilibrium entropy. Class Quant. Gravity 27, 025,006 (2010). https://doi.org/10.1088/0264-9381/27/2/025006. arXiv:0906.4787

    Article  MathSciNet  MATH  Google Scholar 

  86. Rubin, V.C.: A brief history of dark matter. In: Livio, M. (ed.) The Dark Universe: Matter, Energy and Gravity, Symposium Series: 15, pp. 1–13. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  87. Ruggiero, M.L.: Gravitomagnetic field of rotating rings. Astrophys. Space Sci. 361(4), 140 (2016). https://doi.org/10.1007/s10509-016-2723-2

    Article  ADS  MathSciNet  Google Scholar 

  88. Ryden, B.: Introduction to Cosmology. The Ohio State University, Columbus (2006)

    Google Scholar 

  89. Sarkar, S., Vaz, C., Wijewardhana, L.C.R.: Gravitationally bound Bose condensates with rotation. Phys. Rev. D 97, 103,022 (2018). https://doi.org/10.1103/PhysRevD.97.103022

    Article  Google Scholar 

  90. Sbitnev, V.I.: Passage of polarized neutrons through magnetic media. Depolarization by magnetized inhomogeneities. Z. Phys. B 74(3), 321–327 (1989). https://doi.org/10.1007/BF01307879

    Article  ADS  MathSciNet  Google Scholar 

  91. Sbitnev, V.I.: Particle with spin in a magnetic field—the Pauli equation and its splitting into two equations for real functions. Quantum Magic 5(2), 2112–2131 (2008). http://quantmagic.narod.ru/volumes/VOL522008/p2112.html (In Russian)

  92. Sbitnev, V.I.: Matter waves in the Talbot–Lau interferometry (2010). arXiv:1005.0890

  93. Sbitnev, V.I.: Hydrodynamics of the physical vacuum: dark matter is an illusion. Mod. Phys. Lett. A 30(35), 1550,184 (2015). https://doi.org/10.1142/S0217732315501849. arXiv:1507.03519

    Article  MathSciNet  MATH  Google Scholar 

  94. Sbitnev, V.I.: Physical vacuum is a special superfluid medium. In: Pahlavani, M.R. (ed.) Selected Topics in Applications of Quantum Mechanics, Chap. 12, pp. 345–373. InTech, Rijeka (2015). https://doi.org/10.5772/59040

  95. Sbitnev, V.I.: Hydrodynamics of the physical vacuum: I. Scalar quantum sector. Found. Phys. 46(5), 606–619 (2016). https://doi.org/10.1007/s10701-015-9980-8

    Article  ADS  MathSciNet  MATH  Google Scholar 

  96. Sbitnev, V.I.: Hydrodynamics of the physical vacuum: II. Vorticity dynamics. Found. Phys. 46(10), 1238–1252 (2016). https://doi.org/10.1007/s10701-015-9985-3. http://rdcu.be/kdon; arXiv:1603.03069

  97. Sbitnev, V.I.: Hydrodynamics of superfluid quantum space: de Broglie interpretation of the quantum mechanics. Quantum Stud. 5(2), 257–271 (2018). https://doi.org/10.1007/s40509-017-0116-z. http://rdcu.be/un41

  98. Sbitnev, V.I.: Hydrodynamics of superfluid quantum space: particle of spin-1/2 in a magnetic field. Quantum Stud. 5(2), 297–314 (2018). https://doi.org/10.1007/s40509-017-0119-9. http://rdcu.be/uRfJ

  99. Sbitnev, V.I., Fedi, M.: Superfluid quantum space and evolution of the universe. In: Capistrano de Souza, A.J. (ed.) Trends in Modern Cosmology, Chap. 5, pp. 89–112. InTech, Rijeka (2017). https://doi.org/10.5772/68113

  100. Scolarici, G., Solombrino, L.: Notes on quaternionic group representations. Int. J. Theor. Phys. 34(12), 2491–2500 (1995). https://doi.org/10.1007/BF00670780

    Article  MathSciNet  MATH  Google Scholar 

  101. Sekine, A.: Chiral gravitomagnetic effect in topological superconductors and superfluids. Phys. Rev. B 93, 094,510 (2016). https://doi.org/10.1103/PhysRevB.93.094510

    Article  Google Scholar 

  102. Shipov, G.I.: A Theory of Physical Vacuum: A New Paradigm, First English Edition edn. International Institute for Theoretical and Applied Physics, Moscow (1998)

    Google Scholar 

  103. Sternberg, S.: The interaction of spin and torsion. II. The principle of general covariance. Ann. Phys. 162(1), 85–99 (1985). 10.1016/0003-4916(85)90229-5

    Article  ADS  MathSciNet  MATH  Google Scholar 

  104. Talbot, M.: The Holographic Universe. Harper Collins Publ., New York (1991)

    Google Scholar 

  105. Tsatsos, M.C., Tavares, P.E.S., Cidrim, A., Fritsch, A.R., Caracanhas, M.A., dos Santos, F.E.A., Barenghi, C.F., Bagnato, V.S.: Quantum turbulence in trapped atomic Bose–Einstein condensates. Phys. Rep. 622, 1–52 (2016). https://doi.org/10.1016/j.physrep.2016.02.003

    Article  ADS  MathSciNet  Google Scholar 

  106. Ummarino, G.A., Gallerati, A.: Superconductor in a weak static gravitational field. Eur. Phys. J. C 77, 549 (2017). https://doi.org/10.1140/epjc/s10052-017-5116-y

    Article  ADS  Google Scholar 

  107. Vigier, J.P.: Alternative interpretation of the cosmological redshift in terms of vacuum gravitational drag. In: Bertola, F., Madore, B., Sulentic, J. (eds.) New Ideas in Astronomy, pp. 257–274. Cambridge University Press, Cambridge (1988)

    Google Scholar 

  108. Vinen, W.F.: An introduction to quantum turbulence. J. Low Temp. Phys. 145(1–4), 7–24 (2006). https://doi.org/10.1007/s10909-006-9240-6

    Article  ADS  Google Scholar 

  109. Vlahovic, B.: A new cosmological model for the visible universe and its implications (2011). arXiv:1005.4387

  110. Volovik, G.E.: Vacuum energy: quantum hydrodynamics vs. quantum gravity. JETP Lett. 82(6), 319–324 (2005). https://doi.org/10.1134/1.2137368

    Article  ADS  Google Scholar 

  111. Volovik, G.E.: The superfluid universe. In: Bennemann, K.H., Ketterson, J.B. (eds.) Novel Superfluids, vol. 1, chap. 11, pp. 570–618. Oxford University Press, Oxford (2013). arXiv:1004.0597

  112. Wikipedia: cosmic microwave background (2018)

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Acknowledgements

It is pleasant to note useful discussions with Marco Fedi concerning the Universe. The author thanks Mike Cavedon and Sridhar Bulusu for useful and valuable discussions, remarks, and offers.

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  1. St. Petersburg B. P. Konstantinov Nuclear Physics Institute, NRC Kurchatov Institute, Leningrad District, Gatchina, Russia, 188350

    Valeriy I. Sbitnev

  2. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, 94720, USA

    Valeriy I. Sbitnev

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Sbitnev, V.I. Quaternion Algebra on 4D Superfluid Quantum Space-Time: Gravitomagnetism. Found Phys 49, 107–143 (2019). https://doi.org/10.1007/s10701-019-00236-4

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