Colella-Overhauser-Werner test of the weak equivalence principle: A low-energy window to look into the noncommutative structure of space-time?

We construct the quantum mechanical model of the Colella-Overhauser-Werner (COW) experiment assuming that the underlying space time has a granular structure, described by a canonical noncommutative algebra of coordinates $x^{\mu }$. The time-space sector of the algebra is shown to add a mass-dependent contribution to the gravitational acceleration felt by neutron de Brogli waves measured in a COW experiment. This makes time-space noncommutativity a potential candidate for an apparent violation of the weak equivalence principle even if the ratio of the inertial mass $m_i$ and gravitational mass $m_g$ is a universal constant. The latest experimental result based on the COW principle is shown to place an upper bound several orders of magnitude stronger than the existing one on the time-space noncommutative parameter. We argue that the evidence of noncommutative structure of space-time may be found if the COW-type experiment can be repeated with several particle species.