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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.)?

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The discrete form of the Feynman path integral is just time-dependent perturbation theory. This is not worked out so great, I'll try to write an answer when I get a chance, but the paths are labelled by the time of discrete state transitions, and the integral over intermediate time labels introduces the phase factors of the time-dependent perturbation theory. Something like this is covered in an obscure early 1950s article by Feynman, where he introduces a time-ordered product and mathematical tricks to quickly derive time-independent perturbation theory from PI like formalism –  Ron Maimon May 16 '12 at 20:26
    
@RonMaimon that sounds like the kind of explanation I'm looking for - I look forward to it. –  Nathaniel May 17 '12 at 7:49

2 Answers 2

Check out the original paper by Feynman, google "The Space-Time Formulation of Nonrelativistic Quantum Mechanics".

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I guess you might be interested in this kind of an introduction,

http://www.math.sunysb.edu/~tony/whatsnew/column/feynman-1101/feynman1.html

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Thanks, that's very helpful - skimming it's given me some idea of what kind of mathematics is involved - but it takes my request for "a minimum of physics" a bit too far, in that it never mentions Hamiltonians at all, leaving me unsure for the moment of how to connect these ideas to the scenario I described in my question... –  Nathaniel May 17 '12 at 7:48

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