量子化导致波动：平滑可计算函数是仿射的
Quantization causes waves:Smooth finitely computable functions are affine
摘要：
给定一个自动机（一个字母到字母转换器，$p$-进整数的空间$\mathbb Z_p$上的动态1-Lipschitz系统）$\mathfrak A$，其输入和输出字母是$\mathbb F_p=\{0,1,\ldots,p-1\}$，人们通过实平面$\mathbb R^2$中的点集$\mathbf P(\mathfrak A)$来可视化$\mathfrak A$执行的单词转换。对于有限状态自动机$\mathfrak A$，证明了当$\mathbf P(\mathfrak A)$中的某些点构成$\mathbb R^2$中的光滑曲线（属于一类$C^2$）时，该曲线是一段具有有理斜率的直线，并且只有1000条直线的线段在$\mathbf{P}(\mathfrak A)$中.此外，当识别具有2维环面$\mathbb T^2\subset\mathbb R^3$的子集的$\mathbf P(\mathfrak A)$时（在实单位正方形$[0,1]^2$到$\mathbb T^2$的自然映射下），来自$\mathbf P(\mathfrak A)$的平滑曲线构成环面绕组的集合。在圆柱坐标系中，任一绕组都可以归因于复值函数$ψ(x)=e^{i(Ax-2πB(t))}$ $(x\in\mathbb R)$，以获得合适的有理$A,B(t)$。由于$ψ(x)$是量子理论中物质波的标准表达式（其中$B(t)=tB(t_0)$），并且由于换能器可以被看作是因果离散系统的数学形式化，因此本文可以作为量子系统中波动现象固有的数学推理：这是因为因果原理和物质的离散性。
英文摘要：
Given an automaton (a letter-to-letter transducer, a dynamical 1-Lipschitz system on the space $\mathbb Z_p$ of $p$-adic integers) $\mathfrak A$ whose input and output alphabets are $\mathbb F_p=\{0,1,\ldots,p-1\}$, one visualizes word transformations performed by $\mathfrak A$ by a point set $\mathbf P(\mathfrak A)$ in real plane $\mathbb R^2$. For a finite-state automaton $\mathfrak A$, it is shown that once some points of $\mathbf P(\mathfrak A)$ constitute a smooth (of a class $C^2$) curve in $\mathbb R^2$, the curve is a segment of a straight line with a rational slope; and there are only finitely many straight lines whose segments are in $\mathbf{P}(\mathfrak A)$. Moreover, when identifying $\mathbf P(\mathfrak A)$ with a subset of a 2-dimensional torus $\mathbb T^2\subset\mathbb R^3$ (under a natural mapping of the real unit square $[0,1]^2$ onto $\mathbb T^2$) the smooth curves from $\mathbf P(\mathfrak A)$ constitute a collection of torus windings. In cylindrical coordinates either of the windings can be ascribed to a complex-valued function $ψ(x)=e^{i(Ax-2πB(t))}$ $(x\in\mathbb R)$ for suitable rational $A,B(t)$. Since $ψ(x)$ is a standard expression for a matter wave in quantum theory (where $B(t)=tB(t_0)$), and since transducers can be regarded as a mathematical formalization for causal discrete systems, the paper might serve as a mathematical reasoning why wave phenomena are inherent in quantum systems: This is because of causality principle and the discreteness of matter.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
