I've decided to put a very limited amount of my time into understanding the path integral formulation of quantum mechanics. I'm interested in the mathematical formalism more than the physics, so I'd like to understand how the following abstract and fairly general scenario for a discrete quantum system can be translated into the path integral formalism, assuming it can.

A system starts has a time-dependent state that can be represented by an $n\times n$ density matrix $\rho(t)$. It begins in state $\rho_0$ and evolves over time according to
$$ i\hbar \frac{\partial \rho}{\partial t} = H\rho - \rho H,$$
where the Hamiltonian $H$ is some constant Hermitian matrix, yielding
$$
\rho(t) = e^{-iHt/\hbar}\rho_0 e^{iHt/\hbar}.
$$
At time $t'$ I make a complete set of orthogonal measurements, which amounts to (optionally) doing a basis change on $\rho(t')$ and then interpreting its diagonal elements as probabilities.

My question is, can someone show me how to reinterpret the above in path integral terms? For example, what is a "path" in a discrete system like this, and, given an arbitrary Hamiltonian, how does one assign each path an action?

Alternatively, can anyone recommend a treatment that summarises the path integral formalism with a minimum of physics (by which I mean details about how specific Hamiltonians behave and where they come from, etc.)?

This post imported from StackExchange Physics at 2014-04-08 05:14 (UCT), posted by SE-user Nathaniel