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

Abstract

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

24 pages, 0 figures. Changes to v2: Corrected introductory formula for H_I. v3: Corrected Eq. (35). v4: Corrected Eqs. (26--29, 60, 94, 95); definition of ψ' in Section 8; n to m+1 in Section 6, 2nd paragraph; m to n in Section 6, 4th paragraph, L^2 formulas; X_{N+1} to X_N above Eq. (100)