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Dirac’s Point Electron in the Zero-Gravity Kerr–Newman World

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Abstract

The results of a study of the Dirac Hamiltonian for a point electron in the zero-gravity Kerr–Newman spacetime are reported; here, “zero-gravity” means G → 0, where G is Newton’s constant of universal gravitation, and the limit is effected in the Boyer–Lindquist coordinate chart of the maximal analytically extended, topologically nontrivial, Kerr–Newman spacetime. In a nutshell, the results are: the essential self-adjointness of the Dirac Hamiltonian; the reflection symmetry about zero of its spectrum; the location of the essential spectrum, exhibiting a gap about zero; and (under two smallness assumptions on some parameters) the existence of a point spectrum in this gap, corresponding to bound states of Dirac’s point electron in the electromagnetic field of the zero-G Kerr–Newman ring singularity. The symmetry result of the spectrum extends to the Dirac Hamiltonian for a point electron in a generalization of the zero-G Kerr–Newman spacetime with different ratio of electric-monopole to magnetic-dipole moment. The results are discussed in the context of the general-relativistic hydrogen problem. Also, some interesting projects for further inquiry are listed in the lastsection.

Mathematics Subject Classification (2010). 81, 83, 35.

This is an expanded version of the talk titled “The Dirac equation and the Kerr-Newman spacetime,” given by the first author at the Quantum Mathematical Physics conference.

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Notes

  1. 1.

    We are not suggesting that experimental physicists should not worry about this academic problem. For the empirically relevant problem to estimate the influence of, say, Earth’s gravitational field on the spectrum of hydrogen in the lab, see Papapetrou [40].

  2. 2.

    We follow the notation of Lieb and Loss [34]; thus \(C_{c}^{\infty }(\mathbb{R}^{3}\setminus \{\boldsymbol{0}\})\) denotes functions which are compactly supported away from the origin in \(\mathbb{R}^{3}\).

  3. 3.

    For a modern semi-classical approach that produces these quantum numbers, see [30].

  4. 4.

    The additive constant mc 2 drops out in the calculation of Rydberg’s empirical formula for the frequencies of the emitted/absorbed radiation, which are proportional to the differences of the discrete energy eigenvalues.

  5. 5.

    This well-known naked singularity is usually not considered to be a counterexample of the (weak) cosmic censorship hypothesis, based on the following reasoning: paraphrasing Freeman Dyson, general relativity is a classical physical theory which applies only to physics in the large (e.g. astrophysical and cosmic scales), not to atomic physics; and so, since cosmic bodies of mass m and charge q must have a ratio G m 2∕q 2 ≫ 1, the Reissner–Nordström spacetime of such a body (assumed spherical), when collapsed, exhibits a black hole, not a naked singularity. While we agree that cosmic bodies (in mechanical virial-equilibrium) must have a ratio G m 2∕q 2 ≫ 1, we don’t see why the successful applications of general relativity theory at astrophysical and cosmic scales would imply that general relativity cannot be successfully applied at atomic, or even sub-atomic scales, where typically G m 2∕e 2 ≪ 1.

  6. 6.

    Interestingly enough, though, Belgiorno–Martellini–Baldicchi [8] proved the existence of bound states of a Dirac point electron equipped with an anomalous magnetic moment in the Reissner–Nordström spacetime with naked singularity, provided the anomalous magnetic moment is large enough; in that case, the Dirac Hamiltonian is essentially self-adjoint.

  7. 7.

    The addition of a positive cosmological constant [6] has not lead to bound states either.

  8. 8.

    For integrable yet infinitely extended astrophysical Kerr–Newman disc sources, see [33].

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Acknowledgements

Many thanks go to Felix Finster, Jürgen Tolksdorf, and Eberhard Zeidler for their kind invitation to present these results at their superbly organized conference, and for the financial support and the impeccable hospitality offered by the organizers and their staff. We also thank Donald Lynden-Bell and Jonathan Gair for the permission to reproduce their field line drawings (Fig. 1). Finally, we thank the referee for a very careful reading of our paper, and for constructive criticisms.

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  1. Department of Mathematics, Rutgers, The State University of New Jersey, Piscataway, NJ, USA

    Michael K.-H. Kiessling & A. Shadi Tahvildar-Zadeh

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Correspondence to Michael K.-H. Kiessling .

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  1. Universität Regensburg, Fakultät für Mathematik, Regensburg, Germany

    Felix Finster

  2. Regensburg, Germany, Fakultät für Mathematik Universität Rege, Regensburg, Germany

    Johannes Kleiner

  3. Regensburg, Germany, Fakultät für Mathematik Universität Rege, Regensburg, Germany

    Christian Röken

  4. Leipzig, Germany, MPI für Mathematik in den Naturwissensch, Leipzig, Germany

    Jürgen Tolksdorf

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Kiessling, M.KH., Tahvildar-Zadeh, A.S. (2016). Dirac’s Point Electron in the Zero-Gravity Kerr–Newman World. In: Finster, F., Kleiner, J., Röken, C., Tolksdorf, J. (eds) Quantum Mathematical Physics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-26902-3_19

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