连续函数的随机向量表示及其在量子力学中的应用
Random Vector Representation of Continuous Functions and Its Applica-tions in Quantum Mechanics
摘要：
本文揭示了连续函数与随机向量之间的关系，其主要含义是：对于任何给定的连续函数，必有一个概率空间序列和一个随机向量序列，其中每个随机向量都定义在其中一个概率空间上，使得由随机向量形成的条件数学期望的序列一致地收敛到连续函数。这是连续函数的随机向量表示，它被认为是在实函数论和概率论之间架起的一座桥梁。利用这一结论，可以得到关于函数逼近论的一个有趣的结果.连续函数的随机向量表示在物理学中有重要的应用。基于这一结论，如果大部分确定性现象可以用连续函数描述，而随机现象也可以用随机变量或向量描述，那么任何确定性现象都必然是随机现象序列的极限状态。然后，在随机向量序列到连续函数的逼近中，通过适当地选择基函数，得到了量子力学的一个重要结论：经典力学和量子力学是统一的。特别是引入了一个有趣而又非常重要的结论，即宏观物体的质点运动具有一种波特征曲线，称为波-质-点对偶性。
英文摘要：
The relation between continuous functions and random vectors is revealed in the paper that the main meaning is described as, for any given continuous function, there must be a sequence of probability spaces and a sequence of random vectors where every random vector is defined on one of these probability spaces, such that the sequence of conditional mathematical expectations formed by the random vectors uniformly converges to the continuous function. This is random vector representation of continuous functions, which is regarded as a bridge to be set up between real function theory and probability theory. By means of this conclusion, an interesting result about function approximation theory can be got. The random vectors representation of continuous functionsis is of important applications in physics. Based on the conclusion, if a large proportion of certainty phenomena can be described by continuous functions and random phenomenon can also be described by random variables or vectors, then any certainty phenomenon must be the limit state of a sequence of random phenomena. And then, in the approximation from a sequence of random vectors to a continuous function, the base functions are appropriately selected by us, an important conclusion for quantum mechanics is deduced: classical mechanics and quantum mechanics are unified. Particularly, an interesting and very important conclusion is introduced as the fact that the mass point motion of a macroscopical object possesses a kind of wave characteristic curve, which is called wave-mass-point duality.
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