从二元序列中涌现出的空间、时间与光速

中文摘要 AI 辅助

我们证明，狭义相对论的完整运动学结构可从长度为$n$的二元序列对出发，通过逐位异或（XOR）这一单一操作涌现产生。物理事件是一对编码时钟与量尺信息的二元序列，它们的异或差值定义了与观测者无关的世界线映射。我们赋予生成时空变换的计数（时钟位与量尺位）相等的信息权重，证明闵可夫斯基间隔、庞加莱对称性以及色散关系$E^2-p^2=m^2$可直接由此导出。光速的有限性源于世界线的时间排序要求，而其普适性与参考系不变性则源于$\boldsymbol{\text{Z}_2}$算术，并非独立公设。有限序列长度$n$意味着离散的普朗克尺度时空，且具有精确的洛伦兹不变性；当$n\to\boldsymbol{\text{Z}_\text{Z}}$时，连续统极限可恢复标准狭义相对论。结合文献[\text{Powers:2021rfg}]的前期工作，可见二元序列框架为物质与时空提供了单一基底：粒子是具有确定自旋量子数$(j,m_j)$的序列，而时空位移是序列间的映射。该框架产生可检验的预测：洛伦兹快度取离散值，长度为$n$的单帧间映射实现的洛伦兹因子最大仅为$\boldsymbol{\text{γ}_\text{max}=O(\boldsymbol{\text{Z}_\text{Z}})}$，而在连续统极限下可恢复任意大的$\boldsymbol{\text{γ}}$。最引人注目的是，处于信息容量最大值的有限系统（如黑洞、德西特空间）必须呈现有限$n$修正。这些结果表明，空间与时间可能并非基本范畴，而是从离散二元信息中涌现而来。

英文摘要

We construct an information theory framework in which the fundamental objects are binary sequences of length $n$, equipped with the bitwise XOR operation. The only physical observables are counts of XOR-generated symbol classes, while the exact locations of symbols are inaccessible. Averaging over those locations ultimately yields the Minkowski interval as an invariant object that maximizes (information) entropy. Correlations between two binary sequences are base-4 sequences that we label as ``events'', and events are connected with maps. The entire kinematic structure of Special Relativity is recovered under minimal assumptions, i.e. counts that represent space and time increments carry equal informational weight. The central claim is that Lorentz symmetry is the typical large-n behaviour of XOR counting. At finite $n$ the framework yields a discrete rapidity spectrum, a bound $γ_{\max} = O(\sqrt{n})$, and interval fluctuations of relative size $O(n^{-1/2})$, with standard special relativity recovered as $n \to \infty$. However, the light cone and one null coordinate are exact for every microscopic configuration, so the symmetry group is undeformed and dispersion relations are unmodified. Thus, the theory, though discrete, implies no Lorentz violation of the standard phenomenological kind. Ultra-high-energy cosmic rays already require $n \gtrsim 10^{23}$; interferometry excludes the variant in which $n$ scales linearly with system size, leaving a holographic area law. The most striking prediction of the framework concerns systems at the maximum of their information capacity, where $n$ is necessarily finite: black holes and de Sitter space. There the corrections are of order one within a Planck proper length of the horizon, regardless of the horizon's size, and the spacetime description fails altogether at the endpoint of black-hole evaporation, where $n$ itself is of order unity.

发表机构

• HEPCOS, Department of Physics, SUNY at Buffalo（纽约州立大学布法罗分校物理系高能粒子与弦理论中心）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

calculations and presentation updated to emphasis that Lorentz symmetry is the typical large-$n$ behaviour of XOR counting