On the Relativity of Quantumness as Implied by Relativity of Arithmetic and Probability

TL;DR

The work develops a hierarchical arithmetic framework in which a bijection $g_{\mathbb{R}}$ induces a family of isomorphic arithmetics and a corresponding ladder of probabilities $p_k=g^{k}(p)$, enabling a relativity of quantumness across levels. A concrete choice $g(p)=\sin^2(\frac{\pi}{2}p)$ yields a quantum level at $k=1$ that interrelates with hidden and macroscopic levels via generalized product rules, trees, and non-Newtonian calculus. Singlet-state probabilities admit a local-hidden-variable representation within this hierarchy and the Fubini–Study geodesic distance appears as a natural hidden variable connecting geometric and arithmetic structures. The framework also develops non-Newtonian integration, hierarchical laws of large numbers, and an effective truncation to a finite band of levels, suggesting new perspectives on Bell-type results and the relativity of quantumness, with open questions about dynamics and broader implications.

Abstract

A hierarchical structure of isomorphic arithmetics is defined by a bijection $g_\mathbb{R}:\mathbb{R}\to \mathbb{R}$. It entails a hierarchy of probabilistic models, with probabilities $p_k=g^k(p)$, where $g$ is the restriction of $g_\mathbb{R}$ to the interval $[0,1]$, $g^k$ is the $k$th iterate of $g$, and $k$ is an arbitrary integer (positive, negative, or zero; $g^0(x)=x$). The relation between $p$ and $g^k(p)$, $k>0$, is analogous to the one between probability and neural activation function. For \mbox{$k\ll -1$}, $g^k(p)$ is essentially white noise (all processes are equally probable). The choice of $k=0$ is physically as arbitrary as the choice of origin of a line in space, hence what we regard as experimental binary probabilities, $p_{\rm exp}$, can be given by any $k$, $p_{\rm exp}=g^k(p)$. Quantum binary probabilities are defined by $g(p)=\sin^2\fracπ{2}p$. With this concrete form of $g$, one finds that any two neighboring levels of the hierarchy are related to each other in a quantum--subquantum relation. In this sense, any model in the hierarchy is probabilistically quantum in appropriate arithmetic and calculus. And the other way around: any model is subquantum in appropriate arithmetic and calculus. Probabilities involving more than two events are constructed by means of trees of binary conditional probabilities. We discuss from this perspective singlet-state probabilities and Bell inequalities. We find that singlet state probabilities involve simultaneously three levels of the hierarchy: quantum, hidden, and macroscopic. As a by-product of the analysis, we discover a new (arithmetic) interpretation of the Fubini--Study geodesic distance.

Figures (4)

• Figure 1: The relation between $\alpha$ and $\theta$ as given by (\ref{['41']}). There are three fixed points: $\alpha(0)=0$, $\alpha(\pi/2)=\pi/2$, $\alpha(\pi)=\pi$. Here, $\alpha$ is the geometric angle between the two Stern--Gerlach devices, whereas $\theta$ is a hidden parameter.
• Figure 2: A total of 1, 2, 5 and 15 iterations of $g(p)=\sin^2\frac{\pi}{2}p$ (upper plots). All the curves cross at $p=1/2$ and are of the sigmoidal form, analogously to activation functions occurring in learning algorithms. Is it just a coincidence, or are there deeper connections to the problem of measurement, learning, or consciousness? Iterates $g^k$ with $k>15$ are practically indistinguishable within the precision of the plot: They all look like the red step function. An analogous phenomenon occurs for the negative iterates: $k=-1,-2,-5,-15$, but here almost all events described by $g^{-15}(p)$ are equally probable, hence indistinguishable for level-0 observers (lower plots). Effectively, even though the number of levels is infinite, the distinguishable ones are restricted to a finite "band" $k_{\rm min}\le k\le k_{\rm max}$. Of course, the Copernican aspect of the hierarchy means that the same happens in a neighborhood of any $l$, and not only $l=0$ depicted here.
• Figure 3: The upper diagram: An $\mathbb{R}_l$-valued branch of a binary tree of conditional probabilities. This is how one can include events with more results than just two. Assuming independent events and the same value of all $k_j$ (the lower diagram), we can derive a hierarchical analog of the Bernoulli law of large numbers. Laws of large numbers are the places where theory and experiment meet.
• Figure 4: Hierarchical law of large numbers in action. Upper bound on probability of disagreement between theory and experiment in $N$ tosses of a symmetric coin for four different arithmetics $\mathbb{R}_l$ of the observer. Plot of the right-hand side of (\ref{['139""']}) with $\varepsilon=0.1$, for the four iterates $g^l$, $l=1,2,3,4$, of $g(x)=\sin^2\frac{\pi}{2}x$. The number of coin tosses $25\le N\le 75$. Plots are made in the arithmetic $\mathbb{R}_0$, implicitly assumed in Wolfram Mathematica 14.

Theorems & Definitions (11)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• proof
• Lemma 6
• Lemma 7
• proof
• Lemma 8