Here's my layman hypothesis: Time has a wave-like nature, not the particles. Since space and time are related through the Lorentz transformations, you could

Here's my layman hypothesis: Time has a wave-like nature, not the particles. Since space and time are related through the Lorentz transformations, you could substitute a wave-like property of time and make the time dependent wave equation into a space dependent equation. The results would be the same, but it's a different interpretation. This is why you cannot measure a particles momentum and position at the same moment - they do not both exist at the same moment and depend on the current phase of the wave.