Interior-Boundary Conditions for Many-Body Dirac Operators and Codimension-1 Boundaries

Abstract

We are dealing with boundary conditions for Dirac-type operators, i.e., first order differential operators with matrix-valued coefficients, including in particular physical many-body Dirac operators. We characterize (what we conjecture is) the general form of reflecting boundary conditions (which includes known boundary conditions such as the one of the MIT bag model) and, as our main goal, of interior-boundary conditions (IBCs). IBCs are a new approach to defining UV-regular Hamiltonians for quantum field theories without smearing particles out or discretizing space. For obtaining such Hamiltonians, the method of IBCs provides an alternative to renormalization and has been successfully used so far in non-relativistic models, where it could be applied also in cases in which no renormalization method was known. A natural next question about IBCs is how to set them up for the Dirac equation, and here we take first steps towards the answer. For quantum field theories, the relevant boundary consists of the surfaces in $n$-particle configuration space ${\mathbb{R}}^{3n}$ on which two particles have the same location in ${\mathbb{R}}^3$. While this boundary has codimension 3, we focus here on the more basic situation in which the boundary has codimension 1 in configuration space. We describe specific examples of IBCs for the Dirac equation, we prove for some of these examples that they rigorously define self-adjoint Hamiltonians, and we develop the general form of IBCs for Dirac-type operators.

arXiv comment33 pages LaTeX, no figures; v2: Section 2.4 and Appendix A added

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• Bohmian Mechanics at Space-Time Singularities. I. Timelike Singularities
• Interior-Boundary Conditions for the Dirac Equation at Point Sources in 3 Dimensions
• Hamiltonians for polaron models with subcritical ultraviolet singularities
• Boundary Conditions that Remove Certain Ultraviolet Divergences
• Creation Rate of Dirac Particles at a Point Source