What is the Relationship Between Quantum and Classical Path Integrals?

[sourena]

What is the relationship between the quantum mechanical path integral and
classical mechanics path integral?

 

[Born2bwire]

They are basically just analogues of each other. The Schroedinger equation is based around a quantum version of classical Hamiltonian mechanics. By the same token, the path integral is built around a quantum version of Lagrangian mechanics. Another thing to consider is that the classical limit of the path integral is the traditional Lagrangian path. This can be found by taking Planck's constant, \hbar, to the limit of zero. Or, in another way of looking at it, the classical path is the stationary path of the path integral.

There are a lot of these little relationships that you can make between the two but most of these (like the classical limit) are obviously expected for the quantum path integral to make sense. And there is a bunch of formalism behind this. Can't recall a good reference that explains this though...

 

[sourena]

Can the relationship between the quantum mechanical path integral and
classical mechanics be stated as this?
A path integral involves an exponential of the action S.