Dirac Waves in the Kerr Geometry: Integral Representations, Mass Oscillation Property and the Hawking Effect

在我们的研究项目中，我们研究了克尔几何中狄拉克波动力学的各个方面，目的是获得霍金效应的新描述。我们构建了一个视界穿透坐标系，它涵盖了外部和内部克尔几何，并且在事件视界和柯西视界上是规则的。在这个坐标系中，我们证明了有质量的狄拉克方程是可分的。此外，我们给出了严格的渐近分析，包括衰减性质。我们还介绍了一类非一致椭圆边值问题Dirac Hamilton算子的自伴扩张的构造方法。这是至关重要的狄拉克哈密顿在克尔几何未能椭圆的地平线。在此基础上，利用预解式方法，我们得到了Kerr几何中有质量Dirac传播子在视界穿透坐标系下的积分表示，它完整地描述了Dirac波在视界外、视界内、视界内直至柯西视界的动力学过程。我们想通过引入克尔几何中有质量狄拉克方程的解空间的一般“质量分解”来扩展“质量振荡性质”的概念，并借助于上述给出一个明确的构造-狄拉克传播子这就产生了一个典型的分解的解决方案空间成两个子空间。利用这些结构，我们打算建立Fock空间，并研究由此产生的多粒子量子态。在非极端Kerr几何的最大扩展中构造的真空态应该与热霍金态一致，直到由于自旋、角动量和引力的相互耦合而进行校正，我们计划对其进行量化。

In our research project, we study various aspects of the dynamics of Dirac waves in the Kerr geometry, with the goal of obtaining a novel description of the Hawking effect. We constructed a horizon-penetrating coordinate system that covers both the exterior and interior Kerr geometry and is regular across the event and Cauchy horizons. In this coordinate system, we showed that the massive Dirac equation is separable. Moreover, we gave a rigorous asymptotic analysis including decay properties. We also introduced a new method for the construction of a self-adjoint extension of the Dirac Hamiltonian for a class of non-uniformly elliptic boundary value problems. This was of paramount importance as the Dirac Hamiltonian in Kerr geometry fails to be elliptic at the horizons. On the basis of the above and a resolvent method, we worked out an integral representation for the massive Dirac propagator in Kerr geometry in horizon-penetrating coordinates, which completely describes the dynamics of Dirac waves outside, across, and inside the event horizon, up to the Cauchy horizon.In the continuation of our research project, we want to extend the notion of the ``mass oscillation property'' by introducing a general ``mass decomposition'' of the solution space of the massive Dirac equation in the Kerr geometry and give an explicit construction with the help of the above-mentioned Dirac propagator. This yields a canonical decomposition of the solution space into two subspaces. Using those structures, we intend to setup Fock spaces and study the resulting many-particle quantum states. The vacuum state constructed in the maximal extension of the non-extreme Kerr geometry should coincide with the thermal Hawking state up to corrections due to mutual couplings of spin, angular momentum and gravity, which we plan to quantify.