  Some ideas from quantum theory are just beginning to percolate back to
classical probability theory. For example, there is a widely used and
successful theory of "chemical reaction networks", which describes the
interactions of molecules in a stochastic rather than quantum way. Computer
science and population biology use the same ideas under a different name:
"stochastic Petri nets". But if we look at these theories from the perspective
of quantum theory, they turn out to involve creation and annihilation
operators, coherent states and other well-known ideas - but in a context where
probabilities replace amplitudes. We explain this connection as part of a
detailed analogy between quantum mechanics and stochastic mechanics. We use
this analogy to present new proofs of two major results in the theory of
chemical reaction networks: the deficiency zero theorem and the
Anderson-Craciun-Kurtz theorem. We also study the overlap of quantum mechanics
and stochastic mechanics, which involves Hamiltonians that can generate either
unitary or stochastic time evolution. These Hamiltonians are called "Dirichlet
forms", and they arise naturally from electrical circuits made only of
resistors.
