半相对论量子物理中的非线性 PDE 模型
Nonlinear PDE models in semi-relativistic quantum physics
AI总结：
本文提出了自洽 Pauli 方程作为带电自旋 1/2 粒子的半相对论模型，探讨了其作为 N 体 Pauli 方程平均场极限的合理性，并通过 Wigner 方法推导了其向 Vlasov-Poisson 方程的半经典极限。
摘要：
我们提出了自洽 Pauli 方程，这是一个用于描述带电自旋 1/2 粒子与电磁场存在自相互作用的半相对论模型。Pauli 方程是作为相对论 Dirac 方程的 O(1/c) 近似而出现的。完全相对论的自洽模型是 Dirac-Maxwell 方程，其中自旋和磁场的描述是自然产生的。在非相对论设定下，正确的自洽方程是 Schrödinger-Poisson 方程，它不描述自旋和磁场，且其自相互作用仅存在于电场中。Schrödinger-Poisson 方程也是带有 Coulomb 相互作用的 N 体 Schrödinger 方程的平均场极限。我们提出，Pauli-Poisson 方程是作为带有 Coulomb 相互作用的线性 N 体 Pauli 方程在平均场极限 N → ∞ 下出现的，此时必须额外关注 Pauli 方程的费米子性质。我们通过 Wigner 方法展示了 Pauli-Poisson 方程向带有 Lorentz 力并耦合到 Poisson 方程的 Vlasov 方程的半经典极限，这也与自洽 Vlasov 方程在 1/c 层级上保持一致。这是对 Lions & Paul 以及 Markowich & Mauser 开创性工作的非平凡推广，其中我们需要诸如磁性 Lieb-Thirring 估计等方法。
英文摘要：
We present the self-consistent Pauli equation, a semi-relativistic model for charged spin-$1/2$-particles with self-interaction with the electromagnetic field. The Pauli equation arises as the $O(1/c)$ approximation of the relativistic Dirac equation. The fully relativistic self-consistent model is the Dirac-Maxwell equation where the description of spin and the magnetic field arises naturally. In the non-relativistic setting the correct self-consistent equation is the Schrödinger-Poisson equation which does not describe spin and the magnetic field and where the self-interaction is with the electric field only. The Schrödinger-Poisson equation also arises as the mean field limit of the $N$-body Schrödinger equation with Coulomb interaction. We propose that the Pauli-Poisson equation arises as the mean field limit $N \rightarrow \infty$ of the linear $N$-body Pauli equation with Coulomb interaction where one has to pay extra attention to the fermionic nature of the Pauli equation. We present the semiclassical limit of the Pauli-Poisson equation by the Wigner method to the Vlasov equation with Lorentz force coupled to the Poisson equation which is also consistent with the hierarchy in $1/c$ of the self-consistent Vlasov equation. This is a non-trivial extension of the groundbreaking works by Lions & Paul and Markowich & Mauser, where we need methods like magnetic Lieb-Thirring estimates.
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