Abstract
Classical electrodynamics based on the Maxwell–Born–Infeld field equations coupled with a Hamilton–Jacobi law of point charge motion is partially quantized. The Hamilton–Jacobi phase function is supplemented by a dynamical amplitude field on configuration space. Both together combine into a single complex wave function satisfying a relativistic Klein–Gordon equation that is self-consistently coupled to the evolution equations for the point charges and the electromagnetic fields. Radiation-free stationary states exist. The hydrogen spectrum is discussed in some detail. Upper bounds for Born's “aether constant” are obtained. In the limit of small velocities of and negligible radiation from the point charges, the model reduces to Schrödinger's equation with Coulomb Hamiltonian, coupled with the de Broglie–Bohm guiding equation.
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REFERENCES
R. Bartnik, Existence of maximal surfaces in asymptotically flat spacetimes, Commun. Math. Phys. 94:155–175 (1984).
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics(Cambridge University Press, Cambridge, UK, 1987).
K. Berndl, D. Dürr, S. Goldstein, G. Peruzzi, and N. Zanghì, On the global existence of Bohmian mechanics, Commun. Math. Phys. 173:647–673 (1995).
K. Berndl, D. Dürr, S. Goldstein, and N. Zangh$#x00EC;, Nonlocality, Lorentz invariance, and Bohmian quantum theory, Phys. Rev. A 53:2062–2073 (1996).
I. Bialynicki-Birula, Nonlinear electrodynamics: Variations on a theme by Born and Infeld, in Quantum Theory of Particles and Fields, Special Volume in Honor of Jan Lopusza$#x0144;ski, B. Jancewicz and J. Lukierski, eds. (World Scientific, Singapore, 1983), pp. 31–48.
D. Bohm, A tentative interpretation of the quantum theory in terms of hidden variables. Part I, Phys. Rev. 85:166–179 (1952); Part II, ibid. 85:180-193 (1952).
G. Boillat, Nonlinear electrodynamics: Lagrangians and equations of motion, J. Math. Phys. 11:941–951 (1970).
M. Born, Zur Quantenmechanik der Stossvorgänge, Z. Phys. 37:863–867 (1926a); Quantenmechanik der Stossvorgänge, Z. Phys. 38:803-827 (1926b).
M. Born, Modified field equations with a finite radius of the electron, Nature 132:282 (1933).
M. Born, On the quantum theory of the electromagnetic field, Proc. Roy. Soc. London Ser. A 143:410–437 (1934).
M. Born, Théorie non-linéare du champ électromagnétique, Ann. Inst. H. Poincaré 7:155–265 (1937).
M. Born, Atomic Physics, 8th rev. ed. (Blackie & Son, Glasgow, 1969).
M. Born and L. Infeld, Electromagnetic mass, Nature 132:970 (1933).
M. Born and L. Infeld, Foundation of the new field theory, Nature 132:1004 (1933); Proc. Roy. Soc. London Ser. A 144:425-451 (1934).
M. Born and L. Infeld, On the quantization of the new field equations. Part I, Proc. Roy. Soc. London Ser. A 147:522–546 (1934); Part II, Proc. Roy. Soc. London Ser. A 150: 141-166 (1935).
L. V. de Broglie, La structure de la matie`re et du rayonnement et la mécanique ondulatoire, Comptes Rendus 184:273–274 (1927); La mécanique ondulatoire et la structure atomique de la matie`re et du rayonnement, J. Phys. et Rad. 8:225-241 (1927); see also Reports on the 1927 Solvay Congress(Gauthier-Villars, Paris, 1927).
L. V. de Broglie, An Introduction to the Study of Wave Mechanics(E. P. Dutton and Co., New York, 1930).
L. V. de Broglie, La physique quantique restera t-elle indeterministique?(Gauthier-Villars, Paris, 1953).
P. A. M. Dirac, A reformulation of the Born-Infeld electrodynamics, Proc. Roy. Soc. London Ser. A 257:32–43 (1960).
D. Dürr, S. Goldstein, and N. Zangh$#x00EC;, Quantum equilibrium and the origin of absolute uncertainty, J. Stat. Phys. 67:843–907 (1992).
D. Dürr, S. Goldstein, K. Münch-Berndl, and N. Zangh$#x00EC;, Hypersurface Bohm-Dirac models, Phys. Rev. A 60:2729–2736 (1999).
D. Dürr, S. Goldstein, J. Taylor, and N. Zangh$#x00EC;, Bosons, Fermions, and the Natural Configuration Space of Identical Particles, preprint (Rutgers University, March 2003).
J. Glimm and A. Jaffe, Quantum Physics, 2nd ed. (Springer, New York, 1980).
S. Goldstein, Quantum theory without observers, Part I, Phys. Today(March 1998); Part II, ibid.(April 1998).
S. Goldstein and R. Tumulka, Opposite arrows of time can reconcile relativity and nonlocality, Class. Quant. Grav. 20:557–564 (2003).
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th. ed. (Academic Press, San Diego, 1980).
J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975); 3rd ed. (Wiley, New York, 1999).
S. Keppeler, Semiclassical quantisation rules for the Dirac and Pauli equations, Ann. Phys. (N.Y.) 304:40–71 (2003).
M. K.-H. Kiessling, Electromagnetic field theory without divergence problems. 1. The Born legacy, J. Stat. Phys. 116:1057–1122 (2004).
C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation(Freeman, New York, 1973).
J. Plebań ski, Lecture Notes on Nonlinear Electrodynamics(NORDITA, Copenhagen, 1970) (quoted in ref. 5).
M. H. L. Pryce, On a uniqueness theorem, Proc. Cambridge Phil. Soc. 31:625–628 (1935).
M. H. L. Pryce, Commuting coordinates in the new field theory, Proc. Roy. Soc. London Ser. A 150:166–172 (1935).
M. H. L. Pryce, On the new field theory, Proc. Roy. Soc. London Ser. A 155:597–613 (1936).
M. H. L. Pryce, On the new field theory. II. Quantum theory of field and charges, Proc. Roy. Soc. London Ser. A 159:355–382 (1937).
M. Reed and B. Simon, Analysis of operators, in Methods of Modern Mathematical Physics IV(Academic Press, Orlando, 1978).
E. Schrödinger, Contribution to Born's new theory of the electromagnetic field, Proc. Roy. Soc. London Ser. A 150:465–477 (1935).
E. Schrödinger, Non-linear optics, Proc. Roy. Irish Acad. Sect. A 47:77–117 (1942).
E. Schrödinger, Dynamics and scattering-power of Born's electron, Proc. Roy. Irish Acad. Sect. A 48:91–122 (1942).
E. Schrödinger, A new exact solution in non-linear optics (two-wave system), Proc. Roy. Irish Acad. Sect. A 49:59–66 (1943).
R. F. Streater and A. Wightman, PCT, Spin and Statistics, and All That(Princeton University Press, Princeton, 1964).
W. E. Thirring, Classical Mathematical Physics: Dynamical Systems and Field Theory, 3rd ed. (Springer-Verlag, New York, 1997).
R. Tumulka, Private communications (2004).
S. Weinberg, The Quantum Theory of Fields. Vol. I, Foundations(Cambridge University Press, Cambridge, 1995).
J. A. Wheeler and W. H. Zurek (eds.), Quantum Theory and Measurement(Princeton University Press, Princeton, 1982).
A. S. Wightman, Some comments on the quantum theory of measurement, in Probabilistic Methods in Mathematical Physics, Proceedings of the International Workshop held in Siena, May 6-11, 1991, F. Guerra, M. I. Loffredo, and C. Marchioro, eds. (World Scientific, River Edge, NJ, 1992), pp. 411–438.
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Kiessling, M.KH. Electromagnetic Field Theory Without Divergence Problems 2. A Least Invasively Quantized Theory. Journal of Statistical Physics 116, 1123–1159 (2004). https://doi.org/10.1023/B:JOSS.0000037251.24558.5c
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DOI: https://doi.org/10.1023/B:JOSS.0000037251.24558.5c