# A Time–Space Symmetry Based Cylindrical Model for Quantum Mechanical Interpretations

## Abstract

Following a bi-cylindrical model of geometrical dynamics, our study shows that a 6D-gravitational equation leads to geodesic description in an extended symmetrical time–space, which fits Hubble-like expansion on a microscopic scale. As a duality, the geodesic solution is mathematically equivalent to the basic Klein–Gordon–Fock equations of free massive elementary particles, in particular, the squared Dirac equations of leptons. The quantum indeterminism is proved to have originated from space–time curvatures. Interpretation of some important issues of quantum mechanical reality is carried out in comparison with the 5D space–time–matter theory. A solution of lepton mass hierarchy is proposed by extending to higher dimensional curvatures of time-like hyper-spherical surfaces than one of the cylindrical dynamical geometry. In a result, the reasonable charged lepton mass ratios have been calculated, which would be tested experimentally.

## Keywords

Time–space symmetry Higher-dimensional general relativity Wave-like solution Klein–Gordon–Fock equation Heisenberg inequality Lepton mass hierarchy## Notes

### Acknowledgements

The author is grateful to Dr. Tran Chi Thanh (Vietnam Atomic Energy Institute) for VINATOM’s support. We thank Nguyen Anh Ky and Nguyen Ai Viet (Hanoi Institute of Physics), Do Quoc Tuan (Physics Faculty, Hanoi University of Natural Sciences) for useful discussions. A heart-felt thanks is also extended to N. B. Nguyen (Thang Long University) for valuable technical assistance, T. Thu and M. L. Le-Vo for their continuous support and encouragement.

## References

- 1.Kaluza, T.: Zum unitatsproblem der physik. Sitz. Preuss. Akad. Wiss.
**33**, 966–972 (1921)zbMATHGoogle Scholar - 2.Klein, O.: Quantentheorie und fnfdimensionale relativittstheorie. Z. Phys.
**37**, 895–906 (1926)ADSCrossRefGoogle Scholar - 3.de Broglie, L.: La mecanique ondulatoire et la structure atomique de la matiere et du rayonnement. J. Phys. Radium
**8**, 225–241 (1927)CrossRefzbMATHGoogle Scholar - 4.Bohm, D.: A suggested interpretation of the quantum theory in terms of “Hidden” variables. Phys. Rev.
**85**, 166–179 (1952)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 5.Bell, J.S.: On the Einstein-Podolsky-Rosen paradox. Physics
**1**, 195–200 (1964)Google Scholar - 6.Freedman, S.J., Clauser, J.F.: Experimental test of local hidden-variable theories. Phys. Rev. Lett.
**28**, 938–941 (1972)ADSCrossRefGoogle Scholar - 7.Maldacena, J.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys.
**2**, 231–252 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 8.Randall, L., Sundrum, R.: An alternative to compactification. Phys. Rev. Lett.
**83**, 4690–4693 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 9.Wesson, P.S., Ponce de Leon, J.: Kaluza-Klein equations, Einsteins equations, and an effective energy-momentum tensor. J. Math. Phys.
**33**, 3883–3887 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 10.Mashhoon, B., Liu, H., Wesson, P.S.: Particle masses and the cosmological constant in Kaluza-Klein theory. Phys. Lett. B
**331**, 305–312 (1994). Erratum to “Particle masses and the cosmological constant in Kaluza-Klein theory”. Phys. Lett. B 338, 519 (1994)Google Scholar - 11.Ponce de Leon, J.: Equivalence between space-time-matter and brane-world theories. Mod. Phys. Lett. A
**16**, 2291–2303 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 12.Wesson, P.S.: Vacuum waves. Phys. Lett. B
**722**, 1–4 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 13.Wesson, P.S.: The status of modern five-dimensional gravity. Int. J. Mod. Phys. D
**24**, 1530001–1 (2015)ADSCrossRefzbMATHGoogle Scholar - 14.Vo Van, T.: Revealing extradimensions. Int. J. Mod. Phys. A
**24**, 3545–3551 (2009)CrossRefzbMATHGoogle Scholar - 15.Fock, V.: On the invariant form of wave and motion equations for the charged point mass. Z. Phys.
**39**, 226–232 (1926)ADSCrossRefGoogle Scholar - 16.Vo Van, T.: From microscopic gravitational waves to quantum indeterminism. Comm. Phys.
**25**, 247–255 (2015). From microscopic gravitational waves to quantum indeterminism. arXiv:1507.00251 [gr-qc] - 17.Vo Van, T.: Klein-Gordon-Fock equation from general relativity in a time-space symmetrical model. Comm. Phys.
**26**, 181–192 (2016). Derivation of Klein-Gordon-Fock equation from general relativity in a time-space symmetrical model. arXiv:1604.05164v2 [physics.gen-ph] - 18.Vo Van, T.: Time-space symmetry as a solution to the mass hierarchy of charged lepton generations. arXiv:1510.04126 [physics.gen-ph]
- 19.Koide, Y.: A fermion-boson composite model of quarks and leptons. Phys. Lett. B
**120**, 161–165 (1983)ADSCrossRefGoogle Scholar - 20.Hollik, W.G., Salazar, U.J.S.: The double mass hierarchy pattern: simultaneously understanding quark and lepton mixing. Nucl. Phys. B
**892**, 364–389 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 21.Wesson, P.S.: General relativity and quantum mechanics in five dimensions. Phys. Lett. B
**701**, 379–383 (2011)ADSMathSciNetCrossRefGoogle Scholar - 22.Schrodinger, E.: On the free movement in relativistic quantum mechanics. Sitz. Preuss. Akad. Wiss. Phys. Math. Kl.
**24**, 418–428 (1930)zbMATHGoogle Scholar - 23.Beringer, J., et al.: (Particle data group): review of particle physics. Phys. Rev. D
**86**, 010001 (2012)ADSCrossRefGoogle Scholar - 24.Kocik, J.: The Koide lepton mass formula and geometry of circles. arXiv:1201.2067v1 [physics.gen-ph]
- 25.Linde, A.D.: Chaotic inflation. Phys. Lett. B
**129**, 177–181 (1983)ADSCrossRefGoogle Scholar - 26.Seahra, S.S., Wesson, P.S.: Application of the Campbell-Magaard theorem to higher-dimensional physics. Class. Quant. Grav.
**20**, 1321–1339 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 27.Mashhoon, B., Wesson, P.: An embedding for general relativity and its implications for new physics. Gen. Relativ. Gravit.
**39**, 1403–1412 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 28.Overduin, J.M., Wesson, P.S.: Decaying dark energy in higher-dimensional gravity. Astron. Astrophys.
**473**, 727–731 (2007)ADSCrossRefzbMATHGoogle Scholar - 29.Wesson, P.S.: The cosmological constant and quantization in five dimensions. Phys. Lett. B
**706**, 1–5 (2011)ADSMathSciNetCrossRefGoogle Scholar