Layman here. It is often said that the pure fundamental theories do not contain any arrow of time, they are fully reversible in time. But regarding Shrodinger's equation, it describes the evolution of the wavefunction in time. As I understand it, if we consider a static particle after a measurement, let's say event A, its wavefunction will start spreading from that spacetime event location in a spherical shape at the speed of light until it undergoes a new measurement, at which the wavefunction sphere will collapse at some point given probabilistically by the equation. Is such a sphere expanding at light speed truly spherical, pointing also towards the past as well as to the future? So that there would be a probability of the next measurement finding that particle in a spacetime coordinate which lies in the past of event A, as much as finding it in a spacetime coordinate which lies in the future of event A? Because if the answer is no, that the wavefunction evolution is not really a sphere, but only a hemisphere from event A towards the future but not towards the past, wouldn't it mean that the very arrow of time is already contained in such a wavefunction evolution equation? Selecting that only coordinates in the future can be canditates to finding the particle from event A, but never coordinates in the past? Thanks !