The S-Matrix calculates transition probabilities from 1 quantum state to another quantum state over an infinite time of interaction, with interactions taking place over an infinite space. This is claimed to make the math easy, but actually leads to infinity probabilities and the need for renormalization.
Is it possible to go the opposite way and just calculate a quantum state evolution after a really small amount of time, like a picosecond or some plank scale step, calculate the new state (which must be really close to the original state) and the calculate a new step over and over until some significant evolution is computed?
I imagine most of these subatomic interactions are likely over anyway after milliseconds.