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, $Q_p^N$. 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 $Q_p^N$. 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 $p^{-1}$. 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. The $p$-adic quantum mechanics is an analog (or model) of the standard one under the hypothesis of the existence of a Planck length. The precise connection between these two theories is an open problem. Finally, we propose the conjecture that the classical de Broglie wave-particle duality is a manifestation of the discreteness of space–time.