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Bohmian Trajectories for Kerr–Newman Particles in Complex Space-Time

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Abstract

Complexified Liénard–Wiechert potentials simplify the mathematics of Kerr–Newman particles. Here we constrain them by fiat to move along Bohmian trajectories to see if anything interesting occurs, as their equations of motion are not known. A covariant theory due to Stueckelberg is used. This paper deviates from the traditional Bohmian interpretation of quantum mechanics since the electromagnetic interactions of Kerr–Newman particles are dictated by general relativity. A Gaussian wave function is used to produce the Bohmian trajectories, which are found to be multi-valued. A generalized analytic continuation is introduced which leads to an infinite number of trajectories. These include the entire set of Bohmian trajectories. This leads to multiple retarded times which come into play in complex space-time. If one weights these trajectories by their natural Bohmian weighting factors, then it is found that the particles do not radiate, that they are extended, and that they can have a finite electrostatic self energy, thus avoiding the usual divergence of the charged point particle. This effort does not in any way criticize or downplay the traditional Bohmian interpretation which does not assume the standard electromagnetic coupling to charged particles, but it suggests that a hybridization of Kerr–Newman particle theory with Bohmian mechanics might lead to interesting new physics, and maybe even the possibility of emergent quantum mechanics.

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Notes

  1. The static Kerr–Newman solution to the electromagnetic field is double valued when the observation point is continued along paths in space-time. This is well known and studied. Here we are talking about multi-valued analytic functions of the world time variable when the trajectory is a function of this time and not static.

  2. The non-timelike trajectories in the Bohmian set constitute a negligibly small fraction and are ignored here.

  3. Q here is the actual physical charge of the particle.

  4. For elementary particles, these metrics have a naked singularity due to the electromagnetic field energy. If the particle becomes effectively an extended jellium by the GAN transformation, then this naked singularity would generally disappear.

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Acknowledgements

I would like to thank the two reviewers for this paper who provided valuable insights and suggestions. I would also like to thank Ezra Newman for his helpful correspondence.

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    Mark Davidson

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Davidson, M. Bohmian Trajectories for Kerr–Newman Particles in Complex Space-Time. Found Phys 48, 1590–1616 (2018). https://doi.org/10.1007/s10701-018-0217-5

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