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W. A. Zúñiga-Galindo

Publications and source records attributed to W. A. Zúñiga-Galindo.

At least 19 recordsLinked to original sources
arXiv2026

p-Adic Dirac Equations, Continuous-Time Quantum Walks, and Quantum Networks

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.

quant-ph↗
arXiv2026

Continuous-Time Quantum Markov Chains And Discretizations Of p-Adic Schrödinger Equations: Comparisons And Simulations

The continuous-time quantum walks (CTQWs) are a fundamental tool in the development of quantum algorithms. Recently, it was shown that discretizations of p-adic Schrödinger equations give rise to continuous-time quantum Markov chains (CTQMCs); this type of Markov chain includes the CTQWs constructed using adjacency matrices of graphs as a particular case. Unlike classical continuous-time Markov chains, the CTQMCs studied here do not satisfy the classical Chapman-Kolmogorov equation; this is a genuine quantum effect, due to the fact that the underlying unitary evolution composes at the level of probability amplitudes rather than probabilities, and we make precise the sense in which the term ``quantum Markov chain'' is used. In this paper, we study a large class of p-adic Schrödinger equations and the associated CTQMCs by comparing them with p-adic heat equations and the associated continuous-time Markov chains (CTMCs). The comparison is done by a mathematical study of the mentioned equations, which requires, for instance, solving the initial value problems attached to the mentioned equations, and through numerical simulations. We conducted multiple simulations, including numerical approximations of the limiting distribution. Our simulations show that the limiting distribution of quantum Markov chains is greater than the stationary probability of their classical counterparts, for a large class of CTQMCs.

quant-ph↗
arXiv2026

Pattern Formation in Quantum Hierarchical Cellular Neural Networks

We present a new class of quantum neural networks (QNNs) whose states are solutions of $p$-adic Schrödinger equations with a non-local potential that controls the interaction between the neurons. These equations are obtained as Wick rotations of the state equations of $p$-adic cellular neural networks (CNNs). The CNNs are continuous limits of discrete hierarchical neural networks (NNs). The CNNs are bio-inspired by the Wilson-Cowan model, which describes the macroscopic dynamics of large populations of neurons. We provide a detailed study of the discretization of the new $p$-adic Schrödinger equations, which allows the construction of new QNNs on simple graphs. We also conduct detailed numerical simulations, offering a clear insight into the functioning of the new QNNs. At a mathematical level, we show the existence of local solutions for the new $p$ -adic Schrödinger equations.

quant-ph↗
arXiv2026

Wavefunctions localization, and the Wigner's Friend Paradox in a Framework of Discrete-Space Hypothesis

We present a resolution of the Wigner's Friend paradox within a framework of quantum mechanics (QM) on the hybrid space RxQ_{p}, where Q_{p} denotes the field of p-adic numbers, regarded as a model of discrete microscopic space at the Planck-Bronstein scale. In this framework, wavefunction collapse is not an independent postulate but a dynamical consequence of the Schrödinger equation with non-local Hamiltonians: wavefunctions localize onto compact supports during measurement interactions, producing definite pointer readings without the intervention of observers or the exchange of information between subsystems. We model both Wigner and his Friend as classical apparatuses and show that each produces a definite reading through independent applications of the collapse mechanism, thereby eliminating the conflict between their descriptions of reality. The framework is consistent with the principal no-go theorems in finite- and infinite-dimensional Hilbert spaces associated with extended Wigner's Friend scenarios -- including those of Frauchiger-Renner, Brukner, Bong et al., and Guérin et al. -- since it requires no agents capable of recording or reasoning about outcomes, thereby vacating the observer-dependent assumptions that drive those theorems. We illustrate the collapse mechanism explicitly through a toy model of a particle in a box, comparing the standard description with the new one. The non-locality intrinsic to QM on L2(RxQ_{p}) permits realism at the cost of locality, and the Absoluteness of Observed Events holds in our framework without requiring observer independence.

quant-ph↗
arXiv2026

Deep Neural Networks: A Formulation Via Non-Archimedean Analysis

We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified using numbers from the ring of integers of non-Archimdean local fields. These rings have a natural hierarchical organization as infinite rooted trees. Natural morphisms on these rings allow us to construct finite multilayered architectures. The new DNNs are robust universal approximators of real-valued functions defined on the mentioned rings. We also show that the DNNs are robust universal approximators of real-valued square-integrable functions defined in the unit interval.

cs.NE↗
arXiv2026

Quantum mechanics, non-locality, and the space discreteness hypothesis

The space discreteness hypothesis asserts that the nature of space at short distances is radically different from that at large distances. Based on the Bronstein inequality, here, we use a totally disconnected topological space $\mathcal{X}$ as a model for the space. However, we consider the time as a real variable. In this framework, the formalism of Dirac-von Neumann can be used. This discreteness hypothesis implies that given two different points in space, there is no continuous curve (a world line) joining them. Consequently, this hypothesis is not compatible with the theory of relativity. We propose $\mathbb{R}\times(\mathbb{R}\times\mathcal{X})^{3}$ as a model of a space-time. For simplicity, we work out our models using $\mathbb{R}\times(\mathbb{R}\times\mathcal{X})$ as the configuration space. Quantum mechanics (QM), in the sense of Dirac-von Neumann, on the Hilbert space $L^{2}(\mathbb{R}\times\mathcal{X})$ is a non-local theory: the Hamiltonians are non-local operators, and thus, spooky action at a distance is allowed. The paradigm asserting that the universe is non-locally real implies that the proposed version of QM admits realism. This version of QM can be specialized to standard QM by using Hamiltonians acting on wavefunctions supported on the region $\mathbb{R}\times\mathbb{R}$. We apply the developed formalism to the measurement problem. We propose a new mechanism for the collapse of the wavefunction. The mechanism resembles the one proposed by Ghirardi, Ramini, and Weber, but there are significant differences. The most important feature is that the Schrödinger equation describes the dynamics at all times, even at the moment of measurement. We also discuss a model for the two-slit experiment, where bright and dark states of light (proposed recently) naturally occur.

quant-ph↗
arXiv2026

Critical Organization of Deep Neural Networks, and p-Adic Statistical Field Theories

We rigorously study the thermodynamic limit of deep neural networks (DNNS) and recurrent neural networks (RNNs), assuming that the activation functions are sigmoids. A thermodynamic limit is a continuous neural network, where the neurons form a continuous space with infinitely many points. We show that such a network admits a unique state in a certain region of the parameter space, which depends continuously on the parameters. This state breaks into an infinite number of states outside the mentioned region of parameter space. Then, the critical organization is a bifurcation in the parameter space, where a network transitions from a unique state to infinitely many states. We use p-adic integers to codify hierarchical structures. Indeed, we present an algorithm that recasts the hierarchical topologies used in DNNs and RNNs as p-adic tree-like structures. In this framework, the hierarchical and the critical organizations are connected. We study rigorously the critical organization of a toy model, a hierarchical edge detector for grayscale images based on p-adic cellular neural networks. The critical organization of such a network can be described as a strange attractor. In the second part, we study random versions of DNNs and RNNs. In this case, the network parameters are generalized Gaussian random variables in a space of quadratic integrable functions. We compute the probability distribution of the output given the input, in the infinite-width case. We show that it admits a power-type expansion, where the constant term is a Gaussian distribution.

cs.LG↗
arXiv2026

Pseudo-Traveling Waves and Bumps in Quantum and Classical Hierarchical Cellular Neural Networks

We study the existence of pseudo-traveling waves and bump solutions for two classes of hierarchical cellular neural networks (CNNs) defined over the ring of $p$-adic integers $\mathbb{Z}_{p}$. The first type is a $p$-adic CNN described by a reaction-diffusion equation, while the second type is its quantum analog obtained via Wick rotation. The $p$-adic CNNs are hierarchical versions of the classical Chua-Yang CNNs; these networks have a tree-like hierarchical architecture with infinitely many cells and hidden layers. The states are governed by integro-differential equations on $% \mathbb{Z}_{p}$. The $p$-adic traveling waves behave fundamentally differently from their Archimedean counterparts. A traveling wave restricted to a $p$-adic sphere yields a countably infinite collection of independent patterns. We introduce the notion of pseudo-traveling waves as finite truncations of this structure and prove their existence for both the classical and quantum networks. We further establish the existence of time-independent solutions (bumps) for both models. Our theoretical results are complemented by numerical simulations that approximate pseudo-traveling-wave solutions for quantum CNNs.

math-ph↗
arXiv2026

Koba-Nielsen local zeta functions, convex subsets, and generalized Selberg-Mehta-Macdonald and Dotsenko-Fateev-like integrals

The Koba-Nielsen local zeta functions are integrals depending on several complex parameters, used to regularize the Koba-Nielsen string amplitudes. These integrals are convergent and admit meromorphic continuations in the complex parameters. In the original case, the integration is carried out on the n-dimensional Euclidean space. In this work, the integration is over a variety of (bounded or unbounded) convex subsets; the resulting integrals also admit meromorphic continuations in the complex parameters. We describe the meromorphic continuation's polar locus explicitly, using the technique of embedded resolution. This result can be reinterpreted as saying that the meromorphic continuations are weighted sums of Gamma functions, evaluated at linear combinations of the complex parameters, where the weights are holomorphic functions. The integrals announced in the title of this paper occur as a particular case of these new Koba-Nielsen local zeta functions, or of a further generalization to arbitrary hyperplane arrangements.

math-ph↗
arXiv2026

p-Adic Dirac Equations and the Jackiw-Rebbi Model

We present a new p-adic version of the Jackiw-Rebbi model. In the new model, the real numeric line is replaced by a p-adic line (the field of p-adic numbers Q_{p}), and the Dirac Hamiltonian is replaced by a non-local operator acting on complex-valued functions defined on Q_{p}. These Hamiltonians admit localized wavefunctions and allow long-range interactions, so spooky action at a distance is allowed. These features are not present in the original model. The new model gives the same predictions as the standard one. The p-adic line serves as a discrete model for the physical space; in this type of space, non-locality emerges naturally.

quant-ph↗
arXiv2025

2-Adic quantum mechanics, continuous-time quantum walks, and the space discreteness

We show that a large class of 2-adic Schrödinger equations is the scaling limit of certain continuous-time quantum Markov chains (CTQMCs). Practically, a discretization of such an equation gives a CTQMC. As a practical result, we construct new types of continuous time quantum walks (CTQWs) on graphs using two symmetric matrices. One matrix describes the transport between nodes in one direction, while the second describes the transport between nodes in the opposite direction. This construction includes, as a particular case, the CTQWs constructed using adjacency matrices. The final goal of this work is to contribute to the understanding of the foundations of quantum mechanics (QM) and the role of the hypothesis of the discreteness of space. The connection between 2-adic QM and CTQWs shows that 2-adic QM has a physical meaning. 2-Adic QM is a nonlocal theory because the Hamiltonians used are nonlocal operators, and consequently, spooky actions at a distance are allowed. However, this theory is not a mathematical toy. The experimental confirmation of the violation of Bell's inequality implies that this theory allows realism. We pointed out several new research problems connected with the foundations of quantum mechanics.

quant-ph↗
arXiv2025

Dynamic Mean-Field Theory for Continuous Random Networks

This article studies the dynamics of the mean-field approximation of continuous random networks. These networks are stochastic integrodifferential equations driven by Gaussian noise. The kernels in the integral operators are realizations of generalized Gaussian random variables. The equation controls the time evolution of a macroscopic state interpreted as neural activity, which depends on position and time. The position is an element of a measurable space. Such a network corresponds to a statistical field theory (STF) given by a momenta-generating functional. Discrete versions of the mentioned networks appeared in spin glasses and as models of artificial neural networks (NNs). Each of these discrete networks corresponds to a lattice SFT, where the action contains a finite number of neurons and two scalar fields for each neuron. In this article, we develop mathematically rigorous, continuous versions of the mean-field theory approximation and the double-copy system that allow us to derive a condition for the criticality of continuous stochastic networks via the largest Lyapunov exponent. We use two basic architectures; in the first one, the space of neurons is the real line, and then the neurons are organized in one layer; in the second one, the space of neurons is the p-adic line, and then the neurons are organized in an infinite, fractal, tree-like structure. We also studied a toy model of a continuous Gaussian network with a continuous phase transition. This behavior matches the critical brain hypothesis, which states that certain biological neuronal networks work near phase transitions.

cond-mat.dis-nn↗
arXiv2024

p-Adic quantum mechanics, infinite potential wells, and continuous-time quantum walks

This article discusses a p-adic version of the infinite potential well in quantum mechanics (QM). This model describes the confinement of a particle in a p-adic ball. We rigorously solve the Cauchy problem for the Schrödinger equation and determine the stationary solutions. The p-adic balls are fractal objects. By dividing a p-adic ball into a finite number of sub-balls and using the wavefunctions of the infinite potential well, we construct a continuous-time quantum walk (CTQW) on a fully connected graph, where each vertex corresponds to a sub-ball in the partition of the original ball. In this way, we establish a connection between p-adic QM and quantum computing.

quant-ph↗
arXiv2024

p-Adic Quantum Mechanics, the Dirac Equation, and the violation of Einstein causality

This article studies the breaking of the Lorentz symmetry at the Planck length in quantum mechanics. We use three-dimensional p-adic vectors as position variables, while the time remains a real number. In this setting, the Planck length is 1/p, where p is a prime number, and the Lorentz symmetry is naturally broken. In the framework of the Dirac-von Neumann formalism for quantum mechanics, we introduce a new p-adic Dirac equation that predicts the existence of particles and antiparticles and charge conjugation like the standard one. The discreteness of the p-adic space imposes substantial restrictions on the solutions of the new equation. This equation admits localized solutions, which is impossible in the standard case. We show that an isolated quantum system whose evolution is controlled by the p-adic Dirac equation does not satisfy the Einstein causality, which means that the speed of light is not the upper limit for the speed at which conventional matter or energy can travel through space. The new p-adic Dirac equation is not intended to replace the standard one; it should be understood as a new version (or a limit) of the classical equation at the Planck length scale.

quant-ph↗
arXiv2024

The p-Adic Schrödinger Equation and the Two-slit Experiment in Quantum Mechanics

p-Adic quantum mechanics is constructed from the Dirac-von Neumann axioms identifying quantum states with square-integrable functions on the N-dimensional p-adic space. This choice is equivalent to the hypothesis of the discreteness of the space. The time is assumed to be a real variable. The p-adic quantum mechanics is motivated by the question: what happens with the standard quantum mechanics if the space has a discrete nature? The time evolution of a quantum state is controlled by a nonlocal Schrödinger equation obtained from a p-adic heat equation by a temporal Wick rotation. This p-adic heat equation describes a particle performing a random motion in the N-dimensional p-adic space. The Hamiltonian is a nonlocal operator; thus, the Schrödinger equation describes the evolution of a quantum state under nonlocal interactions. In this framework, the Schrödinger equation admits complex-valued plane wave solutions, which we interpret as p-adic de Broglie waves. These mathematical waves have all wavelength 1/p. In the p-adic framework, the double-slit experiment cannot be explained using the interference of the de Broglie waves. The wavefunctions can be represented as convergent series in the de Broglie waves, but the p-adic de Broglie waves are just mathematical objects. Only the square of the modulus of a wave function has a physical meaning as a time-dependent probability density. These probability densities exhibit interference patterns similar to the ones produced by `quantum waves.' In the p-adic framework, in the double-slit experiment, each particle goes through one slit only.

quant-ph↗
arXiv2024

Hierarchical Neural Networks, p-Adic PDEs, and Applications to Image Processing

The first goal of this article is to introduce a new type of p-adic reaction-diffusion cellular neural network with delay. We study the stability of these networks and provide numerical simulations of their responses. The second goal is to provide a quick review of the state of the art of p-adic cellular neural networks and their applications to image processing.

cs.NE↗
arXiv2023

Turing patterns in a p-adic FitzHugh-Nagumo system on the unit ball

We introduce discrete and p-adic continuous versions of the FitzHugh-Nagumo system on the one-dimensional p-adic unit ball. We provide criteria for the existence of Turing patterns. We present extensive simulations of some of these systems. The simulations show that the Turing patterns are traveling waves in the p-adic unit ball.

nlin.PS↗
arXiv2023

A Correspondence Between Deep Boltzmann Machines and p-Adic Statistical Field Theories

There is a strong interest in studying the correspondence between Euclidean quantum fields and neural networks. This correspondence takes different forms depending on the type of networks considered. In this work, we study this correspondence in the case of deep Boltzmann machines (DBMs) having a tree-like topology. We use p-adic numbers to encode this type of topology. A p-adic continuous DBM is a statistical field theory (SFT) defined by an energy functional on the space of square-integrable functions defined on a p-adic N-dimensional ball. The energy functionals are non-local, meaning they depend on the interaction of all the neurons forming the network. Each energy functional defines a probability measure. A natural discretization process attaches to each probability measure a finite-dimensional Boltzmann distribution, which describes a hierarchical DBM. We provide a mathematically rigorous perturbative method for computing the correlation functions. A relevant novelty is that the general correlation functions cannot be calculated using the Wick-Isserlis theorem. We give a recursive formula for computing the correlation functions of an arbitrary number of points using certain 3-partitions of the sets of indices attached to the points.

hep-th↗