Random Vector Representation of Continuous Functions and Its Applica-tions in Quantum Mechanics

TL;DR

This paper establishes a novel connection between continuous functions and random vectors, providing a new perspective for function approximation and unifying classical and quantum mechanics through probabilistic representations.

Contribution

It introduces the random vector representation of continuous functions and applies this framework to unify classical and quantum mechanics, revealing wave characteristics of macroscopic objects.

Findings

Random vectors can uniformly approximate continuous functions.

The approach unifies classical and quantum mechanics.

Mass point motion exhibits wave-like properties, termed wave-mass-point duality.

Abstract

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.