论文标题
走向Madelung方程的数学理论
Towards a mathematical Theory of the Madelung Equations
论文作者
论文摘要
即使Madelung方程对于Bohmian和随机力学等量子力学基础的许多“经典”方法至关重要，但迄今为止，对于这种部分微分方程系统，尚未开发出一致的数学理论。沃尔斯特罗姆（Wallstrom）突出提出了针对马德隆方程的反对意见，目的是表明，没有这样的理论在该方程式中，并且在没有施加额外的“ Ad Hoc量化条件”的情况下恢复了Schrödinger方程，就像Takabayasi提出的那样。我们作品的主要目的是阐明沃尔斯特罗姆的反对意义是合理的，而在哪种意义上没有意义，并观察到现有文献。我们发现，尽管需要更多的数学研究，但在上述意义上可能会构建Madelung方程的数学理论。
Even though the Madelung equations are central to many 'classical' approaches to the foundations of quantum mechanics such as Bohmian and stochastic mechanics, no coherent mathematical theory has been developed so far for this system of partial differential equations. Wallstrom prominently raised objections against the Madelung equations, aiming to show that no such theory exists in which the system is well-posed and in which the Schrödinger equation is recovered without the imposition of an additional 'ad hoc quantization condition'--like the one proposed by Takabayasi. The primary objective of our work is to clarify in which sense Wallstrom's objections are justified and in which sense they are not, with a view on the existing literature. We find that it may be possible to construct a mathematical theory of the Madelung equations which is satisfactory in the aforementioned sense, though more mathematical research is required.
