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Vladimir Anashin
Quantization causes waves: Smooth finitely computable functions are affine
Abstract
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 lettertoletter transducer)
$\mathfrak A$ whose both input and output alphabets are $\mathbb F_p=\{0,1,\ldots,p1\}$
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 nonempty word $v=\gamma_{m1}\gamma_{m2}\ldots\gamma_0$
there corresponds a number $0.v\in\mathbb R$ with base$p$ expansion
$0.\gamma_{m1}\gamma_{m2}\ldots\gamma_0$; so
to every $m$letter input word $w=\alpha_{m1}\alpha_{m2}\cdots\alpha _0$ of $\mathfrak
A$ and to the
respective $m$letter output word $\mathfrak a(w)=\beta_{m1}\beta_{m2}\cdots \beta_0$
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 nonempty words over the alphabet $\mathbb F_p$.
For a finitestate 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 2dimensional 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 complexvalued functions $\psi(x,t)=e^{i(Ax2\pi Bt)}$ $(x,t\in\mathbb
R)$. 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 spacelike 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. 169227.
