# The Lagrangian and Hamiltonian Aspects of the Electrodynamic Vacuum-Field Theory Models

### Abstract

The important classical Ampère’s and Lorentz laws derivations are revisited and their relationships with the modern vacuum field theory approach to modern relativistic electrodynamics are demonstrated. The relativistic models of the vacuum field medium and charged point particle dynamics as well as related classical electrodynamics problems jointly with the fundamental principles, characterizing the electrodynamical vacuum-field structure, based on the developed field theory concepts are reviewed and analyzed detail. There is also described a new approach to the classical Maxwell theory based on the derived and newly interpreted basic equations making use of the vacuum field theory approach. There are obtained the main classical special relativity theory relationships and their new explanations. The well known Feynman approach to Maxwell electromagnetic equations and the Lorentz type force derivation is also. A related charged point particle dynamics and a hadronic string model analysis is also presented. We also revisited and reanalyzed the classical Lorentz force expression in arbitrary non-inertial reference frames and present some new interpretations of the relations between special relativity theory and its quantum mechanical aspects. Some results related with the charge particle radiation problem and the magnetic potential topological aspects are discussed. The electromagnetic Dirac-Fock-Podolsky problem of the Maxwell and Yang-Mills type dynamical systems is analyzed within the classical Dirac-Marsden-Weinstein symplectic reduction theory. Based on the Gelfand-Vilenkin representation theory of infinite dimensional groups and the Goldin-Menikoff-Sharp theory of generating Bogolubov type functionals the problem of constructing Fock type representations and retrieving their creation-annihilation operator structure is analyzed. An application of the suitable current algebra representation to describing the non-relativistic Aharonov-Bohm paradox is demonstrated. The current algebra coherent functional representations are constructed and their importance subject to the linearization problem of nonlinear dynamical systems in Hilbert spaces is also presented.

### References

References

Abraham R. and Marsden J. Foundations of Mechanics, Second Edition, Benjamin Cummings, NY, 1978

Aharonov Y. and Bohm D. Phys. Rev. 115, 1959, p. 485

Albeverio S., Kondratiev Y.G. and Streit L. How to generalize white noice analysis to non-gaussian measures. Preprint Bi-Bo-S, Bielefeld, 1992

Anderson J. L. Comment on Feynman's proof of the Maxwell equations by F.J. Dyson [Am.J.Phys.58,209-211(1990)]. Am. J. Phys. 59, 1991, p. 86

Aquino F. Mathematical foundations of the relativistic theory of quantum gravity. The Pacific Journal of Science and Techmology, 11(1, 2010, p. 173-232

Amelino-Camelia G., Freidel L., Kowalski-Glikman and Smolin Lee. Relative locality: a deepening of the relativity principle. arXiv:11106.0313v1; arXiv:1101.0931v2;

Arnold V.I. Mathematical Methods of Classical Mechanics., Springer, NY, 1978

Barbashov B.M. On the canonical treatment of Lagrangian constraints. arXiv:hepth/0111164, 2001

Barbashov B.M., Chernikov N.A. Solution and quantization of a nonlinear Born-Infeld type model. Journal of Theoret. Mathem. Physics, 60, N 5, 1966, 1296-1308 in Russian)

Barbashov B. M., Nesterenko V. V. Introduction to the relativistic string theory. World Scientific, Singapore, 1990

Barbashov B.M, Pervushin V.N., Zakharov A.F. and Zinchuk V.A. The Hamiltonian approach to general relativity and CNB-primordal spectrum. arXiv:hep-th/0606054, 2006

Berezansky Yu. M. Eigenfunctions expansions related with selfadjoint operators, Kiev, Nauk. Dumka Publ., 1965 in Russian)

Berezansky Y.M. A generalization of white noice analysis by means of theory of hypergroups. Reports on Math. Phys., 38, N3, 289-300, 1996

Berezansky Y.M. and Kondratiev Y.G. Spectral methods in infinite dimensional analysis, v.1 and 2, Kluwer, 1995.

Berezin F.A. The second quantization method. Nauka Puplisher, Moscow, 1986 in Russian)

Berezin F.A. and Marinov M.S. Particle dynamics as the Grassmann variant of classical mechanics. Annales of Physics, 104, 1977, p. 336-362

Berezin F.A. and Shubin M.A. Schrodinger equation, Moscow, the Moscow University Publisher, 1983 in Russian)

Bjorken J.D. and Drell S.D. Relativistic Quantum Fields. Mc Graw-Hill Book Co., NY, 1965

Bogolubov N.N. jr. and Prykarpatsky A.K. The Analysis of Lagrangian and Hamiltonian Properties of the Classical Relativistic Electrodynamics Models and Their Quantization. Found Phys 40, 2010, p. 469--493

Blackmore D., Prykarpatsky A.K. and Samoylenko V.Hr. Nonlinear dynamical systems of mathematical physics: spectral and differential-geometrical integrability analysis. World Scientific Publ., NJ, USA, 2011

Bogolubov N.N. and Bogolubov N.N. jr. Introduction into quantum statistical mechanics. World Scientific, NJ, 1986

Bogolubov N.N., Logunov A.A., Oksak A.I. and Todorov I.T. Introduction to Axiomatic Field Theory, Massachussetts: W.A.Benjamin, Inc. Advanced Book Progarm, 1975

Bogolubov N.N. Jr., Mykytyuk I.V. and Prykarpatsky A.K. Verma moduli over the quantum Lie current algebra on the circle. Soviet Mathematical Doklady. 314(2, 1990, p. 268-272 - 349 In Russian)

Bogolubov N.N. Jr. and Prykarpatsky A.K. The Lagrangian and Hamiltonian formalisms for the classical relativistic electrodynamical models revisited. arXiv:0810.4254v1 [gr-qc] 23 Oct 2008

Bogolubov N.N. Jr., Prykarpatsky A.K. Quantum method of generating Bogolubov functionals in statistical physics: current Lie algebras, their representations and functional equations. Physics of Elementary Particles and Atomique Nucleus, v.17, N4, 791-827, 1986 in Russian)

Bogolubov N.N., Prykarpatsky A.K. The Lagrangian and Hamiltonian formalisms for the classical relativistic electrodynamical models revisited. arXiv:0810.4254v1 [gr-qc] 23 Oct 2008

Bogolubov Jr., Prykarpatsky A.K., Taneri U., and Prykarpatsky Y.A. The electromagnetic Lorentz condition problem and symplectic properties of Maxwell- and Yang--Mills-type dynamical systems. J. Phys. A: Math. Theor. 42 2009 165401 16pp)

Bogolubov N.N. and Shirkov D.V. Introduction to quantized field theory . Moscow, Nauka, 1984 in Russian)

Bogolubov N.N., Shirkov D.V. Quantum Fields. "Nauka", Moscow, 1984

Bolotovskii B. M. and Stolyarov S. N. Radiation from and energy loss by charged particles in moving media. Sov. Phys. Usp. 35, N 2, 1992, p. 143--150

Boyer T.H. Phys. Rev. D 8(6, 1973, p. 1679

Brans C.H. and Dicke R.H. Mach's principle and a relativistic theory of gravitation. Phys. Rev., 124, 1961, 925

Brillouin L. Relativity reexamined. Academic Press Publ., New York and London, 1970

Bialynicky-Birula I. Phys Rev., 155, 1967, 1414; 166,1968, 1505

Brehme R.W. Comment on Feynman's proof of the Maxwell equations by F.J. Dyson [Am.J.Phys.58, 209-211(1990)]. Am. J. Phys. 59, 1991, p. 85-86

Bulyzhenkov-Widicker I.E. Einstein's gravitation for Machian relativism of nonlocal energy charges. Int. Journal of Theor. Physics, 47, 2008, 1261-1269

Bulyzhenkov I.E. Einstein's curvature for nonlocal gravitation of Gesamt energy carriers. arXiv:math-ph/0603039, 2008

Deser S., Jackiw R. Time travel? arXiv:hep-th/9206094, 1992

Di Bartolo B. Classical theory of electromagnetism. The Second Edition, World Scientific, NJ, 2004

Dirac P.A.M. Proc. Roy. Soc. Ser. A, 167, 1938, p. 148-169

Dirac P.A.M. Generalized Hamiltonian dynamics. Canad. J. of Math., 2, N2, 129-148, 1950

Dirac P.A.M, Fock W.A. and Podolsky B. Phys. Zs. Soviet. 1932, 2, p. 468

Dombey N. Comment on Feynman's proof of the Maxwell equations. by F.J. Dyson [Am.J.Phys. 58,209-211(1990)]." Am. J. Phys. 59, 1991, p. 85

Donaldson S.K. An application of gauge theory to four dimansional topology. J. Diff. Geom., 17, 279-315, 1982

Drinfeld V.G. Quantum Groups. Proceedings of the International Congress of Mathematicians, MRSI Berkeley, p. 798, 1986

Dubrovin B.A., Novikov S.P. and Fomenko A.T. Modern geometry Nauka, Moscow, 1986 in Russian)

Dunner G., Jackiw R. "Peierles substitution" and Chern-Simons quantum mechanics. arXiv:hep-th/92004057, 1992

Dyson F.J. Feynman's proof of the Maxwell equations, Am. J. Phys. 58, 1990, p. 209-211

Dyson F. J. Feynman at Cornell. Phys. Today 42 2, 32-38 1989.

Faddeev L.D. Energy problem in the Einstein gravity theory. Russian Physical Surveys, 136, N 3, 1982, 435-457 in Russian)

Faddeev L.D., Quantum inverse scattering problem II, in Modern problems of mathematics, M: VINITI Publ.,3, 93-180, 1974 in Russian)

Faddeev L.D., Sklyanin E.K. Quantum mechanical approach to completely integrable field theories. Proceed. of the USSR Academy of Sciences DAN, 243, 1430-1433, 1978 in Russian)

Farquar I. E. Comment on Feynman's proof of the Maxwell equations by F.J. Dyson [Am.J.Phys.58,209-211(1990)],. Am. J. Phys. 59, 1991, p. 87; Fermi E. Nuovo Cimento, 25, 1923, p. 159;

Fedorchenko A.M. Classical mechanics and electrodynamics. Kiev, Vyshcha shkola, 1992

Feynman R. Quantum mechanical computers. Found. Physics, 16, 1986, p. 507-531

Feynman R., Leighton R. and Sands M. The Feynman Lectures on Physics. Electrodynamics, v. 2, Addison-Wesley, Publ. Co., Massachusetts, 1964

Feynman R., Leighton R. and Sands M. The Feynman lectures on physics. The modern science on the Nature. Mechanics. Space, time, motion. v. 1, Addison-Wesley, Publ. Co., Massachusetts, 1963

Fock V.A. Konfigurationsraum und zweite Quantelung. Zeischrift Phys. Bd. 75, 622-647, 1932

Galtsov D.V. Theoretical physics for mathematical students. Moscow State University Publisher, 2003 in Russian)

Gelfand I., Vilenkin N. Generalized functions, 4, Academic Press, New York, 1964

Gillemin V., Sternberg S. On the equations of motion of a classical particle in a Yang-Mills field and the principle of general covariance. Hadronic Journal, 1978, 1, p.1-32

Glauber R.J. Quantum Theory of Optical Coherence. Selected Papers and Lectures, Wiley-VCH, Weinheim 2007

Godbillon C. Geometrie Differentielle et Mecanique Analytique. Hermann Publ., Paris, 1969

Goldin G.A. Nonrelativistic current algebras as unitary representations of groups. Journal of Mathem. Physics, 12(3, 462-487, 1971

Goldin G.A. Grodnik J. Powers R.T. and Sharp D. Nonrelativistic current algebra in the N/V limit. J. Math. Phys. 15, 88-100, 1974

Goldin G.A., Menikoff R., Sharp F.H. Diffeomorphism groups, gauge groups, and quantum theory. Phys. Rev. Lett. 51, 1983, p. 2246-2249

Goldin G.A., Menikoff R., Sharp F.H. Representations of a local current algebra in nonsimply connected space and the Aharonov-Bohm effect. J. Math. Phys., 22(8, 1981, p. 1164-1668

Goto T. Relativistic quantum mechanics of one-dimensional mechanical continuum and subsidiary condition of dual resonance model. Prog. Theor. Phys. 46, N 5, 1971, p.1560-1569

Green B. The fabric of the Cosmos. Vintage Books Inc., New York, 2004

Grover L.K. Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325-328, 1997

Hajra S. Classical interpretations of relativistic fenomena. Journal of Modern Physics, 3, 2012, p. 187-189

Hammond R.T. Electrodynamics and Radiation reaction. Found. Physics, 43, 2013, p. 201-209

Hammond R.T. Relativistic Particle Motion and Radiation Reaction. Electronic Journal of Theoretical Physics, No. 23, 2010, p. 221--258

Hojman S. and Shepley L. C. No Lagrangian? No quantization! J. Math. Phys. 32, 142-146 1991)

Holm D. and Kupershmidt B. Poisson structures of superfluids. Phys. Lett., 91A, 425-430, 1982

Holm D., and Kupershmidt B. Superfluid plasmas: multivelocity nonlinear hydrodynamics of superfluid solutions with charged condensates coupled electromagnetically. Phys. Rev., 1987, 36A, N8, p. 3947-3956

Holm D., Marsden J., Ratiu T. and Weinstein A. Nonlinear stability of fluid and plasma equilibria. Physics Reports, 123/(1 and 2, 1-116, 19851

t'Hooft G. Introduction to General Relativity, Rinton, Princeton, N. J. 2001); http://www.phys.uu.nl/ thooft/lectures/genrel.pdf

Hughes R. J. On Feynman's proof of the Maxwell equations. Am. J. Phys. 60, 1992, p.301-306

Jackiw R., Polychronakos A.P. Dynamical Poincare symmetry realized by field-dependent diffeomorhisms. arXiv:hep-th/9809123, 1998

Jackiw R. Lorentz violation in a diffeomorphism-invariant theory. arXiv:hep-th/0709.2348, 2007

Jackson J.D. Classical electrodynamics. 3-rd Edition., NY, Wiley, 1999

Jaffe A. and Quinn F. Theoretical mathematics: toward a cultural synthesis of mathematics and theoretical physics. Bull. Amer. Math. Soc. 29, 1-13, 1993

Jefimenko O.D. Causality Electromagnetic Induction and Gravitation. Electret Scientific Company, Star City, 2000

Kaya A. Hubble's law and fasrter than light expansion speeds. Am. J. Phys., 79(11, 2011, p. 1151-1154

Kibble T.W.B. Lorentz invariance and the gravitational field. Journal of Math. Physics, 2(2, 1961, p. 212-22

Kleinert H. Path integrals. World Scientific. Second Edition, 1995, 685

Klymyshyn I.A. Relativistic astronomy. in Ukrainian.Kyiv, "Naukova Dumka" Publisher, 1980

Kondratiev Y.G., Streit L., Westerkamp W. and Yan J.-A. Generalized functions in infinite dimensional analysis. II AS preprint, 1995

Kosyakov B.P. Radiation in electrodynamics and in Yang-Mills theory. Sov. Phys. Usp. 35(2 135--142 1992)

Kowalski K. and Steeb W.-H. Symmetries and first integrals for nonlinear dynamical systems: Hilbert space approach. I and II. Progress of Theoretical Physics, 85, N4, 713-722, 1991, and 85, N4, 975-983, 1991

Kowalski K. Methods of Hilbert spaces in the theory of nonlinear dynamical systems. World Scientific, 1994

Kowalski K. and Steeb W.-H. Non linear dynamical systems and Carleman linearization. World Scientific, 1991

Kulish P.I., Sklyanin E.K. Lecture Notes in Physics, 151, 1982a, 61-119, Berlin: Springer

Kummer M. On the construction of the reduced phase space of a Hamiltonian system with symmetry. Indiana University Mathem. Journal, 1981, 30,N2, p.281-281.

Kupershmidt B.A. Infinite-dimensional analogs of the minimal coupling principle and of the Poincare lemma for differential two-forms. Diff. Geom. & Appl. 1992, 2,p. 275-293

Landau L.D. and Lifshitz E.M. Field theory. v. 2, " Nauka" Publisher, Moscow, 1973

Landau L.D. and Lifshitz E.M. Quantum mechanics. v. 6, " Nauka" Publisher, Moscow, 1974

Lee C. R. The Feynman-Dyson Proof Of The Gauge Field Equations. Phys. Lett. A 148, 1990, p. 146-148

Logunov A. A. The Theory of Gravity. Moscow. Nauka Publisher, 2000

Logunov A.A. and Mestvirishvili M.A. Relativistic theory of gravitation. Moscow, Nauka, 1989 In Russian)

Lytvynov E.W., Rebenko A.L. and Shchepaniuk G.V. Wick calculus on spaces of generalized functions compound Poisson white noise. Reports on Math. Phys. 39, N2, 1997, p. 219-247

Marsden J. and Chorin A. Mathematical foundations of the mechanics of liquid. Springer, New York, 1993

Marsden J. and Weinstein A. The Hamiltonian structure of the Maxwell-Vlasov equations. Physica D, 1982, 4, p. 394-406

Martins Alexandre A. and Pinheiro Mario J. On the electromagnetic origin of inertia and inertial mass. Int J Theor Phys., 47, 2008. p. 2706-2715

Medina R. Radiation reaction of a classical quasi-rigid extended particle. J. Phys. A: Math. Gen., 2006, p. 3801-3816

Mermin N.D. Relativity without light. Am. J. Phys., 52, 1984, 119-124.

Mermin N.D. It's About Time: Understanding Einstein's Relativity, Princeton, NJ., Princeton University Press, 2005

Mitropolsky Yu., Bogolubov N. jr., Prykarpatsky A. and Samoylenko V. Integrable dynamical system: spectral and differential-geometric aspects. Kiev, "Naukova Dunka", 1987. in Russian)

Moor J.D. Lectures on Seiberg-Witten invariants. Lect. Notes in Math., N1629, Springer, 1996.

Morozov V.B. On the question of the electromagnetic momentum of a charged body. Physics Uspekhi, 181(4, 2011, p. 389 - 392

Nambu Y. Strings, monopoles, and guage fields. Phys. Rev. D., 10, N 12, 1974, 4262-4268

Neumann J. von. Mathematische Grundlagen der Quanten Mechanik. J. Springer, Berlin, 1932

Newman R.P. The Global Structure of Simple Space-Times. Comm. Mathem. Physics, 123, 1989, 17-52

Owczarek R. Topological defects in superfluid Helium. Int. J. Theor. Phys., 30/12, 1605-1612, 1991

Pauli W. Theory of Relativity. Oxford Publ., 1958

Pegg D.T. Rep. Prog. Phys. 1975. V. 38. P. 1339

Penrose R. Twistor algebra. Journal of Math. Physics, 8(2, 1966, p. 345-366

Prykarpatsky A. and Mykytyuk I. Algebraic Integrability of nonlinear dynamical systems on manifolds: classical and quantum aspects. Kluwer Academic Publishers, the Netherlands, 1998, 553p.

Peradzy ski Z. Helicity theorem and vertex lines in superfluid He. Int. J. Theor. Phys., 29/11, 1277-1284, 1990

Prykarpatsky A.K. and Bogolubov N.N. Jr. The Maxwell Electromagnetic Equations and the Lorentz Type Force Derivation---The Feynman Approach Legacy. Int J Theor Phys, 2012 51, p. 237--245

Prykarpatsky A., Bogolubov N jr., Golenia J. A symplectic generalization of the Peradzynski Helicity Theorem and Some Applications. Int J Theor Phys 47, 1919--1928, 2008

Prykarpatsky A., Bogoluboiv N. jr., Golenia J., Taneri U. Introductive backgrounds of modern quantum mathematics with application to nonlinear dynamical systems. Available at: http://publications.ict it Preprint ICTP-IC/2007/108

Prykarpatsky A., Bogolubov N. jr., Golenia J., Taneri U. Introductive Backgrounds to Modern Quantum Mathematics with Application to Nonlinear Dynamical Systems. Int J Theor Phys., 47, 2882--2897, 2008

Prykarpatsky A.K., Bogolubov N.N. Jr. and Taneri U. The vacuum structure, special relativity and quantum mechanics revisited: a field theory no-geometry approach. Theor. Math. Phys. 160(2, 2009, p. 1079--1095; arXiv lanl: 0807.3691v.8 [gr-gc] 24.08.2008 )

Prykarpatsky A.K., Bogolubov N.N. Jr. and Taneri U. The field structure of vacuum, Maxwell equations and relativity theory aspects. Preprint ICTP, Trieste, IC/2008/051 http://publications.ictp.it)

Prykarpatsky A.K., Bogolubov N.N. Jr. and Taneri U. The Relativistic Electrodynamics Least Action Principles Revisited: New Charged Point Particle and Hadronic String Models Analysis. Int. J. Theor. Phys. 2010 49, p. 798--820

Prykarpatsky A.K. and Kalenyuk P.I. Gibbs representation of current Lie algebra and complete system of quantum Bogolubov functional equations. Soviet Mathematical Doklady. 300(2, 1988, p. 346 - 349 1988. In Russian)

Prykarpatsky A.K., Taneri U. and Bogolubov N.N. Jr. Quantum field theory and application to quantum nonlinear optics. World Scientific, NY, 2002

Prytula M., Prykarpatsky A., Mykytyuk I. Fundamentals of the Theory of Differential-Geometric Structures and Dynamical Systems. Kiev, the Ministry of Educ. Publ., 1988 in Ukrainian)

Prykarpatsky Ya.A., Samoylenko A.M. and Prykarpatsky A.K. The geometric properties of reduced symplectic spaces with symmetry, their relationship with structures on associated principle fiber bundles and some applications. Part 1. Opuscula Mathematica, Vol. 25, No. 2, 2005, p. 287-298

Prykarpatsky Ya.A. Canonical reduction on cotangent symplectic manifolds with group action and on associated principal bundles with connections. Journal of Nonlinear Oscillations, Vol. 9, No. 1, 2006, p. 96-106

Prykarpatsky A. and Zagrodzinski J. Dynamical aspects of Josephson type media. Ann. of Inst. H. Poincaré, Physique Theorique, v. 70, N5, p. 497-524

Putterman S.J. Superfluid Hydrodynamics, North Holland, Amsterdam, 1974

Ratiu T., Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body. Proc. NAS of USA, 1981, 78, N3, p. 1327-1328

Repchenko O. Field physics. Moscow, Galeria Publ., 2005 in Russian)

Rohrlich F. Classical Charged Particles. Reading, Mass.: Addison-Wesley, 1965

Samoilenko A., Prykarpatsky Ya., Taneri U., Prykarpatsky A., Blackmore D. A geometrical approach to quantum holonomic computing algorithms. Mathematics and Computers in Simulation, 66, 1--20, 2004

S awianowski J.J. Geometry of Phase Spaces. John Wiley and Sons, 1991

Santilli R.M. The etherino and/or neutrino hypothesis. Foundations of Physics, 37 415, 2007, p. 670-694

Schott G.A. Electromagnetic Radiation. Cambridge Univ. Publ., Cambridge, 1912

Silagadze Z.K. Feynman derivation of Maxwell equations and extra dimensions. arXiv:hep-ph/0106235, 2002

Simulik V.M. The electron as a system of classical electromagnetic and scalar fields. In book: What is the electron? P.109-134. Edited by V.M. Simulik. Montreal: Apeiron, 2005

Sidharth B.G. Dark Energy and Electrons. Int J Theor Phys 48, 2009, p. 2122-2128

Smolin L. Fermions and Topology, arXive:9404010/gr-qc arXiv:hep-ph/0106235, 2002

Tanimura S. Relativistic generalization and extension to the nonabelian gauge theory of Feynman's proof of the Maxwell equations. Annals Phys. 220, 1992, p. 229-247

Teitelboim C. Phys. Rev. 1970, D1, p. 1512

Thirring W. Classical Mathematical Physics. Springer, Third Edition, 1992

Trammel G.T. Phys. Rev. 134(5B, 1964, B1183-N1184

Urban M., Couchot F., Sarazin X., Djannati-Atai A. The quantum vacuum as the origin of the speed of light. arxiv.org/1302.6165v1, 2013

Vaidya A. and Farina C. Can Galilean mechanics and full Maxwell equations coexist peacefully? Phys. Lett. A 153, 1991, p. 265-267

Weyl H. The Theory of Groups and Quantum Mechanics. Dover, New York, 1931

Witten E. Nonabelian bozonization in two dimensions. Commun. Mathem. Physics, 92, 455-472, 1984

Weinstock R. New approach to special relativity. Am. J. Phys., 33, 1965, 640-645

Wilczek F. QCD and natural phylosophy. Ann. Henry Poincare, 4, 2003, 211-228

Wheeler J.B. and Feynman R.P. Interaction with the Absorber as the Mechanism of Radiation. Rev. Modern Phys., 17, N2-3, p. 157-181

*Boson Journal of Modern Physics*,

*2*(2), 105-196. Retrieved from http://scitecresearch.com/journals/index.php/bjmp/article/view/642

Copyright (c) 2016 Boson Journal of Modern Physics

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

**TRANSFER OF COPYRIGHT**

BJMP is pleased to undertake the publication of your contribution to **Boson** **Journal of Modern Physics.**

The copyright to this article is transferred to BJMP(including without limitation, the right to publish the work in whole or in part in any and all forms of media, now or hereafter known) effective if and when the article is accepted for publication thus granting BJMP all rights for the work so that both parties may be protected from the consequences of unauthorized use.