  A recently proposed approach for avoiding the ultraviolet divergence of
Hamiltonians with particle creation is based on interior-boundary conditions
(IBCs). The approach works well in the non-relativistic case, that is, for the
Laplacian operator. Here, we study how the approach can be applied to Dirac
operators. While this has been done successfully already in 1 space dimension,
and more generally for codimension-1 boundaries, the situation of point sources
in 3 dimensions corresponds to a codimension-3 boundary. One would expect that,
for such a boundary, Dirac operators do not allow for boundary conditions
because they are known not to allow for point interactions in 3d, which also
correspond to a boundary condition. And indeed, we confirm this expectation
here by proving that there is no self-adjoint operator on (a truncated) Fock
space that would correspond to a Dirac operator with an IBC at configurations
with a particle at the origin. However, we also present a positive result
showing that there are self-adjoint operators with IBC (on the boundary
consisting of configurations with a particle at the origin) that are, away from
those configurations, given by a Dirac operator plus a sufficiently strong
Coulomb potential.
