Fractal Approximation of Motion and Its Implications in Quantum Mechanics

Affiliation(s)

Physics Department, Faculty of Machine Manufacturing and Industrial Management,.

Physics Department, “Al. I. Cuza” University, Iasi, Romania.

Department of Natural and Synthetic Polymers, Faculty of Chemical Engineering and Environmental Protection, “Gheorghe.

Physics Department, Faculty of Machine Manufacturing and Industrial Management,.

Physics Department, “Al. I. Cuza” University, Iasi, Romania.

Department of Natural and Synthetic Polymers, Faculty of Chemical Engineering and Environmental Protection, “Gheorghe.

ABSTRACT

Inconsistencies of some standard quantum mechanical models (Madelung’s, de Broglie’s models) are eliminated as- suming the micro particle movements on continuous, but non-differentiable curves (fractal curves). This hypothesis, named by us the fractal approximation of motion, will allow an unitary approach of the phenomena in quantum me-chanics (separation of the physical motion of objects in wave and particle components depending on the scale of resolution, correlated motions of the wave and particle,* i.e*. wave-particle duality, the mechanisms of duality, by means of both phase wave-particle coherence and wave-particle incoherence, the particle as a clock, particle incorporation into the wave and the implications of such a process). Moreover, correspondences with standard gravitational models (Einstein’s model, string theory) can be also distinguished.

Inconsistencies of some standard quantum mechanical models (Madelung’s, de Broglie’s models) are eliminated as- suming the micro particle movements on continuous, but non-differentiable curves (fractal curves). This hypothesis, named by us the fractal approximation of motion, will allow an unitary approach of the phenomena in quantum me-chanics (separation of the physical motion of objects in wave and particle components depending on the scale of resolution, correlated motions of the wave and particle,

Cite this paper

M. Agop, D. Magop and E. Bacaita, "Fractal Approximation of Motion and Its Implications in Quantum Mechanics,"*Open Journal of Microphysics*, Vol. 2 No. 3, 2012, pp. 33-45. doi: 10.4236/ojm.2012.23005.

M. Agop, D. Magop and E. Bacaita, "Fractal Approximation of Motion and Its Implications in Quantum Mechanics,"

References

[1] L. Nottale and J. Schneider, “Fractals and Non-Standard Analysis,” Journal of Mathematical Physics, Vol. 25, No. 12, 1984, pp. 96-300.

[2] L. Nottale, “Fractals and the Quantum Theory of Space-Time,” International Journal of Modern Physics, Vol. A4, No. 50, 1989, pp. 47-117.

[3] L. Nottalle, “Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity,” World Scientific, Singapore, 1993.

[4] L. Nottale, “Scale Relativity and Fractal Space-Time: Applications to Quantum Physics, Cosmology and Chaotic Systems,” Chaos, Solitons & Fractals, Vol. 7, No. 6, 1996, pp. 877-938.

[5] L. Nottale, “The Theory of Scale Relativity: Non-Differentiable Geometry, Fractal Space-Time and Quantum Mechanics,” Computing Anticipatory Systems: CASYS’ 03-33 Sixth International Conference, Liège, 11-16 August 2003, pp. 68-95.

[6] J. Chaline, L. Nottale and P. Grou, “The Trees of Evolution: Universe S,” Life Companies Societies, Paris, 2000.

[7] J. Chaline, L. Nottale and P. Grou, “Flowers for Scrodinger Sscale Relativity and Its Applications,” Editure Ellipses Marketing, Paris, 2009.

[8] A. Harabagiu and M. Agop, “Hydrodyamic Model of Scale Relativity Theory,” Bulletin of the Polytechnic Institute of Iasi, Vol. LV, No. 3-4, 2005, pp. 77-82.

[9] A. Harabagiu, O. Niculescu, M. Colotin, T. D. Bibere, I. Gottlieb and M. Agop, “Particle in a Box by Means of Fractal Hydrodynamic Model,” Romanian Reports in Physics, Vol. 61, No. 3, 2009, pp. 395-400.

[10] A. Harabagiu, D. Magop and M. Agop, “Fractality and Quantum Mechanics,” Ars Longa Publishing House, Ia?i, 2010.

[11] M. Agop, L. Chico?, P. Nica and A. Harabagiu, “Euler’s Fluids and Non-Differentiable Space-Time,” Far East Journal of Dynamical Systems, Vol. 10, No. 1, 2008, pp. 93-106.

[12] M. Agop, A. Harabagiu and P. Nica, “Wave-Particle Duality through a Hydrodynamic Model of the Fractal Space Time Theory,” Acta Physica Polonica A, Vol. 113, No. 6, 2008, pp. 1557-1574.

[13] S. ?i?eica, “Quantum Mechanics,” Academiei Publishing House, Bucure?ti, 1984.

[14] B. Felsager, “Geometry, Particles and Fields,” Odense University Press, Odense, 1981.

[15] A. Peres, “Quantum Theory: Concepts and Methods,” Kluwer Academic Publishers, Boston, 1993.

[16] J. J. Sakurai and T. S. Fu, “Modern Quantum Mechanics,” Addison-Wesley, Boston, 1994.

[17] F. Halbwacs, “Theorie Relativiste Des Fluids a Spin,” Gauthier-Villars, Paris, 1960.

[18] R. Madelung, “A Hydrodynamic Interpretation of Quantum Mechanics,” Journal for Physics, Vol. 40, No. 3-4, 1927, pp. 322-326.

[19] D. Bohm and B. J. Hiley, “The Undivided Universe: An Ontological Interpretation of Quantum Theory,” Routledge and Kegan Paul, London, 1993.

[20] D. D?rr, S. Goldstein and N. Zanghi, “Quantum Mechanics, Randomness and Deterministic Reality,” Physics Letters A, Vol. 172, No. 1-2, 1992, pp. 6-12. doi:10.1016/0375-9601(92)90181-K

[21] D. D?rr, S. Goldstein and N. Zanchi, “A Global Equilibrium as the Foundation of Quantum Randomness,” Foundations of Physics, Vol. 23, 1993, pp. 712-738.

[22] D. Z. Albert, “Bohm’s Alternative to Quantum Mechanics”, Scientific American, Vol. 270, No. 2, 1994, pp. 32-39. doi:10.1038/scientificamerican0594-58

[23] J. S. Bell, “Speakable and Unspeakable in Quantum Mechanics,” Cambridge University Press, Cambridge, 1987.

[24] K. Berndl, D. D?rr, S. Goldstein, G. Peruzzi and N. Zanchi, “Existance of Trajectories for Bohmian Mechanics,” International Journal of Theoretical Physics, Vol. 32, No. 12, 1993, pp. 2245-2251. doi:10.1007/BF00672996

[25] K. Berndl, D. D?rr, S. Goldstein, N. Zanchi, “Selfa- djointness and the Existance of Deterministic Trajectories in Quantum Theory,” In: M. Fannes, C. Maes and A. Verbeure, Eds., On Three Levels: Micro-, Meso-, and Macroscopic Approches in Physics, (NATO ASI Series B: Physics), Plenum, New York, 1994, pp. 429-434.

[26] P. R. Holland, “The Quantum Theory of Motion,” Camb- ridge University Press, Cambridge, 1993. doi:10.1017/CBO9780511622687

[27] L. de Broglie, “Causal Interpretation Attempt and Non-Linear Wave Mechanics: The Theory of Double Solution,” Gauthier-Villars, Paris, 1956.

[28] L. de Broglie, “The Measure Theory in Wave Mechanics,” Gauthier-Villars, Paris, 1957.

[29] L. de Broglie, “Wave Mechanics Interpretation,” Journal of Physics Radium, Vol. 20, No. 12, 1959, pp. 963-979. doi:10.1051/jphysrad:019590020012096300

[30] L. de Broglie, “Critical Study of the Current Foundations Interpretation of Wave Mechanics,” Gauthier-Villars, Paris, 1963.

[31] L. de Broglie, “Thermodynamics of Isolated Particle (Hidden Thermodynamics of Particles),” Gauthier-Villars, Paris, 1964.

[32] L. de Broglie, “Certainity and Uncertainity in Science,” Politic? Publishing House, Bucure?ti, 1980.

[33] D. Bohm, “A Suggested Interpretation of Quantum Theory in Terms of ‘Hidden’ Variables I,” Physical Review, Vol. 85, No. 2, 1952, pp. 166-179. doi:10.1103/PhysRev.85.166

[34] D. Bohm, “A Suggested Interpretation of Quantum Theory in Terms of ‘Hidden Variables’: Part II,” Physical Review, Vol. 85, No. 2, 1952, pp. 180-193. doi:10.1103/PhysRev.85.180

[35] D. Bohm, “Proof that Probability Density Approaches in Causal Interpretation of Quantum Theory,” Physical Review, Vol. 89, No. 2, 1953, pp. 458-466. doi:10.1103/PhysRev.89.458

[36] L. Budei, “Fractal Models: Applications in Enviroment Arhitecture,” Publishing House University, Gh. Asachi, Ia?i, 2000.

[37] M. Barnsley, “Fractals Everywhere: The First Course in Deterministic Fractal Geometry,” Academic Press, Boston, 1988.

[38] A. Le Mehante, “Les Geometries Fractales,” Hermes, Paris, 1990.

[39] A. Heck and J. M. Perdang, “Applying Fractals in Astronomy,” Springer Verlag, Berlin, 1991.

[40] J. Feder and A. Aharony, “Fractals in Physics,” North-Holland, Amsterdam, 1990.

[41] P. Berge, Y. Pomeau and C. Vidal, “The Order in Chaos,” Wiley, New York, 1984.

[42] J. F. Gouyet, “Physical Structures and Fractals,” Masson, Paris, 1992.

[43] M. S. El Naschie, O. E. R?ssler and I. Prigogine, “Quantum Mechanics, Diffusion and Chaotic Fractals”, Elsevier, Oxford, 1995.

[44] P. Weibel, G. Ord and O. E. R?ssler, “Space Time Physics and Fractality: Festschroft in Honer of Mohamad El Naschie Vienna,” Springer, New York, 2005.

[45] E. Nelson, “Quantum Fluctuations,” Princeton University Press, Princeton, 1985.

[46] M. Agop, M. Colotin and V. P?un, “Haoticity, Fractality and Fields: Elements of Fractal Theory,” Publishing House ArsLonga, Ia?i, 2009, pp. 12-46.

[47] V. Chioroiu, L. Munteanu, P. ?tiuc? and S. Donescu, “Introduction in Nanomechanics,” Publishing House Academiei Romane, Bucure?ti, 2005.

[48] D. K. Ferry and S. M. Goodnick, “Transport in Nanostructures,” Cambridge University Press, Cambridge, 2001.

[49] Y. Imry, “Introduction to Mesoscopic Physics,” Oxford University Press, Oxford, 2002.

[50] D. Benoit, “Physics of Semiconductor Microcavities from Fundamentals to Nano-Scale Decretes,” Wiley-VCH Verlag GmbH, Weinheim, 2007.

[51] C. Ciuti and I. Camsotto, “Quantum Fluids Effects and Parametric Instabilities in Microcavities,” Physica Status Solidi B, Vol. 242, No. 11, 2005, p. 2224.

[52] C. P. Poole, K. A. Farach and R. Creswick, “Super-Conductivity,” Academic Press, San Diego, 1995.

[53] G. Gr?ssing, “Diffusion Waves in Sub-Quantum Thermodynamics: Resolution of Einstein’s ‘Particle-in-a-Box’ Objection,” 2008. http://arxiv.org/abs/0806.4462

[54] A. Mandelis, L. Nicolaides and Y. Chen, “Structure and the Reflectionless/Refractionless Nature of Parabolic Diffusion-Wave Fields, “ Physical Review Letters, Vol. 87, No. 2, 2001, 4 pages. doi:10.1103/PhysRevLett.87.020801

[55] A. Mandelis, “Diffusion Waves and Their Uses,” Physics Today, Vol. 53, No. 8, 2000, p. 29. doi:10.1063/1.1310118

[56] S. Popescu, “Actual Issues in Self-Structuring Systems,” Publishing House Tehnopress, Iasi, 2004.

[57] E. R. Bittner, “Quantum Tunneling Dynamics Using Hydrodynamic Trajectories,” Journal of Chemical Physics, Vol. 112, No. 22, 2000, p. 9703. doi:10.1063/1.481607

[58] D. Ruelle and F. Takens, “On the Nature of Turbulence,” Communications in Mathematical Physics, Vol. 20, No. 23, 1971, pp. 167-192,

[59] D. Ruelle, “Strange Attractors,” The Mathematical Intelligencer, Vol. 2, No. 3, 1979, pp. 126-137.

[60] C. V. Munceleanu, D. Magop, C. Marin and M. Agop, “Fractal Models in Polymer Physics,” Ars Longa Publishing House, Iasi, 2010.

[61] F. J. Ernst, “New Formulation of the Axially Symemetric Gravitational Fielf Problem I,” Physical Reviews, Vol. 167, No. 5, 1968, pp. 1175-1178. doi:10.1103/PhysRev.167.1175

[62] F. J. Ernst, “Exterior Algebraic Derivation of Einstein Field Equation Employing a Generalized Basis,” Journal of Mathematical Physics, Vol. 12, No. 11, 1971, pp. 2395-2398. doi:10.1063/1.1665549

[63] M. Agop and N. Mazilu, “Fundaments of Modern Physics,” Junimea Publishing House, Iasi, 1989.

[64] M. Agop and N. Mazilu, “The Crossroads of the Theories between Newton and Einstein Barbilian’s Univers,” Ars Longa, Ia?i, 2010.

[65] M. B. Green, J. H. Schwarz and E. Witten, “Superstring Theory,” Cambridge University Press, Cambridge, 1998.

[1] L. Nottale and J. Schneider, “Fractals and Non-Standard Analysis,” Journal of Mathematical Physics, Vol. 25, No. 12, 1984, pp. 96-300.

[2] L. Nottale, “Fractals and the Quantum Theory of Space-Time,” International Journal of Modern Physics, Vol. A4, No. 50, 1989, pp. 47-117.

[3] L. Nottalle, “Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity,” World Scientific, Singapore, 1993.

[4] L. Nottale, “Scale Relativity and Fractal Space-Time: Applications to Quantum Physics, Cosmology and Chaotic Systems,” Chaos, Solitons & Fractals, Vol. 7, No. 6, 1996, pp. 877-938.

[5] L. Nottale, “The Theory of Scale Relativity: Non-Differentiable Geometry, Fractal Space-Time and Quantum Mechanics,” Computing Anticipatory Systems: CASYS’ 03-33 Sixth International Conference, Liège, 11-16 August 2003, pp. 68-95.

[6] J. Chaline, L. Nottale and P. Grou, “The Trees of Evolution: Universe S,” Life Companies Societies, Paris, 2000.

[7] J. Chaline, L. Nottale and P. Grou, “Flowers for Scrodinger Sscale Relativity and Its Applications,” Editure Ellipses Marketing, Paris, 2009.

[8] A. Harabagiu and M. Agop, “Hydrodyamic Model of Scale Relativity Theory,” Bulletin of the Polytechnic Institute of Iasi, Vol. LV, No. 3-4, 2005, pp. 77-82.

[9] A. Harabagiu, O. Niculescu, M. Colotin, T. D. Bibere, I. Gottlieb and M. Agop, “Particle in a Box by Means of Fractal Hydrodynamic Model,” Romanian Reports in Physics, Vol. 61, No. 3, 2009, pp. 395-400.

[10] A. Harabagiu, D. Magop and M. Agop, “Fractality and Quantum Mechanics,” Ars Longa Publishing House, Ia?i, 2010.

[11] M. Agop, L. Chico?, P. Nica and A. Harabagiu, “Euler’s Fluids and Non-Differentiable Space-Time,” Far East Journal of Dynamical Systems, Vol. 10, No. 1, 2008, pp. 93-106.

[12] M. Agop, A. Harabagiu and P. Nica, “Wave-Particle Duality through a Hydrodynamic Model of the Fractal Space Time Theory,” Acta Physica Polonica A, Vol. 113, No. 6, 2008, pp. 1557-1574.

[13] S. ?i?eica, “Quantum Mechanics,” Academiei Publishing House, Bucure?ti, 1984.

[14] B. Felsager, “Geometry, Particles and Fields,” Odense University Press, Odense, 1981.

[15] A. Peres, “Quantum Theory: Concepts and Methods,” Kluwer Academic Publishers, Boston, 1993.

[16] J. J. Sakurai and T. S. Fu, “Modern Quantum Mechanics,” Addison-Wesley, Boston, 1994.

[17] F. Halbwacs, “Theorie Relativiste Des Fluids a Spin,” Gauthier-Villars, Paris, 1960.

[18] R. Madelung, “A Hydrodynamic Interpretation of Quantum Mechanics,” Journal for Physics, Vol. 40, No. 3-4, 1927, pp. 322-326.

[19] D. Bohm and B. J. Hiley, “The Undivided Universe: An Ontological Interpretation of Quantum Theory,” Routledge and Kegan Paul, London, 1993.

[20] D. D?rr, S. Goldstein and N. Zanghi, “Quantum Mechanics, Randomness and Deterministic Reality,” Physics Letters A, Vol. 172, No. 1-2, 1992, pp. 6-12. doi:10.1016/0375-9601(92)90181-K

[21] D. D?rr, S. Goldstein and N. Zanchi, “A Global Equilibrium as the Foundation of Quantum Randomness,” Foundations of Physics, Vol. 23, 1993, pp. 712-738.

[22] D. Z. Albert, “Bohm’s Alternative to Quantum Mechanics”, Scientific American, Vol. 270, No. 2, 1994, pp. 32-39. doi:10.1038/scientificamerican0594-58

[23] J. S. Bell, “Speakable and Unspeakable in Quantum Mechanics,” Cambridge University Press, Cambridge, 1987.

[24] K. Berndl, D. D?rr, S. Goldstein, G. Peruzzi and N. Zanchi, “Existance of Trajectories for Bohmian Mechanics,” International Journal of Theoretical Physics, Vol. 32, No. 12, 1993, pp. 2245-2251. doi:10.1007/BF00672996

[25] K. Berndl, D. D?rr, S. Goldstein, N. Zanchi, “Selfa- djointness and the Existance of Deterministic Trajectories in Quantum Theory,” In: M. Fannes, C. Maes and A. Verbeure, Eds., On Three Levels: Micro-, Meso-, and Macroscopic Approches in Physics, (NATO ASI Series B: Physics), Plenum, New York, 1994, pp. 429-434.

[26] P. R. Holland, “The Quantum Theory of Motion,” Camb- ridge University Press, Cambridge, 1993. doi:10.1017/CBO9780511622687

[27] L. de Broglie, “Causal Interpretation Attempt and Non-Linear Wave Mechanics: The Theory of Double Solution,” Gauthier-Villars, Paris, 1956.

[28] L. de Broglie, “The Measure Theory in Wave Mechanics,” Gauthier-Villars, Paris, 1957.

[29] L. de Broglie, “Wave Mechanics Interpretation,” Journal of Physics Radium, Vol. 20, No. 12, 1959, pp. 963-979. doi:10.1051/jphysrad:019590020012096300

[30] L. de Broglie, “Critical Study of the Current Foundations Interpretation of Wave Mechanics,” Gauthier-Villars, Paris, 1963.

[31] L. de Broglie, “Thermodynamics of Isolated Particle (Hidden Thermodynamics of Particles),” Gauthier-Villars, Paris, 1964.

[32] L. de Broglie, “Certainity and Uncertainity in Science,” Politic? Publishing House, Bucure?ti, 1980.

[33] D. Bohm, “A Suggested Interpretation of Quantum Theory in Terms of ‘Hidden’ Variables I,” Physical Review, Vol. 85, No. 2, 1952, pp. 166-179. doi:10.1103/PhysRev.85.166

[34] D. Bohm, “A Suggested Interpretation of Quantum Theory in Terms of ‘Hidden Variables’: Part II,” Physical Review, Vol. 85, No. 2, 1952, pp. 180-193. doi:10.1103/PhysRev.85.180

[35] D. Bohm, “Proof that Probability Density Approaches in Causal Interpretation of Quantum Theory,” Physical Review, Vol. 89, No. 2, 1953, pp. 458-466. doi:10.1103/PhysRev.89.458

[36] L. Budei, “Fractal Models: Applications in Enviroment Arhitecture,” Publishing House University, Gh. Asachi, Ia?i, 2000.

[37] M. Barnsley, “Fractals Everywhere: The First Course in Deterministic Fractal Geometry,” Academic Press, Boston, 1988.

[38] A. Le Mehante, “Les Geometries Fractales,” Hermes, Paris, 1990.

[39] A. Heck and J. M. Perdang, “Applying Fractals in Astronomy,” Springer Verlag, Berlin, 1991.

[40] J. Feder and A. Aharony, “Fractals in Physics,” North-Holland, Amsterdam, 1990.

[41] P. Berge, Y. Pomeau and C. Vidal, “The Order in Chaos,” Wiley, New York, 1984.

[42] J. F. Gouyet, “Physical Structures and Fractals,” Masson, Paris, 1992.

[43] M. S. El Naschie, O. E. R?ssler and I. Prigogine, “Quantum Mechanics, Diffusion and Chaotic Fractals”, Elsevier, Oxford, 1995.

[44] P. Weibel, G. Ord and O. E. R?ssler, “Space Time Physics and Fractality: Festschroft in Honer of Mohamad El Naschie Vienna,” Springer, New York, 2005.

[45] E. Nelson, “Quantum Fluctuations,” Princeton University Press, Princeton, 1985.

[46] M. Agop, M. Colotin and V. P?un, “Haoticity, Fractality and Fields: Elements of Fractal Theory,” Publishing House ArsLonga, Ia?i, 2009, pp. 12-46.

[47] V. Chioroiu, L. Munteanu, P. ?tiuc? and S. Donescu, “Introduction in Nanomechanics,” Publishing House Academiei Romane, Bucure?ti, 2005.

[48] D. K. Ferry and S. M. Goodnick, “Transport in Nanostructures,” Cambridge University Press, Cambridge, 2001.

[49] Y. Imry, “Introduction to Mesoscopic Physics,” Oxford University Press, Oxford, 2002.

[50] D. Benoit, “Physics of Semiconductor Microcavities from Fundamentals to Nano-Scale Decretes,” Wiley-VCH Verlag GmbH, Weinheim, 2007.

[51] C. Ciuti and I. Camsotto, “Quantum Fluids Effects and Parametric Instabilities in Microcavities,” Physica Status Solidi B, Vol. 242, No. 11, 2005, p. 2224.

[52] C. P. Poole, K. A. Farach and R. Creswick, “Super-Conductivity,” Academic Press, San Diego, 1995.

[53] G. Gr?ssing, “Diffusion Waves in Sub-Quantum Thermodynamics: Resolution of Einstein’s ‘Particle-in-a-Box’ Objection,” 2008. http://arxiv.org/abs/0806.4462

[54] A. Mandelis, L. Nicolaides and Y. Chen, “Structure and the Reflectionless/Refractionless Nature of Parabolic Diffusion-Wave Fields, “ Physical Review Letters, Vol. 87, No. 2, 2001, 4 pages. doi:10.1103/PhysRevLett.87.020801

[55] A. Mandelis, “Diffusion Waves and Their Uses,” Physics Today, Vol. 53, No. 8, 2000, p. 29. doi:10.1063/1.1310118

[56] S. Popescu, “Actual Issues in Self-Structuring Systems,” Publishing House Tehnopress, Iasi, 2004.

[57] E. R. Bittner, “Quantum Tunneling Dynamics Using Hydrodynamic Trajectories,” Journal of Chemical Physics, Vol. 112, No. 22, 2000, p. 9703. doi:10.1063/1.481607

[58] D. Ruelle and F. Takens, “On the Nature of Turbulence,” Communications in Mathematical Physics, Vol. 20, No. 23, 1971, pp. 167-192,

[59] D. Ruelle, “Strange Attractors,” The Mathematical Intelligencer, Vol. 2, No. 3, 1979, pp. 126-137.

[60] C. V. Munceleanu, D. Magop, C. Marin and M. Agop, “Fractal Models in Polymer Physics,” Ars Longa Publishing House, Iasi, 2010.

[61] F. J. Ernst, “New Formulation of the Axially Symemetric Gravitational Fielf Problem I,” Physical Reviews, Vol. 167, No. 5, 1968, pp. 1175-1178. doi:10.1103/PhysRev.167.1175

[62] F. J. Ernst, “Exterior Algebraic Derivation of Einstein Field Equation Employing a Generalized Basis,” Journal of Mathematical Physics, Vol. 12, No. 11, 1971, pp. 2395-2398. doi:10.1063/1.1665549

[63] M. Agop and N. Mazilu, “Fundaments of Modern Physics,” Junimea Publishing House, Iasi, 1989.

[64] M. Agop and N. Mazilu, “The Crossroads of the Theories between Newton and Einstein Barbilian’s Univers,” Ars Longa, Ia?i, 2010.

[65] M. B. Green, J. H. Schwarz and E. Witten, “Superstring Theory,” Cambridge University Press, Cambridge, 1998.