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Diffraction for the Dirac–Coulomb Propagator

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Abstract

The Dirac equation in \(\mathbb {R}^{1,3}\) with potential \({\textsf{Z}}/r\) is a relativistic field equation modeling the hydrogen atom. We analyze the singularity structure of the propagator for this equation, showing that the singularities of the Schwartz kernel of the propagator are along an expanding spherical wave away from rays that miss the potential singularity at the origin, but also may include an additional spherical wave of diffracted singularities emanating from the origin. This diffracted wavefront is \(1-{\epsilon }\) derivatives smoother than the main singularities, for all \({\epsilon }>0,\) and is a conormal singularity.

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Notes

  1. Readers consulting other references should be aware that there are at least two conventions in the literature. Indeed, many physics texts (e.g., Akhiezer and Berestetsky [1] and Rose [37]) ask that the gamma matrices satisfy a Riemannian anticommutation relation and then set \(x_{0} = ict\).

  2. More generally, we remark that we can replace the smooth term by a term that is smooth on the blowup of the origin with no change in the arguments of this section.

  3. Recall that \({\textrm{b}}\)-Sobolev spaces are by default defined with respect to the b-density rather than the metric density.

  4. We assume our quantization is arranged so that it yields properly supported operators.

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Authors and Affiliations

  1. Texas A & M University, College Station, United States

    Dean Baskin

  2. Northwestern University, Evanston, United States

    Jared Wunsch

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  1. Dean Baskin
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  2. Jared Wunsch
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Correspondence to Jared Wunsch.

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Communicated by Jan Derezinski.

Publisher's Note

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The authors are grateful to Christian Gérard and Michał Wrochna for suggesting the problem and providing helpful insight into its importance, as well as for helpful comments on an early version of the manuscript. They are also grateful to Richard Melrose, András Vasy, and especially Oran Gannot for many helpful conversations. Two anonymous referees made a number of corrections and suggestions that improved the presentation. The research for this paper began during a Research in Paris stay at the Institut Henri Poincaré. Part of this material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2019 semester. DB was supported by NSF CAREER grant DMS-1654056. JW was supported by NSF grant DMS-1600023 and Simons Foundation Grant 631302.

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Baskin, D., Wunsch, J. Diffraction for the Dirac–Coulomb Propagator. Ann. Henri Poincaré 24, 2607–2659 (2023). https://doi.org/10.1007/s00023-023-01279-0

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