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

Mads J. Damgaard

Abstract

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

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 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.
Paper Structure (12 sections, 1 theorem, 133 equations)

This paper contains 12 sections, 1 theorem, 133 equations.

Table of Contents

  1. Introduction
  2. A simplified Dirac interaction operator
  3. A domain on which $\hat{H}_\mathrm{I}$ is self-adjoint
  4. Introducing a set $W \subset V$
  5. Showing that $W$ is a subset of $U$
  6. The density of $\mathrm{Dom}(\hat{H}_\mathrm{I})$ in $\mathbf{H}$
  7. The symmetry of $\hat{H}_\mathrm{I}$
  8. The self-adjointness of $\hat{H}_\mathrm{I}$
  9. Conclusion
  10. Symmetrizing the Hilbert space
  11. Adding a free energy to the operator
  12. Handling the perturbed vacuum

Key Result

Proposition 1

Suppose that $\hat{H}_\mathrm{I}$ and $\mathrm{Dom}(\hat{H}_\mathrm{I})$ are defined as above. Then $\mathrm{Dom}(\hat{H}_\mathrm{I})$ is dense in $\mathbf{H}$ and $\hat{H}_\mathrm{I}$ is self-adjoint.

Theorems & Definitions (1)

  • Proposition 1