发表机构
University of Texas Rio Grande Valley School of Mathematical & Statistical Sciences（德克萨斯大学里奥格兰德河谷分校数学与统计科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在量子力学框架下提出新型p进狄拉克方程，离散化得到两种连续时间量子行走，证明其转移概率符合随机矩阵性质，可作为具相对论型内部自由度的量子网络基础，为相关领域提供新方向。
AI 中文摘要
我们在量子力学的标准公理框架中引入了一类新型p进狄拉克方程，其中普通空间导数被由任意可积核构建的非局域算子取代。我们在动量空间中对角化所得的自由狄拉克哈密顿量，构造其平面波解，确定其能谱，并建立了联系粒子与反粒子部分的p进电荷共轭对称性。随后，我们以两种方式离散化该自由方程，两种方式均得到真正的连续时间量子行走，而非现有文献中占主导地位的离散时间硬币型行走：第一种是在基础p进空间的可数覆盖上的构造，第二种是在有限树状图上的更明确构造，对于后者，我们证明当结合波函数的内部（粒子/反粒子）分量时，转移概率在每一个时间瞬间都构成一组真正的、适当归一化的转移概率；换言之，忽略行走的内部结构，其行为完全如同该图上的普通随机行走。基于这一随机矩阵性质，我们讨论了所得构造如何可作为具有真正相对论型内部自由度的量子网络的基础，补充了早期的非相对论p进量子神经网络。据我们所知，这是首个自由动力学与分层图上的狄拉克方程完全一致的连续时间量子行走。最后，我们讨论了该构造提出的开放数学与计算问题。
英文摘要
We introduce a new class of p-adic Dirac equations, formulated in the standard axiomatic framework of quantum mechanics, in which the ordinary spatial derivatives are replaced by non-local operators built from arbitrary integrable kernels. We diagonalize the resulting free Dirac Hamiltonian in momentum space, construct its plane-wave solutions, determine its spectrum, and establish a p-adic charge-conjugation symmetry relating particle and antiparticle sectors. We then discretize the free equation in two ways, both giving genuine continuous-time quantum walks rather than the discrete-time, coined walks that dominate the existing literature: a first construction on a countable covering of the underlying p-adic space, and a second, more explicit construction on a finite, tree-structured graph, for which we prove that the transition probabilities, once the internal (particle/antiparticle) components of the wavefunction are combined, form a genuine, properly normalized set of transition probabilities at every instant of time; in other words, ignoring the internal structure of the walk, it behaves exactly like an ordinary random walk on that graph. Building on this stochastic-matrix property, we discuss how the resulting construction can serve as the foundation of a quantum network with genuinely relativistic-type internal degrees of freedom, complementing earlier, non-relativistic p-adic quantum neural networks. To the best of our knowledge, this is the first continuous-time quantum walk whose free dynamics coincides exactly with a Dirac equation on a hierarchical graph. We close with a discussion of the open mathematical and computational problems raised by this construction.
