关于波函数的几何假设及其产生动力学
On the geometrical hypotheses underlying wave functions and their emerging dynamics
摘要：
单个物理系统的经典力学和非相对论粒子的量子力学被证明是用两个流形之间的映射描述的广义动力学的例外情况，源是位形空间。目标空间被认为是二维的和平坦的，它们的正交方向的物理解释。映射方程中的所有项在目标空间中具有几何意义，并且其旋转Killing 1-形式的拉回允许在位形空间中构造速度场。用源空间中的切向量来识别这个速度场，就得到了运动的动力学定律。对于映射方程中存在的任意标量函数的特定选择，并且使用目标空间中的笛卡尔坐标，映射方程变为线性的并且可以简化为薛定谔方程。我们将目标空间的二维性与量子力学的基本非定域性联系起来。这里提出的框架的许多扩展是直接的，其深刻的后果还有待探索。
英文摘要：
Classical mechanics for individual physical systems and quantum mechanics of non-relativistic particles are shown to be exceptional cases of a generalized dynamics described in terms of maps between two manifolds, the source being configuration space. The target space is argued to be 2-dimensional and flat, and their orthogonal directions are physically interpreted. All terms in the map equation have a geometrical meaning in the target space, and the pull-back of its rotational Killing one-form allows the construction of a velocity field in configuration space. Identification of this velocity field with tangent vectors in the source space leads to the dynamical law of motion. For a specific choice of an arbitrary scalar function present in the map equation, and using Cartesian coordinates in the target space, the map equation becomes linear and can be reduced to the Schrödinger equation. We link the bi-dimensionality of the target space with the essential non-locality of quantum mechanics. Many extensions of the framework here presented are immediate, with deep consequences yet to be explored.
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