  Non-interferometric experiments have been successfully employed to constrain
models of spontaneous wave function collapse, which predict a violation of the
quantum superposition principle for large systems. These experiments are
grounded on the fact that, according to these models, the dynamics is driven by
a noise that, besides collapsing the wave function in space, generates a
diffusive motion with characteristic signatures, which, though small, can be
tested. The non-interferometric approach might seem applicable only to those
models which implement the collapse through a noisy dynamics, not to any model,
which collapses the wave function in space. Here we show that this is not the
case: under reasonable assumptions, any collapse dynamics (in space) is
diffusive. Specifically, we prove that any space-translation invariant dynamics
which complies with the no-signaling constraint, if collapsing the wave
function in space, must change the average momentum of the system, and/or its
spread.
