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We are dealing with boundary conditions for Dirac-type operators, i.e., firstorder differential operators with matrix-valued coefficients, including inparticular physical many-body Dirac operators. We characterize (what weconjecture is) the general form of reflecting boundary conditions (whichincludes 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 newapproach to defining UV-regular Hamiltonians for quantum field theories withoutsmearing particles out or discretizing space. For obtaining such Hamiltonians,the method of IBCs provides an alternative to renormalization and has beensuccessfully used so far in non-relativistic models, where it could be appliedalso in cases in which no renormalization method was known. A natural nextquestion about IBCs is how to set them up for the Dirac equation, and here wetake first steps towards the answer. For quantum field theories, the relevantboundary 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 themore basic situation in which the boundary has codimension 1 in configurationspace. We describe specific examples of IBCs for the Dirac equation, we provefor some of these examples that they rigorously define self-adjointHamiltonians, and we develop the general form of IBCs for Dirac-type operators.

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