Functional Integration pp 97-130 | Cite as

# Functional Integration and Wave Propagation

Chapter

## Abstract

We present a pedagogical introduction, including several new results, to Kac’s path integral solution for the telegrapher equation, with emphases on: 1) wave propagation especially waveform (signal) restoration (reconstruction), designing and prediction; and 2) the underlying Poissonian random walk (essentially an asynchronous telegraph signal), especially the measure, and the connection to Brownian motion.

## Keywords

Wave Propagation Brownian Motion Random Walk Green Function Diffusion Equation
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## References

- [1]M. Kac, “Some stochastic problems in physics and mathematics,” colloquium lectures in the pure and applied sciences, No. 2, Magnolia Petroleum Co. and Socony Mobil Oil Co. (1956), Hectographed. (Translations in Polish and Russian are available in book form.); Rocky Mountain J. of Math. 4, 497-509 (1974)Google Scholar
- [2]S. Goldstein, “On diffusion by discontinuous movements, and on the telegraph equation,” Quant. J. Mech. Appl. Math.
**4**, 129–156 (1951)zbMATHCrossRefGoogle Scholar - [3]S. Kaplan, “Differential equations in which the Poisson process plays a role,” Bull. Am. Math. Soc.
**70**, 264–268 (1964)zbMATHCrossRefGoogle Scholar - [4]R. P. Feynman and A. R. Hibbs,
*Quantum Mechanics and Path Integrals*(McGraw-Hill, 1965)Google Scholar - [5]R. Hersh, “Random evolutions: A survey of results and problems,” Rocky Mountain J. of Math. 4 (1974), 443–477; and “Stochastic solutions of hyperbolic equations,” in Partial Differential Equations and Related Topics, eds. A. Dold and B. Eckmann, Lecture Notes in Mathematics #446 (Springer-Verlag, Berlin, 1975), p. 283-300MathSciNetzbMATHCrossRefGoogle Scholar
- [6]B. Gaveau, T. Jacobson, M. Kac, and L. S. Schulman, “Relativistic Extension of the Analogy between Quantum Mechanics and Brownian Motion,” Phys. Rev. Lett.
**53**, 419–422 (1984)MathSciNetADSCrossRefGoogle Scholar - [7]T. Ichinose, “Path Integral for the Dirac Equation in two space-time dimensions,” Proc. of the Japan Academy,
**58**, Ser. A, No. 7 (1982)Google Scholar - [8]T. Jacobson and L. S. Schulman, “Quantum Stochastics: The passage from a relativistic to a non-relativistic path integral,” J. Phys. A 17, 375 (1984)MathSciNetADSCrossRefGoogle Scholar
- [9]S. N. Ethier and T. G. Kurtz, Markov Processes Characterization and Convergence (John Wiley and Sons, New York, 1986), p. 468–491zbMATHGoogle Scholar
- [10]C. DeWitt-Morette and S. K. Foong, “Path integral solutions of wave equations with dissipation” Phys. Rev. Lett. 62, 2201–2204 (1989)MathSciNetADSCrossRefGoogle Scholar
- [11]C. DeWitt-Morette and S. K. Foong, “Kac’s solution of the telegrapher equation, revisited: Part I,” in Developments in General Relativity, Astrophysics and Quantum Theory, A Jubilee Volume in Honour of Nathan Rosen, eds. F. I. Cooperstock et. al. (I. O. P. Publishing, Bristol, 1990), p.351–366.Google Scholar
- [12]S. K. Foong, “Kac’s solution of the telegrapher equation, revisited: Part II,” in Developments in General Relativity, Astrophysics and Quantum Theory, A Jubilee Volume in Honour of Nathan Rosen, eds. F, I. Cooperstock et. al. (I. O. P. Publishing, Bristol, 1990), p.351–366.Google Scholar
- [13]E. Orsingher, “ Probability law, flow function, maximum distribution of wave-governed random motions and their connection with Kirchoff’s laws,” Stochastic Process. Appl. 34, 49–66 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
- [14]B. Gaveau and L. S. Schulman, “Charged polymer in an electric field,” Phys. Rev. A 42, 3470–3475 (1990)MathSciNetADSCrossRefGoogle Scholar
- [15]T. Nakamura, “A nonstandard representation of Feynman’s path integrals,” J. Math. Phys. 32(2), 457–463 (1991)MathSciNetADSzbMATHCrossRefGoogle Scholar
- [16]S. K. Foong and U. van Kolck, “Poisson random walk for solving wave equations,” Prog. Theor. Phys.
**87**, 285–292 (1992)ADSCrossRefGoogle Scholar - [17]D. Mugnai, A. Ranfagni, R. Ruggeri and A. Agresti, “Path integral solution of the telegrapher equation: An application to tunneling time determination,” Phys. Rev. Lett.
**68**, 259–262 (1992)ADSCrossRefGoogle Scholar - [18]D. G. C. McKeon and G. N. Ord, “Time Reversal in Stochastic Processes and the Dirac Equation” Phys. Rev. Lett.
**69**, 3–4 (1992)MathSciNetADSzbMATHCrossRefGoogle Scholar - [19]S. K. Foong, “Path integral solution of the telegrapher equation” in
*Lectures on Path Integration: Trieste 1991 Adriatico Research Conference, Trieste, Italy, 26 August–6 September 1991*, edited by H. Cerdeira et. al., 427-449 (World Scientific, Singapore, 1993)Google Scholar - [20]T. Zastawniak, “Evaluation of the distribution of the random variable \(S\left( t \right) = \int_0^t {{{\left( { - 1} \right)}^{N\left( u \right)}}du} \)]” in
*Lectures on Path Integration: Trieste 1991 Adriatico Research Conference, Trieste, Italy, 26 August–6 September 1991*, edited by H. Cerdeira et. al., 446-448 (World Scientific, Singapore, 1993) (Appendix to Ref. [19])Google Scholar - [21]P. Enders and D. D. Cogan, “The correlated random walk as a general discrete model and as numerical routine,” School of Information Systems, University of East Anglia, England, preprintGoogle Scholar
- [22]S. K. Foong and F. E. Nakamura, “Path integral solutions of telegrapher’s equation: Numerical approach,” in
*Path Integral from meV to MeV: Tutzing’ 92*(Proceedings of the 4th International Conference on Path Integrals from meV to MeV, Tutzing 1992), edited by H. Grabert, et. al., 268-275 (World Scientific, 1993)Google Scholar - [23]J. Masoliver, J. M. Porra, G. H. Weiss, “SoIutions of the telegrapher’s equation in the presence of reflecting and partly reflecting boundaries,” Phys. Rev. E,
**48**, 939–944 (1993)ADSCrossRefGoogle Scholar - [24]S. K. Foong and S. Kanno, “Properties of the telegrapher’s random process with or without a trap,” Stochastic Processes and their Appl.,
**53**, p.147–173 (1994)MathSciNetzbMATHCrossRefGoogle Scholar - [25]S. K. Foong, “Overcoming dissipative distortions by a waveform restorer and designer,” J. Math. Phys. (Special Issue on Functional Integration),
**36**, p.2324–2340 (1995)MathSciNetADSzbMATHCrossRefGoogle Scholar - [26]C. DeWitt-Morette and P. Cartier, “A new perspective on functional integration” Path Integrals: Dubna’ 96 (Proceedings of the Dubna Joint Meeting of Int. Seminar “Path Integrals: Theory& Applications” and “5th Int. Conf. “Path Integrals from meV to MeV”), Edited by V. S. Yarunin and M. A. Smondyrev, (JINR, Dubna, 96.5.27-31)Google Scholar
- [27]L. H. Kauffman and H. P. Noyes, “Discrete physics and the Dirac equation,” To appear in Physics Letters A.Google Scholar
- [28]S. Kanno and S. K. Foong, “Predicting telegraph waveform,” presentation at the Inaugural Conference of APCTP (Asia Pacific Center for Theoretical Physics) (Seoul, 96.6.4-10)Google Scholar
- [29]J. A. Stratton, Electromagnetic Theory (McGraw Hill, 1941)Google Scholar
- [30]D. D. Joseph and L. Preziosi, “Heat waves,” Rev. Mod. Phys.
**61**, 41–73 (1989)MathSciNetADSzbMATHCrossRefGoogle Scholar - D. D. Joseph and L. Preziosi, “Addendum to the paper ‘Heat waves’,” Rev. Mod. Phys.
**62**, 375–391 (1990)MathSciNetADSCrossRefGoogle Scholar - [31]W. Band,
*Introduction to Mathematical Physics*(D. Van Nostrand Company, Inc. 1959)Google Scholar - [32]S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys., 1-89 (1943). Reprinted in
*Selected papers on Noise and Stochastic Processes*, Ed. N. Wax, p.3-91 (Dover, 1954)Google Scholar - [33]L. S. Schulman, Techniques and Applications of Path Integration (John Wiley, New York, 1981)zbMATHGoogle Scholar
- [34]K. L. Chung, Elementary Probability Theory with Stochastic Processes, 3rd Ed. (Springer-Verlag, New York, 1979)zbMATHCrossRefGoogle Scholar
- [35]Mathematica, Wolfram Research, Inc. (1988)Google Scholar
- [36]D. Zwillinger, Handbook of Differential Equations (Academic Press, 1989)Google Scholar
- [37]P. M. Morse, and H. Feshbach,
*Methods of Theoretical Physics*(McGraw-Hill, 1953)Google Scholar - [38]I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 1980).Google Scholar

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