$p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

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Vladimir Anashin

Quantization causes waves:
Smooth finitely computable functions are affine


In the talk, we mathematically deduce wave functions of particles from only two principles, the discreteness of matter and causality. Namely, any discrete causal system can be considered as an automaton. Every automaton (a letter-to-letter transducer) $\mathfrak A$ whose both input and output alphabets are $\mathbb F_p=\{0,1,\ldots,p-1\}$ produces a map $f_\mathfrak A$ from the space $\mathbb Z_p$ of $p$-adic integers to $\mathbb Z_p$. The map $f_\mathfrak A$ satisfies Lipschitz condition with a constant 1 w.r.t. $p$-adic metric on $\mathbb Z_p$ and can naturally be plotted into a unit real square $\mathbb I^2\subset\mathbb R^2$ as follows: To an $m$-letter non-empty word $v=\gamma_{m-1}\gamma_{m-2}\ldots\gamma_0$ there corresponds a number $0.v\in\mathbb R$ with base-$p$ expansion $0.\gamma_{m-1}\gamma_{m-2}\ldots\gamma_0$; so to every $m$-letter input word $w=\alpha_{m-1}\alpha_{m-2}\cdots\alpha _0$ of $\mathfrak A$ and to the respective $m$-letter output word $\mathfrak a(w)=\beta_{m-1}\beta_{m-2}\cdots \beta_0$ of $\mathfrak A$ there corresponds a point $(0.w;0.{\mathfrak a(w)})\in\mathbb R^2$. Denote $\mathbf P(\mathfrak A)$ a closure in $\mathbb R^2$ of the point set $(0.w;0.{\mathfrak a(w)})$ where $w$ ranges through all non-empty words over the alphabet $\mathbb F_p$. For a finite-state automaton $\mathfrak A$ we prove that once some points of $\mathbf P(\mathfrak A)$ constitute a $C^2$-smooth curve in $\mathbb R^2$, the curve is a segment of a straight line with a rational slope. Moreover, when identifying $\mathbf P(\mathfrak A)$ with a subset of a 2-dimensional unit torus $\mathbb T^2\subset\mathbb R^3$ (under a natural mapping of the unit square $\mathbb I^2$ onto the surface of the torus $\mathbb T^2$), the smooth curves from $\mathbf P(\mathfrak A)$ constitute a collection of torus windings which can be ascribed to complex-valued functions $\psi(x,t)=e^{i(Ax-2\pi Bt)}$ $(x,t\in\mathbb R)$. From here under assumption on the existence of the smallest time interval (e.g., of Plank time) we derive wave function of a particle. Moreover, we show that then necessarily $p=2$ and therefore input/output words of automata may be treated as sequences of bits; thus the results may be treated as a contribution to the informational interpretation of quantum theory, namely, as an argument in favour of J.~A.~Wheeler's `it from bit' doctrine. The main result remains true for automata with multiple inputs/outputs thus naturally leading to Hilbert space-like formalism of quantum mechanics. As automata are causal discrete systems, the results may serve a sort of mathematical reasoning why wave phenomena are inherent in quantum systems: This is just because of causality principle and discreteness of matter. The talk mainly follows author's paper (of the same title) recently published in \emph{$p$-adic Numbers, Ultrametric Analysis and Applications}, 2015, Vol. 7, No. 3, pp. 169--227.