Is it possible to derive action principle #2 from QM?

[Perturbation]

Wiki says the same thing here:
I still don't see how you can equate the above two actions. The first is the action of a path, and the second is the action of a field. That's pretty much what you say here:

If I take the total mass of a body m, I can express it as the integral over the volume of the body of the mass density. It's the same here: I'm integrating some quantity over the region of space that we're interested into get our Lagrangian. For this reason we call it the Lagrangian density. If I then integrate the mass over some time contour, I'm integrating the mass-density over a four volume. This is an analogous situation to what's causing you problems, but there's nothing wrong with doing it, as the previous example shows.

 

[Perturbation]

that in QFT we assume from the outset the action principle, and the action principle implies Einstein's equation, which means we are assuming Lorentz covariance from the beginning.
David

In elementary formulations of field theory it is indeed assumed. We begin with a classical field, by classical we mean unquantised, and a dynamical equation or a Lagrangian density. In the field theory of the Klein-Gordon and Dirac fields we start with a field equation and then find a Lagrangian density that gives this equation using least action. In general, one begins with a Lagrangian density then constructs the field equations, for example when one builds a general gauge invariant field theory we choose appropriately amicable terms to construct the Lagrangian. We only had the field equations for Dirac and Klein-Gordon first because they both came from the relativistic equation for energy.

In second or cannonical quantisation one would then go about deciding upon a solution for the classical fields of the field equations in terms of operators etc. For this reason the field equations are useful and least action is important. From this method of quantisation propogation amplitudes and all the rules and properties for the field theory follow.

However in the Feynman path integral approach we needn't assume least action to be able to quantise the field theory, all we need do is make the jump from the classical fields to opertors on Hilbert space and have ourselves a Lagrangian. The Feynman rules then follow.