Scheme for measuring experimentally the velocity of pilot waves and the discreteness of time

We consider the following two questions. Suppose that a quantum system suffers a change of the boundary condition or the potential at a given space location. Then (1)when will the wavefunction shows a response to this change at another location? And (2)how does the wavefunction changes?The answer to question (1) could reveal how a quantum system gets information on the boundary condition or the potential. Here we show that if the response takes place immediately, then it can allow superluminal signal transfer. Else if the response propagates in space with a finite velocity, then it could give a simple explanation why our world shows classicality on the macroscopic scale. Furthermore, determining the exact value of this velocity can either clarify the doubts on static experiments for testing Bell's inequality, or support the pilot-wave interpretation of quantum mechanics. We propose a feasible experimental scheme for measuring this velocity, which can be implemented with state-of-art technology, e.g., single-electron biprism interferometry.Question (2) is studied with a square-well potential model, and we find a paradox between the impossibility of superluminal signal transfer and the normalization condition of wavefunctions. To solve the paradox, we predict that when a change of the potential occurs at a given space location, the system will show no response to this change at all, until after a certain time interval. Otherwise either special relativity or quantum mechanics will be violated. As a consequence, no physical process can actually happen within Planck time. Therefore it gives a simple proof that time is discrete, with Planck time being the smallest unit. Combining with the answer to question (1), systems with a larger size and a slower velocity could have a larger unit of time, making it possible to test the discreteness of time experimentally. Our result also sets a limit on the speed of computers, and gives instruction to the search of quantum gravity theories.