Self-adjointness of a simplified Dirac interaction operator without any cutoffs

We show that a simplified version of the Dirac interaction operator given by ${\widehat{H}}_{\mathrm{I}}\propto \int d\mathbf{k}d\mathbf{p}(\widehat{a}(\mathbf{k})+{\widehat{a}}^{†}(-\mathbf{k})){\widehat{b}}^{†}(\mathbf{p}+\mathbf{k})\widehat{b}(\mathbf{p})\mathrm{/}\sqrt{\mathrm{\mid }\mathbf{k}\mathrm{\mid }}$ 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.