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

• URL: http://arxiv.org/abs/2502.16416v1
• Date: Sun, 23 Feb 2025 03:10:55 GMT
• Title: 2-Adic quantum mechanics, continuous-time quantum walks, and the space discreteness
• Authors: W. A. Zúñiga-Galindo,
• Abstract summary: Using techniques of p-adic analysis, it is possible to formulate a rigorous version of the quantum mechanics.<n>The experimental testability of physical theories at the Planck scale is currently impossible.<n>We show that a large class of Schr"odinger equations describes the scaling limits of continuous-time quantum walks on graphs.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Using techniques of p-adic analysis, it is possible to formulate a rigorous version of the quantum mechanics (QM), in the sense of Dirac-von Neumann, consistent with the existence of the Planck length. Such a model cannot be formulated if we use R^{3} as a model for physical space. The experimental testability of physical theories at the Planck scale is currently impossible. Here, we provide an indirect, theoretical argument that shows that the p-adic QM has physical content. We show that a large class of Schr\"odinger equations describes the scaling limits of continuous-time quantum walks on graphs (stochastic automata). These quantum walks appear as fundamental tools in quantum computing. We conjecture that this interpretation is valid in a general framework. The `new theory' does not have Lorentz symmetry, and the Einstein causality is violated. This fact does not contradict the so-called no-communication theorem; such a result requires as a primary hypothesis that R^{4} be a valid model for space-time at the Planck scale. Thus, the no-communication theorem under the discreteness of the space is an open problem.

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