How is Schwinger's quantum action principle related to least action?

Question

The principle of least action says that a body moves in such a way that the action value $S=\int L dt$ is stationary (often minimal). The principle is written as $$\delta S =0 \ .$$

In contrast, Schwinger's quantum action principle between an in and an out state is $$ \langle{\rm out}| \delta S |{\rm in}\rangle = \frac{\hbar}{i} \delta \langle {\rm out}|{\rm in}\rangle \ .$$ I have a simple question: what is minimized in this case? Or, more generally: what does the principle state, if only words are used? That would be my question 1.

Question 2. Least action implies that the actual trajectory is special: it differs from all the others, because it has minimal/stationary action. In what sense does Schwinger's principle make the actual state evolution special from all the others?

Answering a similar, older question, Qmechanic had written:

Schwinger's quantum action principle is not a variational principle in the sense of finding stationary points for a functional. Rather it gives a formula for how a quantum system (typically an overlap/transition amplitude ⟨𝐴|𝐵⟩) changes under a change of external parameters/sources in the action 𝑆.

Can one add a few sentences to complete the answer to question 2? In what sense is the quantum motion special? How does it apply to free particles? Or: how can one express the quantum action principle for free particles in words?

Question 3. How does Schwinger's action principle become the least action principle in the classical limit? Ok, if $\hbar$ is zero, the two equations are very similar. Is that the way the transition is made? What can be said to make it clearer?

Question 4. The right hand side of Schwinger's principle is $-i\hbar$ times a complex number with a magnitude equal or smaller than one. Why does it enter the principle? Is it correct to say that for long times, the overlap is small, and thus the right hand side is zero?

Answer 1

In the quantum variational principle nothing is minimized: there is no variation that is set to zero.

Rather, the principle is the infinitesimal version of Feynman's path integral. Indeed, integrating the variational principle one obtains the usual formula for the functional integral, as very clearly emphasized by Schwinger's student Bryce DeWitt, cf. ref 1 §10. In fact, it is hard to find a better presentation of the variational principle than that of this reference.

As the variational principle is completely equivalent to the functional integral, the physical interpretation is the same: the amplitude for any given field configuration is a phase, determined by the classical action of that configuration. This is a postulate, there is no deeper explanation for where this comes from. One can derive all of quantum mechanics from this principle, but one cannot derive the principle itself. DeWitt has a very good motivation though. Go check it.

Moreover, and for the same reason, the classical limit is obtained in the same way: As usual, the classical configuration – being a critical point – gives a particularly large contribution to the total amplitude, at least as long as $S/\hbar$ is sufficiently large. In the limit $S/\hbar\to\infty$, the classical configuration is the only configuration – the rest all interfere destructively. One should keep in mind that, if $S/\hbar$ is not large, purely quantum effects may dominate, in which case the classical configuration is entirely irrelevant to the dynamics.

I don't understand subquestion 4. But no: the r.h.s. is not (typically) negligible for large times. It is negligible in the classical limit, and only in that limit, by definition. Sometimes, $t\to\infty$ might be equivalent to the classical limit (e.g. if we adiabatically turn off interactions or something like that). But not in general.

As a side comment, one can reformulate the quantum variational principle as a classical variational principle, by replacing the classical action by the effective (quantum) action. In fancier terms one can introduce the quantum BV bracket, etc., all of which replace classical objects. See ref. 1 §24 if interested. This is beyond the scope of this post.

References.

1. DeWitt B.S. - The global approach to quantum field theory Vol.1.