Quantization causes waves:Smooth finitely computable functions are affine

Abstract

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 $\psi(x)=e^{i(Ax-2\pi B(t))}$ $(x\in\mathbb R)$ for suitable rational $A,B(t)$. Since $\psi(x)$ 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.