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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)
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