The talk aims to discuss our recent results on $p$-adic quantum mechanics. In the Dirac-von Neumann formulation of quantum mechanics (QM), the states of a (closed) quantum system are vectors of an abstract complex Hilbert space $\mathcal{H}$. A particular choice of space $\mathcal{H}$ goes beyond the mathematical formulation and belongs to the domain of physical practice and intuition. In practice, selecting a particular Hilbert space also implies choosing a topology for the space. The standard choice $\mathcal{H}=L (\mathbb{R}^{3})$ implies that we are assuming that space ($\mathbb{R}^{3}$) is continuous, i.e., it is an arcwise topological space, meaning a continuous path joins any two points in the space. Let us denote by $\mathbb{% Q}_{p}$ the field of $p$-adic numbers; here, $p$ is a fixed prime number. The space $\mathbb{Q}_{p}^{3}$ is `discrete', i.e., the points and the empty set are the only connected subsets. The Hilbert spaces $L^{2}(\mathbb{R}^{3}) $, $L^{2}(\mathbb{Q}_{p}^{3})$ are isometric, but the geometries of the underlying spaces ($\mathbb{R}^{3}$, $\mathbb{Q}_{p}^{3}$) are radically different. By $p$-adic QM, we mean QM on $L^{2}(\mathbb{Q}_{p}^{3})$; in this case, the time is a real number, but the position is a $p$-adic vector.
We first discuss the physical content of the $p$-adic Schrodinger equation. Recently, we found a connection between the mentioned equation and continuous-time quantum walks (CTQWs). In [1], we established that the $2$-adic Schrodinger equation describes the scaling limit of a continuous-time quantum walk in $\mathbb{Q}_{p}$. Using our results, we can recover the CTQWs introduced by Farhi and Gutmann,[2]. In quantum information theory, quantum walks are used extensively as algorithmic tools for quantum computation. In [3], we constructed a family of CTQWs using the solutions of the $p$-adic Schrodinger equation for some infinite potential wells. In practical terms, the QM on $\mathbb{C}^{2^{l}}$ (which is used in quantum computing) is a type of $2$-adic QM.
The $p$-adic Schrodinger operators are nonlocal, which allows long-distance interactions. Thus, by definition, the $p$-adic QMis a nonlocal theory. Now, by the paradigm that the universe is not locally real, we have to choose between locality and realism. Since $p$-adic QM is nonlocal, we have that it is `real.' Performing a Wick rotation, the $p$ -adic Schrodinger equation becomes a $p$-adic heat equation, which describes a particle performing a random motion in a fractal space.
In $p$-adic QM, by definition, the Lorentz symmetry is broken. In [4], in the framework of the Dirac-von Neumann formalism, we introduce a new $p$-adic Dirac equation that predicts the existence of particles and antiparticles and charge conjugation like the standard one. 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.
In [5], in the $p$-adic framework, we develop the mathematical model for the two-slit experiment. In this model, each particle goes through one slit only. A similar description of the two-slit experiment was given in [6]: "Instead of a quantum wave passing through both slits, we have a localized particle with nonlocal interactions with the other slit." We obtain the same conclusion, but in our framework, the nonlocal interactions result from the discreteness of the space $\mathbb{Q}_{p}^{3}$.
[1] Zuniga-Galindo W. A., Chacon-Cortes L. F., The $2$-adic Schrodinger equation describes the scaling limit of a continuous-time quantum walk. Preprint 2025. In progress.
[2] Farhi E., Gutmann S., Quantum computation and decision trees, Phys. Rev. A (3)58(1998), no.2, 915–928.
[3] Zuniga-Galindo W. A., Mayes Nathaniel P., $p$-Adic quantum mechanics, infinite potential wells, and continuous-time quantum walks. DOI: https://doi.org/10.48550/arXiv.2410.13048.
[4] Zuniga-Galindo W. A., $p$-adic quantum mechanics, the Dirac equation, and the violation of Einstein causality, J. Phys. A 57 (2024), no. 30, Paper No. 305301, 29 pp.
[5] Zuniga-Galindo W. A., The $p$-Adic Schrodinger equation and the two-slit ex- periment in quantum mechanics, Ann. Physics 469 (2024), Paper No. 169747.
[6] Aharonov Y., Cohen E., Colombo F., Landsberger T., Sabadini I., Struppa D., and Tollaksen J., Finally making sense of the double-slit experiment. Proc. Natl. Acad. Sci. U. S. A. 114, 6480 (2017).