论文标题
dirac方程在点源的内部边界条件在3个维度
Interior-Boundary Conditions for the Dirac Equation at Point Sources in 3 Dimensions
论文作者
论文摘要
最近提出的一种避免哈密顿量与粒子产生的紫外线差异的方法是基于内部边界条件（IBC）。该方法在非权利主义情况下，即对于拉普拉斯运营商而言。在这里，我们研究了如何将方法应用于Dirac运营商。尽管这已经在1个空间维度上成功完成，更一般而言，对于Codimension-1界限，在3个维度中的点源的情况对应于Codimension-3边界。人们希望，对于这样的边界，狄拉克操作员不允许在边界条件下，因为已知它们不允许在3D中进行点相互作用，这也对应于边界条件。实际上，我们在这里证实了这一期望，证明了（截断的）fock空间上没有自偶会操作员，该操作员将与具有IBC的Dirac运算符相对应，该操作员具有IBC的配置。但是，我们还提出了一个积极的结果，表明有IBC的自动伴侣运算符（在边界上（由带有粒子的构型组成的边界上），它们远离那些配置，由狄拉克运算符以及足够强的库仑电位给出。
A recently proposed approach for avoiding the ultraviolet divergence of Hamiltonians with particle creation is based on interior-boundary conditions (IBCs). The approach works well in the non-relativistic case, that is, for the Laplacian operator. Here, we study how the approach can be applied to Dirac operators. While this has been done successfully already in 1 space dimension, and more generally for codimension-1 boundaries, the situation of point sources in 3 dimensions corresponds to a codimension-3 boundary. One would expect that, for such a boundary, Dirac operators do not allow for boundary conditions because they are known not to allow for point interactions in 3d, which also correspond to a boundary condition. And indeed, we confirm this expectation here by proving that there is no self-adjoint operator on (a truncated) Fock space that would correspond to a Dirac operator with an IBC at configurations with a particle at the origin. However, we also present a positive result showing that there are self-adjoint operators with IBC (on the boundary consisting of configurations with a particle at the origin) that are, away from those configurations, given by a Dirac operator plus a sufficiently strong Coulomb potential.
