Abstract
We inquire into classical and quantum laws of motion for the point charge sources in the nonlinear Maxwell–Born–Infeld field equations of classical electromagnetism in flat and curved Einstein spacetimes.
Mathematics Subject Classification (2010). 78A02, 78A25, 83A05, 83C10, 83C50, 83C75.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abraham, M., Prinzipien der Dynamik des Elektrons, Phys. Z. 4, 57–63 (1902); Ann. Phys. 10, pp. 105–179 (1903).
Abraham, M., Theorie der Elektrizität, II, Teubner, Leipzig (1905).
Anco, S. C., and The, D., Symmetries, conservation laws, and cohomology of Maxwell’s equations using potentials, Acta Appl. Math. 89, 1–52 (2005).
Appel, W., and Kiessling, M. K.-H., Mass and spin renormalization in Lorentz electrodynamics, Annals Phys. (N.Y.) 289, 24–83 (2001).
Appel, W., and Kiessling, M. K.-H., Scattering and radiation damping in gyroscopic Lorentz electrodynamics, Lett. Math. Phys. 60, 31–46 (2002).
Bartnik, R., Maximal surfaces and general relativity, pp. 24–49 in “Miniconference on Geometry/Partial Differential Equations, 2” (Canberra, June 26-27, 1986), J. Hutchinson and L. Simon, Ed., Proceedings of the Center of Mathematical Analysis, Australian National Univ., 12 (1987).
Bartnik, R., Isolated points of Lorentzian mean-curvature hypersurfaces, Indiana Univ. Math. J. 38, 811–827 (1988).
Bartnik, R., and Simon, L., Spacelike hypersurfaces with prescribed boundary values and mean curvature, Commun. Math. Phys. 87, 131–152 (1982).
Barut, A. O., Electrodynamics and classical theory of fields and particles, Dover, New York (1964).
Bauer, G., Ein Existenzsatz für die Wheeler–Feynman-Elektrodynamik, Doctoral Dissertation, Ludwig Maximilians Universität, München, (1998).
Bauer, G., Deckert, D.-A., and Dürr, D., Wheeler–Feynman equations for rigid charges — Classical absorber electrodynamics. Part II. arXiv:1009.3103 (2010)
Bia_lynicki-Birula, I., Nonlinear electrodynamics: variations on a theme by Born and Infeld, pp. 31–48 in “Quantum theory of particles and fields,” special volume in honor of Jan _Lopusza´nski; eds. B. Jancewicz and J. Lukierski, World Scientific, Singapore (1983).
Boillat, G., Nonlinear electrodynamics: Lagrangians and equations of motion, J. Math. Phys. 11, 941–951 (1970).
Born, M., Modified field equations with a finite radius of the electron, Nature 132, 282 (1933).
Born, M., On the quantum theory of the electromagnetic field, Proc. Roy. Soc. A143, 410–437 (1934).
Born, M., Théorie non-linéaire du champ électromagnétique, Ann. Inst. H. Poincaré 7, 155–265 (1937).
Born, M., and Infeld, L., Electromagnetic mass, Nature 132, 970 (1933).
Born, M., and Infeld, L., Foundation of the new field theory, Nature 132, 1004 (1933).
Born, M., and Infeld, L., Foundation of the new field theory, Proc. Roy. Soc. London A 144, 425–451 (1934).
Born, M., and Schrödinger, E., The absolute field constant in the new field theory, Nature 135, 342 (1935).
Brenier, Y., Hydrodynamic structure of the augmented Born–Infeld equations, Arch. Rat. Mech. Anal., 172, 65–91 (2004).
Carati, A., Delzanno, P., Galgani, L., and Sassarini, J., Nonuniqueness properties of the physical solutions of the Lorentz–Dirac equation, Nonlinearity 8, pp. 65–79 (1995).
Carley, H., and Kiessling, M. K.-H., Nonperturbative calculation of Born–Infeld effects on the Schrödinger spectrum of the Hydrogen atom, Phys. Rev. Lett. 96, 030402 (1–4) (2006).
Carley, H., and Kiessling, M. K.-H., Constructing graphs over ℝn with small prescribed mean-curvature, arXiv:1009.1435 (math.AP) (2010).
Chae, D., and Huh, H., Global existence for small initial data in the Born-Infeld equations, J. Math. Phys. 44, 6132–6139 (2003).
Christodoulou, D., The action principle and partial differential equations, Annals Math. Stud. 146, Princeton Univ. Press, Princeton (2000).
Christodoulou, D., and Klainerman, S., The global nonlinear stability of the Minkowski space, Princeton Math. Ser. 41, Princeton Univ. Press, Princeton (1993).
Deckert, D.-A., Electrodynamic absorber theory: a mathematical study, Ph.D. dissertation LMU Munich, 2010.
Dirac, P. A. M., Classical theory of radiating electrons, Proc. Roy. Soc. A 167, 148–169 (1938).
Dirac, P. A. M., A reformulation of the Born–Infeld electrodynamics, Proc. Roy. Soc. A 257, 32–43 (1960).
Dürr, D., and Teufel, S. Bohmian mechanics as the foundation of quantum mechanics, Springer (2009).
Ecker, K. Area maximizing hypersurfaces in Minkowski space having an isolated singularity, Manuscr. Math. 56, 375–397 (1986).
Einstein, A., Zur Elektrodynamik bewegter Körper, Ann. Phys. 17, 891–921 (1905).
Einstein, A., Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?, Ann. Phys. 18, pp. 639–641 (1905).
Einstein, A., Infeld, L., and Hoffmann, B., The gravitational equations and the problem of motion, Annals Math. 39, 65–100 (1938).
Franklin, J., and Garon, T., Approximate calculations of Born–Infeld effects on the relativistic hydrogen spectrum, Phys. Lett. A 375 1391–1395 (2011).
Gibbons, G. W., Born-Infeld particles and Dirichlet p-branes, Nucl. Phys. B 514, 603–639 (1998).
Gittel, H.-P., Kijowski, J., and Zeidler, E., The relativistic dynamics of the combined particle-field system in renormalized classical electrodynamics, Commun. Math. Phys. 198, 711–736 (1998).
Hawking, S. W., and Ellis, G. F. R., The large scale structure of spacetime, Cambridge Univ. Press, Cambridge (1973).
Holland, P., The quantum theory of motion, Cambridge University Press (1993).
Hoppe, J., Some classical solutions of relativistic membrane equations in 4 space-time dimensions, Phys. Lett. B 329, 10–14 (1994).
Jackson, J. D., Classical electrodynamics, J. Wiley & Sons, New York 2nd ed. (1975).
Jackson, J. D., Classical electrodynamics, J. Wiley & Sons, New York 3rd ed. (1999).
Jackson, J. D., and Okun, L. B., Historical roots of gauge invariance, Rev. Mod. Phys. 73, 663–680 (2001).
Jost, R., Das Märchen vom elfenbeinernen Turm. (Reden und Aufsätze), Springer Verlag, Wien (2002).
Kiessling, M. K.-H., Electromagnetic field theory without divergence problems. 1. The Born legacy, J. Stat. Phys. 116, 1057–1122 (2004).
Kiessling, M. K.-H., Electromagnetic field theory without divergence problems. 2. A least invasively quantized theory, J. Stat. Phys. 116, 1123–1159 (2004).
Kiessling, M. K.-H., On the quasi-linear elliptic PDE \( {-{\nabla}.\left({\nabla}u / \sqrt{1-{\mid{\nabla}u\mid}^2}\right) = 4 \pi \sum\nolimits_ k {a_k}{\delta_s}_k} \) in physics and geometry, Rutgers Univ. preprint (2011).
Kiessling, M. K.-H., Some uniqueness results for stationary solutions of the Maxwell–Born–Infeld field equations and their physical consequences, submitted to Phys. Lett. A (2011).
Kiessling, M. K.-H., Convergent perturbative power series solution of the stationary Maxwell–Born–Infeld field equations with regular sources, J. Math. Phys. 52, art. 022902, 16 pp. (2011).
Klyachin, A. A., and Miklyukov, V. M., Existence of solutions with singularities for the maximal surface equation in Minkowski space, Mat. Sb. 184, 103-124 (1993); English transl. in Russ. Acad. Sci. Sb. Math. 80, 87–104 (1995).
Klyachin, A. A., Solvability of the Dirichlet problem for the maximal surface equation with singularities in unbounded domains, Dokl. Russ. Akad. Nauk 342, 161–164 (1995); English transl. in Dokl. Math. 51, 340–342 (1995).
Landau, L., and Lifshitz, E. M., The theory of classical fields, Pergamon Press, Oxford (1962).
von Laue, M., Die Wellenstrahlung einer bewegten Punktladung nach dem Relativitätsprinzip, Ann. Phys. 28, 436–442 (1909).
Liénard, A., Champ électrique et magnétique produit par une charge concentrée en un point et animée d’un mouvement quelconque, L’´Eclairage électrique 16 p. 5; ibid. p. 53; ibid. p. 106 (1898).
Lorentz, H. A., La théorie électromagnetique de Maxwell et son application aux corps mouvants, Arch. Néerl. Sci. Exactes Nat. 25, 363–552 (1892).
Lorentz, H. A., Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Körpern, Teubner, Leipzig (1909) (orig. Leyden: Brill, 1895.)
Lorentz, H. A., Weiterbildung der Maxwell’schen Theorie: Elektronentheorie., Encyklopädie d. Mathematischen Wissenschaften V2, Art. 14, 145–288 (1904).
Lorentz, H. A., The theory of electrons and its applications to the phenomena of light and radiant heat, 2nd ed., 1915; reprinted by Dover, New York (1952).
Madhava Rao, B. S., Ring singularity in Born’s unitary theory - 1, Proc. Indian Acad. Sci. A4, 355–376 (1936).
Mie, G., Grundlagen einer Theorie der Materie, Ann. Phys. 37, 511–534 (1912); ibid. 39, 1–40 (1912); ibid. 40, 1–66 (1913).
Pippard, A. B., J. J. Thomson and the discovery of the electron, pp. 1–23 in “Electron—a centenary volume,” M. Springford, ed., Cambridge Univ. Press (1997).
Pleba´nski, J., Lecture notes on nonlinear electrodynamics, NORDITA (1970) (quoted in [BiBi1983]).
Poincaré, H., Sur la dynamique de l’électron, Comptes-Rendus 140, 1504–1508 (1905); Rendiconti del Circolo Matematico di Palermo 21, 129–176 (1906).
Pryce, M. H. L., The two-dimensional electrostatic solutions of Born’s new field equations, Proc. Camb. Phil. Soc. 31, 50–68 (1935).
Pryce, M. H. L., On a uniqueness theorem, Proc. Camb. Phil. Soc. 31, 625–628 (1935).
Pryce, M. H. L., On the new field theory, Proc. Roy. Soc. London A 155, 597–613 (1936).
Rohrlich, F., Classical charged particles, Addison Wesley, Redwood City, CA (1990).
Schrödinger, E., Non-linear optics, Proc. Roy. Irish Acad. A 47, 77–117 (1942).
Schrödinger, E., Dynamics and scattering-power of Born’s electron, Proc. Roy. Irish Acad. A 48, 91–122 (1942).
Serre, D., Les ondes planes en électromagnétisme non-linéaire, Physica D31, 227–251 (1988).
Serre, D., Hyperbolicity of the nonlinear models of Maxwell’s equations, Arch. Rat. Mech. Anal. 172, 309–331 (2004).
Speck, J., The nonlinear stability of the trivial solution to the Maxwell–Born–Infeld system, arXiv:1008.5018 (2010).
Speck, J., The global stability of the Minkowski spacetime solution to the Einstein-nonlinear electromagnetic system in wave coordinates, arXiv:1009.6038 (2010).
Spohn, H., Dynamics of charged particles and their radiation field, Cambridge University Press, Cambridge (2004).
Tahvildar-Zadeh, A. S., On the static spacetime of a single point charge, Rev. Math. Phys. 23, 309–346 (2011).
Thirring, W. E., Classical mathematical physics, (Dynamical systems and field theory), 3rd ed., Springer Verlag, New York (1997).
Thomson, J. J., Cathode rays, Phil. Mag. 44, 294–316 (1897).
Wheeler, J. A., and Feynman, R., Classical electrodynamics in terms of direct particle interactions, Rev. Mod. Phys. 21, 425–433 (1949).
Wiechert, E., Die Theorie der Elektrodynamik und die Röntgen’sche Entdeckung, Schriften d. Physikalisch- Ökonomischen Gesellschaft zu Königsberg in Preussen 37, 1-48 (1896).
Wiechert, E., Experimentelles über die Kathodenstrahlen, Schriften d. Physikalisch- Ökonomischen Gesellschaft zu Königsberg in Preussen 38, 3–16 (1897).
Wiechert, E., Elektrodynamische Elementargesetze, Arch. Néerl. Sci. Exactes Nat. 5, 549–573 (1900).
Yaghjian, A. D., Relativistic dynamics of a charged sphere, Lect. Notes Phys. Monographs 11, Springer, Berlin (1992).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Basel AG
About this chapter
Cite this chapter
Kiessling, M.KH. (2012). On the Motion of Point Defects in Relativistic Fields. In: Finster, F., Müller, O., Nardmann, M., Tolksdorf, J., Zeidler, E. (eds) Quantum Field Theory and Gravity. Springer, Basel. https://doi.org/10.1007/978-3-0348-0043-3_14
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0043-3_14
Published:
Publisher Name: Springer, Basel
Print ISBN: 978-3-0348-0042-6
Online ISBN: 978-3-0348-0043-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)