Stochastic vs Quantum

http://math.ucr.edu/home/baez/prob.pdf

A little bit about stochastic vs quantum. Do the same trick of Hermitian generators of unitary transformations but with infinitesimal stochastic operators generating stochastic matrices.

Stochastic matrices list off the probabilities of a transition from state i to state j. Therefore every row has to sum to 1 since state i has to go somewhere. Also the entries are all between 0 and 1 since they are probabilities.

But stochastic matrices aren't as good as unitary matrices, they usually aren't invertible. Think of the stochastic matrix that takes every state to state 1 with probability 1. Clearly you have no information about the original state. For all you know the probabilities were equally distributed in the original configuration.

We can't reverse the configuration, but at least we can propagate forward in time. The example Baez gives is with some number of amoebas that can be born and die with some given rates. Given an infinitesimal stochastic operator giving the probabilities for a bunch of processes like creation of an amoeba or more generally killing k and creating l amoebas, we can create Feynman diagrams which when added up give the full time evolution operator as it usually does.

This can be generalized if you give more vertices that describe more processes between your states. Again draw your Feynman diagrams and compute the time evolution of your initial state.