Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman–von Neumann’s Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.

中文翻译：

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关于普朗克尺度物理引起的经典力学变形

几个量子引力和弦理论思想实验表明，海森堡不确定性关系在普朗克尺度上得到了修改，因此确实出现了最小长度。这种修改可能意味着对规范对易关系的修改，因此也意味着对普朗克尺度的量子力学的修改。经典力学的相应修改通常通过用泊松括号替换修改过的量子交换器来考虑，泊松括号以保留其主要属性（反对称、线性、莱布尼茨规则和雅可比恒等式）的方式适当修改。我们表明存在另一种有趣的可能性。正如 Sudarshan 所说，Koopman-von Neumann 的经典力学希尔伯特空间公式允许将经典力学视为隐变量量子系统。然后，这种量子系统的普朗克尺度修改自然会引起经典衬底中动力学的相应修改。有趣的是，这种诱导修改似乎实际上破坏了经典性：经典的位置和动量算子不再是对易的，隐藏变量确实出现在它们的演化方程中。