Telegraph Equations and Complementary Dirac Equation from Brownian Movement

ABSTRACT

Telegraph equations describing the particle densities in Brownian movement on a lattice site have been derived and it has been shown that the complementary classical Dirac equation appears naturally as the consequence of correlations in particle trajectories in Brownian movement. It has also been demonstrated that Heisenberg uncertainty relation between energy and time is the necessary and sufficient condition to transform this classical equation into usual Dirac’s relativistic quantum equation.

Telegraph equations describing the particle densities in Brownian movement on a lattice site have been derived and it has been shown that the complementary classical Dirac equation appears naturally as the consequence of correlations in particle trajectories in Brownian movement. It has also been demonstrated that Heisenberg uncertainty relation between energy and time is the necessary and sufficient condition to transform this classical equation into usual Dirac’s relativistic quantum equation.

Cite this paper

B. Rajput, "Telegraph Equations and Complementary Dirac Equation from Brownian Movement,"*Journal of Modern Physics*, Vol. 3 No. 9, 2012, pp. 989-993. doi: 10.4236/jmp.2012.39128.

B. Rajput, "Telegraph Equations and Complementary Dirac Equation from Brownian Movement,"

References

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[2] D. G. C. McKeon and G. N. Ord, “Time Reversal in Stochastic Processes and the Dirac Equation,” Physical Review Letters, Vol. 69, No. 1, 1992, pp. 3-4.

[3] G. N. Ord, “A Reformulation of the Feynman Chessboard Model,” Journal of Statistical Physics, Vol. 66, No. 1-2, 1992, pp. 647-659. doi:10.1007/BF01060086

[4] G. N. Ord, “Quantum Interference from Charge Conservation,” Physics Letters A, Vol. 173, No. 4-5, 1993, pp. 343-346. doi:10.1016/0375-9601(93)90247-W

[5] G. N. Ord, “Schr?dinger’s Equation and Discrete Random Walks in a Potential Field,” Annals of Physics, Vol. 250, No. 1, 1996, pp. 63-68. doi:10.1006/aphy.1996.0088

[6] G. N. Ord, “The Schrodinger and Dirac Free Particle Equations without Quantum Mechanics,” Annals of Phy- sics, Vol. 250. No. 1, 1996, pp. 51-62. doi:10.1006/aphy.1996.0087

[7] G. N. Ord and A. S. Deakin, “Random Walks, Continuum Limits, and Schr?dinger’s Equation,” Physical Review A, Vol. 54, No. 5, 1996, 3772-3778. doi:10.1103/PhysRevA.54.3772

[8] M. Kac and Mt. Rocky, “A Stochastic Model Related to the Telegrapher’s Equation,” Rocky Mountain Journal of Mathematics, Vol. 4, No. 3, 1974, pp. 494-510.

[9] M. Kac, B. Gaveau, T. Jacobson and L. Schulman, “Relativistic Extension of the Analogy between Quantum Mechanics and Brownian Motion,” Physical Review Letters, 53, No. 5, 1984, pp. 419-422.

[10] D. G. C. McKeon and G. N. Ord, “Time Reversal and a Stochastic Model of the Dirac Equation in an Electromagnetic Field,” Canadian Journal of Physics, Vol. 82, No. 1, 2004, pp. 19-27.

[11] J. Dunkel, P. Talkner and P. Hanggi, “Relativistic Diffusion Processes and Random Walk Models,” Physical Review D, Vol. 75, No. 4, 2007, Article ID: 043001. doi:10.1103/PhysRevD.75.043001

[12] E. M. Rabei, A.-W. Ajlouni and B. Humam, “Quantization of Brownian Motion,” International Journal of Theoretical Physics, Vol. 45, No. 9, 2006, 1613-1623. doi:10.1007/s10773-005-9001-3

[13] B. S. Rajput, “Telegraph Equations and Complementary Dirac Equation from Classical Approach,” Acta Ciencia Indica, Vol. 36, No. 1, 2010, pp. 81-88.

[14] B. S. Rajput, “Quantum Equations from Brownian Motion,” Canadian Journal of Physics, Vol. 89, No. 2, 2011, pp. 185-191. doi:10.1139/P10-111

[15] B. S. Rajput, “Quantum Equations from Classical Approach,” Indian Journal of Physics, Vol. 85, No. 12, 2010, pp. 1817-1828. doi:10.1007/s12648-011-0195-3

[16] J. Y. Bang and M. S. Berger, “Possible Equilibria of Interacting Dark Energy Models,” Physical Review D, Vol. 74, No. 12, 2006, Article ID: 125012. doi:10.1103/PhysRevD.74.125012

[17] Ed. Seidewitz, “Foundations of a Spacetime Path Formalism for Relativistic Quantum Mechanics,” Journal of Mathematical Physics, Vol. 47, No. 11, 2006, Article ID: 112302.

[1] G. N. Ord, “Classical Analog of Quantum Phase,” International Journal of Theoretical Physics, Vol. 31, No. 7, 1992, pp. 1177-1195. doi:10.1007/BF00673919

[2] D. G. C. McKeon and G. N. Ord, “Time Reversal in Stochastic Processes and the Dirac Equation,” Physical Review Letters, Vol. 69, No. 1, 1992, pp. 3-4.

[3] G. N. Ord, “A Reformulation of the Feynman Chessboard Model,” Journal of Statistical Physics, Vol. 66, No. 1-2, 1992, pp. 647-659. doi:10.1007/BF01060086

[4] G. N. Ord, “Quantum Interference from Charge Conservation,” Physics Letters A, Vol. 173, No. 4-5, 1993, pp. 343-346. doi:10.1016/0375-9601(93)90247-W

[5] G. N. Ord, “Schr?dinger’s Equation and Discrete Random Walks in a Potential Field,” Annals of Physics, Vol. 250, No. 1, 1996, pp. 63-68. doi:10.1006/aphy.1996.0088

[6] G. N. Ord, “The Schrodinger and Dirac Free Particle Equations without Quantum Mechanics,” Annals of Phy- sics, Vol. 250. No. 1, 1996, pp. 51-62. doi:10.1006/aphy.1996.0087

[7] G. N. Ord and A. S. Deakin, “Random Walks, Continuum Limits, and Schr?dinger’s Equation,” Physical Review A, Vol. 54, No. 5, 1996, 3772-3778. doi:10.1103/PhysRevA.54.3772

[8] M. Kac and Mt. Rocky, “A Stochastic Model Related to the Telegrapher’s Equation,” Rocky Mountain Journal of Mathematics, Vol. 4, No. 3, 1974, pp. 494-510.

[9] M. Kac, B. Gaveau, T. Jacobson and L. Schulman, “Relativistic Extension of the Analogy between Quantum Mechanics and Brownian Motion,” Physical Review Letters, 53, No. 5, 1984, pp. 419-422.

[10] D. G. C. McKeon and G. N. Ord, “Time Reversal and a Stochastic Model of the Dirac Equation in an Electromagnetic Field,” Canadian Journal of Physics, Vol. 82, No. 1, 2004, pp. 19-27.

[11] J. Dunkel, P. Talkner and P. Hanggi, “Relativistic Diffusion Processes and Random Walk Models,” Physical Review D, Vol. 75, No. 4, 2007, Article ID: 043001. doi:10.1103/PhysRevD.75.043001

[12] E. M. Rabei, A.-W. Ajlouni and B. Humam, “Quantization of Brownian Motion,” International Journal of Theoretical Physics, Vol. 45, No. 9, 2006, 1613-1623. doi:10.1007/s10773-005-9001-3

[13] B. S. Rajput, “Telegraph Equations and Complementary Dirac Equation from Classical Approach,” Acta Ciencia Indica, Vol. 36, No. 1, 2010, pp. 81-88.

[14] B. S. Rajput, “Quantum Equations from Brownian Motion,” Canadian Journal of Physics, Vol. 89, No. 2, 2011, pp. 185-191. doi:10.1139/P10-111

[15] B. S. Rajput, “Quantum Equations from Classical Approach,” Indian Journal of Physics, Vol. 85, No. 12, 2010, pp. 1817-1828. doi:10.1007/s12648-011-0195-3

[16] J. Y. Bang and M. S. Berger, “Possible Equilibria of Interacting Dark Energy Models,” Physical Review D, Vol. 74, No. 12, 2006, Article ID: 125012. doi:10.1103/PhysRevD.74.125012

[17] Ed. Seidewitz, “Foundations of a Spacetime Path Formalism for Relativistic Quantum Mechanics,” Journal of Mathematical Physics, Vol. 47, No. 11, 2006, Article ID: 112302.