Horizons and non-local time evolution of quantum mechanical systems

According to general relativity, trapping surfaces and horizons are classical causal structures that arise in sys- tems with sharply defined energy and corresponding gravi- tational radius. The latter concept can be extended to a quan- tum mechanical matter state simply by means of the spec- tral decomposition, which allows one to define an associated “horizon wave-function”. Since this auxiliary wave-function contains crucial information about the causal structure of space-time, a new proposal is formulated for the time evolu- tion of quantum systems in order to account for the funda- mental classical property that outer observers cannot receive signals from inside a horizon. The simple case of a massive free particle at rest is used throughout the paper as a toy model to illustrate the main ideas.