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

Abstract: We show that a large class of 2-adic Schr\"odinger 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.