Czachor Marek
Wydział Fizyki Technicznej i Matematyki Stosowanej, Politechnika Gdańska, 80-233 Gdańsk, Poland.
Entropy (Basel). 2025 Sep 2;27(9):922. doi: 10.3390/e27090922.
A hierarchical structure of isomorphic arithmetics is defined by a bijection gR:R→R. It entails a hierarchy of probabilistic models, with probabilities pk=gk(p), where is the restriction of gR to the interval [0,1], gk is the th iterate of , and is an arbitrary integer (positive, negative, or zero; g0(x)=x). The relation between and gk(p), k>0, is analogous to the one between probability and neural activation function. For k≪-1, gk(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, pexp, can be given by any , pexp=gk(p). Quantum binary probabilities are defined by g(p)=sin2π2p. With this concrete form of , 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.
同构算术的层次结构由双射(g_R:R→R)定义。它蕴含着一个概率模型的层次结构，其中概率(p_k = g^k(p))，这里(g)是(g_R)在区间([0,1])上的限制，(g^k)是(g)的(k)次迭代，(k)是任意整数（正整数、负整数或零；(g^0(x)=x)）。(g)与(g^k(p))（(k>0)）之间的关系类似于概率与神经激活函数之间的关系。对于(k\ll -1)，(g^k(p))本质上是白噪声（所有过程的概率相等）。选择(k = 0)在物理上与选择空间中一条线的原点一样具有任意性，因此我们所认为的实验二元概率(p_{exp})可以由任何(g^k(p))给出，即(p_{exp}=g^k(p))。量子二元概率由(g(p)=\sin^2\frac{\pi}{2}p)定义。有了(g)的这种具体形式，人们发现层次结构中任意两个相邻层次在量子 - 亚量子关系中相互关联。从这个意义上说，层次结构中的任何模型在适当的算术和微积分中都是概率性量子的。反之亦然：任何模型在适当的算术和微积分中都是亚量子的。涉及两个以上事件的概率通过二元条件概率树来构建。我们从这个角度讨论单态概率和贝尔不等式。我们发现单态概率同时涉及层次结构的三个层次：量子、隐藏和宏观。作为分析的一个副产品，我们发现了富比尼 - 斯图迪测地线距离的一种新的（算术）解释。
