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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 9th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added a section ([here](https://ncatlab.org/nlab/show/Dirac+field#FreeDiracFieldInCoulombBackground)) highlighting the remarkable result of * [[Martin Klaus]], [[Günter Scharf]], *The regular external field problem in quantum electrodynamics*, Helv. Phys. Acta **50** (1977) $[$[doi:10.5169/seals-114890](http://doi.org/10.5169/seals-114890)$]$ * [[Alan Carey]], [[Charles Angas Hurst]], [[Denis O'Brien]], *Automorphisms of the canonical anticommutation relations and index theory*, Journal of Functional Analysis **48** 3 (1982) 360-393 $[$<a href="https://doi.org/10.1016/0022-1236(82)90092-1">doi:10.1016/0022-1236(82)90092-1</a>$]$ that the vacua of the free Dirac field in a static Coulomb background are characterized by Fredholm operators whose kernel/cokernel are the Hilbert subspaces of inhabited electron/positron states in the resulting charged vacuum. As far as I am aware, this result has not been followed up on (but please drop me a note if you know of developments). At the same time, this is plausibly the result which is needed in order to actually prove the conjectured [[K-theory classification for topological phases of matter]] (which, despite common impression, is far from established) <a href="https://ncatlab.org/nlab/revision/diff/Dirac+field/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/Dirac+field/13">v13</a>, <a href="https://ncatlab.org/nlab/show/Dirac+field">current</a>

   I have added a section (here) highlighting the remarkable result of

   • Martin Klaus, Günter Scharf, The regular external field problem in quantum electrodynamics, Helv. Phys. Acta 50 (1977) $[$doi:10.5169/seals-114890$]$

   • Alan Carey, Charles Angas Hurst, Denis O’Brien, Automorphisms of the canonical anticommutation relations and index theory, Journal of Functional Analysis 48 3 (1982) 360-393 $[$doi:10.1016/0022-1236(82)90092-1$]$

   that the vacua of the free Dirac field in a static Coulomb background are characterized by Fredholm operators whose kernel/cokernel are the Hilbert subspaces of inhabited electron/positron states in the resulting charged vacuum.

   As far as I am aware, this result has not been followed up on (but please drop me a note if you know of developments). At the same time, this is plausibly the result which is needed in order to actually prove the conjectured K-theory classification for topological phases of matter (which, despite common impression, is far from established)

   diff, v13, current