Formulae for Energy and Momentum of Relativistic Particle Regular at Zero-Mass State

Author(s)
Robert M Yamaleev

ABSTRACT

In this paper we substantiate a necessity of introduction of a concept*the counterpart of rapidity* into the framework of relativistic physics. It is shown, formulae for energy and momentum defined via counterpart of rapidity are regular near the zero-mass and speed of light states. The representation for the energy-momentum is realized as a mapping from the massless-state onto the massive one which looks like as a "q"-deformation. Quantization of the energy, momentum and the velocity near the light-speed is presaged. An analogue between the relativistic dynamics and the statistical thermodynamics of a micro-canonical ensemble is brought to light.

In this paper we substantiate a necessity of introduction of a concept

KEYWORDS

Relativistic Dynamics, Complex Algebra, Rapidity, Energy-Momentum, Background Energy, Statistical Thermodynamics.

Relativistic Dynamics, Complex Algebra, Rapidity, Energy-Momentum, Background Energy, Statistical Thermodynamics.

Cite this paper

nullR. Yamaleev, "Formulae for Energy and Momentum of Relativistic Particle Regular at Zero-Mass State,"*Journal of Modern Physics*, Vol. 2 No. 8, 2011, pp. 849-856. doi: 10.4236/jmp.2011.28101.

nullR. Yamaleev, "Formulae for Energy and Momentum of Relativistic Particle Regular at Zero-Mass State,"

References

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[2] Y. Nambu, “Dynamical Model of Elementary Particles Based on an Anal-ogy with Superconductivity,” Physical Review, Vol. 117, 1960, pp. 648-659. doi:10.1103/PhysRev.117.648

[3] J.-M. Levy-Leblond, “A Gedankenexperiment in Science History,” American Journal of Physics, Vol. 48, 1980, pp. 345-354.

[4] V. Varicak, “Ap-plication of Lobachevskian Geometry in the Theory of Relativ-ity,” Physikalische Zeitschrift, Vol. 11, 1910, pp. 93-96.

[5] H. Hergoltz, “Geometrical Aspects of Relativity Theory,” Ann.d.Phys, Vol.10, 1910, pp. 31-45.

[6] A. A. Robb, “Optical Geometry of Motion, a New View of the The-ory of Relativity,” Cambridge, 1911.

[7] A. A. Ungar, “Hy-perbolic Trigonometry and Its Applications in Poincaré Ball Model of Hyperbolic Geometry,” Computers & Mathematics with Applications, Vol. 41, No. 1-2, 2001, pp. 135-147. doi:10.1016/S0898-1221(01)85012-4

[8] A. O. Barut, “Elec-trodynamics and Classical Theory of Fields and Particles,” Dover Publications, Inc., New York, 1980.

[9] R. M. Ya-maleev, “Extended Relativistic Dynamics of Charged Spinning Particle in Quaternionic Formulation,” Advances in Applied Clifford Algebras, Vol. 13, No. 2, 2003, pp. 183-218. doi:10.1007/s00006-003-0015-8

[10] R. M. Yamaleev, “Rela-tivistic Equations of Motion within Nambu’s Formalism of Dy-namics,” Annals of Physics, Vol. 285, No. 2, 2000, pp. 141-160. doi:10.1006/aphy.2000.6075

[11] A. R. Rodriguez-Dominguez, “Ecuaciones de fuerza de Lorentz Como Ecuaciones de Heisenberg,” Revista Mexicana de Fisica, Vol. 53, No. 4, 2007, pp. 270-280.

[12] I. M. Yaglom, “Complex Numbers in Ge-ometry,” Academic Press, New York, 1968.

[13] P. Fjelstad and S. Gal, “Two Dimensional Geometries, Topologies, Trigo-nometries and Physics Generated by Complex-Type Numbers,” Advances in Applied Clifford Algebras, Vol. 11, No. 1, 2001, pp. 81-90. doi:10.1007/BF03042040

[14] R. M. Yamaleev, “Multicom-plex Algebras on Polynomials and Generalized Hamilton Dy-namics,” Journal of Ma- thematical Analysis and Applications, Vol. 322, No. 2, 2006, pp. 815-824. doi:10.1016/j.jmaa.2005.09.073

[15] R. M. Yamaleev, “Geo-metrical and Physical Interpretation of Evolution Governed by General Complex Algebra,” Journal of Mathematical Analysis and Application, Vol. 340, No. 2, 2008, pp. 1046-1057. doi:10.1016/j.jmaa.2007.09.018

[16] R. M. Yamaleev, “Com-plex Algebras on N-order Polynomials and Generalizations of Trigonometry, Oscillator Model and Hamilton Dynamics,” Advances in Applied Clifford Algebras, Vol. 15, No. 1, 2005, pp. 123-125.

[17] V. F. Kagan, “Gostexizdat,” Foundations of Geometry, Part I,” Moscow, 1949, pp. 331-336.

[18] W. Greiner, L. Neise and H. St?cker, “Thermodynamics and Sta-tistical Mechanics,” Springer-Verlag, New York, 1995, pp. 208-214, ISBN 0-387-94299-8.

[19] H. S. Robertson, “Statis-tical Thermodynamics,” PTR Prentice Hall, Englewood Cliffs, 1993, pp. 110-112, ISBN 0-13-845603-8.

[20] R. M. Ya-maleev, “New Representation for Energy- Mo- mentum and Its Applications to Relativistic Dynamics,” Nuclear Physics, Vol. 74, No. 7, 2011, pp. 1-8, (in Russian).

[1] W. Heisenberg, “Zur Theorie de Elementarteilchen,” Zeit Naturforsch, Vol. 14a, 1959, pp. 441-451.

[2] Y. Nambu, “Dynamical Model of Elementary Particles Based on an Anal-ogy with Superconductivity,” Physical Review, Vol. 117, 1960, pp. 648-659. doi:10.1103/PhysRev.117.648

[3] J.-M. Levy-Leblond, “A Gedankenexperiment in Science History,” American Journal of Physics, Vol. 48, 1980, pp. 345-354.

[4] V. Varicak, “Ap-plication of Lobachevskian Geometry in the Theory of Relativ-ity,” Physikalische Zeitschrift, Vol. 11, 1910, pp. 93-96.

[5] H. Hergoltz, “Geometrical Aspects of Relativity Theory,” Ann.d.Phys, Vol.10, 1910, pp. 31-45.

[6] A. A. Robb, “Optical Geometry of Motion, a New View of the The-ory of Relativity,” Cambridge, 1911.

[7] A. A. Ungar, “Hy-perbolic Trigonometry and Its Applications in Poincaré Ball Model of Hyperbolic Geometry,” Computers & Mathematics with Applications, Vol. 41, No. 1-2, 2001, pp. 135-147. doi:10.1016/S0898-1221(01)85012-4

[8] A. O. Barut, “Elec-trodynamics and Classical Theory of Fields and Particles,” Dover Publications, Inc., New York, 1980.

[9] R. M. Ya-maleev, “Extended Relativistic Dynamics of Charged Spinning Particle in Quaternionic Formulation,” Advances in Applied Clifford Algebras, Vol. 13, No. 2, 2003, pp. 183-218. doi:10.1007/s00006-003-0015-8

[10] R. M. Yamaleev, “Rela-tivistic Equations of Motion within Nambu’s Formalism of Dy-namics,” Annals of Physics, Vol. 285, No. 2, 2000, pp. 141-160. doi:10.1006/aphy.2000.6075

[11] A. R. Rodriguez-Dominguez, “Ecuaciones de fuerza de Lorentz Como Ecuaciones de Heisenberg,” Revista Mexicana de Fisica, Vol. 53, No. 4, 2007, pp. 270-280.

[12] I. M. Yaglom, “Complex Numbers in Ge-ometry,” Academic Press, New York, 1968.

[13] P. Fjelstad and S. Gal, “Two Dimensional Geometries, Topologies, Trigo-nometries and Physics Generated by Complex-Type Numbers,” Advances in Applied Clifford Algebras, Vol. 11, No. 1, 2001, pp. 81-90. doi:10.1007/BF03042040

[14] R. M. Yamaleev, “Multicom-plex Algebras on Polynomials and Generalized Hamilton Dy-namics,” Journal of Ma- thematical Analysis and Applications, Vol. 322, No. 2, 2006, pp. 815-824. doi:10.1016/j.jmaa.2005.09.073

[15] R. M. Yamaleev, “Geo-metrical and Physical Interpretation of Evolution Governed by General Complex Algebra,” Journal of Mathematical Analysis and Application, Vol. 340, No. 2, 2008, pp. 1046-1057. doi:10.1016/j.jmaa.2007.09.018

[16] R. M. Yamaleev, “Com-plex Algebras on N-order Polynomials and Generalizations of Trigonometry, Oscillator Model and Hamilton Dynamics,” Advances in Applied Clifford Algebras, Vol. 15, No. 1, 2005, pp. 123-125.

[17] V. F. Kagan, “Gostexizdat,” Foundations of Geometry, Part I,” Moscow, 1949, pp. 331-336.

[18] W. Greiner, L. Neise and H. St?cker, “Thermodynamics and Sta-tistical Mechanics,” Springer-Verlag, New York, 1995, pp. 208-214, ISBN 0-387-94299-8.

[19] H. S. Robertson, “Statis-tical Thermodynamics,” PTR Prentice Hall, Englewood Cliffs, 1993, pp. 110-112, ISBN 0-13-845603-8.

[20] R. M. Ya-maleev, “New Representation for Energy- Mo- mentum and Its Applications to Relativistic Dynamics,” Nuclear Physics, Vol. 74, No. 7, 2011, pp. 1-8, (in Russian).