  Using Kaluza-Klein theory we discuss the quantum mechanics of a particle in
the background of a domain wall (brane) embedded in extra dimensions. We show
that the geometric phases associated with the particle depend on the
topological features of those spacetimes. Using a cohomological modeling
schema, we deduce a random phase sequence composed of the geometric phases
accompanying the periodic evolution over the spacetimes. The random phase
sequence is demonstrated some properties that could be experimental
verification. We argue that it is related to the nonlocality of quantum
entanglement.
