  We show that a simplified version of the Dirac interaction operator given by
$\hat H_\mathrm{I} \propto \int d\mathbf{k}d\mathbf{p}(\hat a(\mathbf{k}) +
\hat a^\dagger(-\mathbf{k})) \hat b^\dagger(\mathbf{p} + \mathbf{k}) \hat
b(\mathbf{p})/\sqrt{|\mathbf{k}|}$ is self-adjoint on a certain domain that is
dense in the Hilbert space, even without any cutoffs. The technique that we use
for showing this can potentially be extended to a much wider range of operators
as well. This technique might therefore potentially lead to more mathematically
well-defined theories of QFT in the future.
