Quantization causes waves:Smooth finitely computable functions are affine

TL;DR

This paper demonstrates that smooth finitely computable functions in automata theory are necessarily affine, revealing a connection between quantization, wave phenomena, and automata behavior, with implications for quantum systems and causality.

Contribution

It proves that smooth curves in automaton-generated point sets are affine lines with rational slopes, linking automata, wave phenomena, and quantum theory.

Findings

Smooth curves are segments of straight lines with rational slopes.

Automaton point sets on a torus form torus windings.

Wave-like behaviors emerge from automata due to quantization and causality.

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.