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Is there any sense in which we can consider Action a physical observable? What would experiments measuring it even look like? I am interested in answers both in classical and quantum mechanics.

I ran across a physics textbook called "Motion Mountain" today, with volumes covering a broad swath of physics, written over the past decade by a dedicated German physicist with support from some foundations for physics outreach. So it appears to be a serious endeavor, but it's approach to many things is non-standard and often just sounds wrong to me. In discussing his approach to some topics the author says:

On action as an observable

Numerous physicists finish their university studies without knowing that action is a physical observable. Students need to learn this. Action is the integral of the Lagrangian over time. It is a physical observable: action measures how much is happening in a system over a lapse of time. If you falsely believe that action is not an observable, explore the issue and convince yourself - especially if you give lectures.

http://www.motionmountain.net/onteaching.html

Further on the author also discusses measurements of this physical observable, saying

No single experiment yields [...] action values smaller than hbar

So I think he does mean it literally that action is physically measurable and is furthermore quantized. But in what sense, if any, can we discuss action as an observable?


My current thoughts:
Not useful in answering the question in general, but hopefully explains where my confusion is coming from.

I will admit, as chastised in the quote, I did not learn this in university and actually it just sounds wrong to me. The textbook covers classical and quantum mechanics, and I really don't see how this idea fits in either.

Classical physics
The system evolves in a clear path, so I guess we could try to measure all the terms in the Lagrangian and integrate them along the path. However, multiple Lagrangians can describe the same evolution classically. The trivial example being scaling by a constant. Or consider the Lagrangian from electrodynamics which includes a term proportional to the vector potential, which is itself not directly measurable. So if Action was actually a "physical observable", one could determine the "correct" Lagrangian, which to me sounds like nonsense. Maybe I'm reading into the phrasing too much, but I cannot figure out how to interpret it in a manner that is actually both useful and correct.

Quantum mechanics
At least here, the constant scaling issue from classical physics goes away. However the way to use the Lagrangian in quantum mechanics is to sum over all the paths. Furthermore, the issue of the vector-potential remains. So I don't see how one could claim there is a definite action, let alone a measurable one. Alternatively we could approach this by asking if the Action can be viewed as a self-adjoint operator on Hilbert space ... but the Action is a functional of a specific path, it is not an operator that acts on a state in Hilbert space and gives you a new state. So at first blush it doesn't even appear to be in the same class of mathematical objects as observables.

Ultimately the comments that experiments have measured action and show it is quantized, make it sound like this is just routine and basic stuff I should have already learned. In what sense, if any, can we discuss action as a physical observable? What would experiments measuring it even look like?

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The book propagates a myth.

Experiments measure angular momentum - not action, though these have the same units. One find that angular momentum in any particular (unit length) direction appears in multiples of $\hbar/2$, due to the fact that its components generate the compact Lie group SO(3) or its double cover U(2).

On the other hand, the action of a system defined by a Lagrangian is a well-defined observable only in the very general and abstract sense of quantum mechanics where every self-adjoint operator on a Hilbert space is called an observable, no matter whether or not we have a way to measure it.

Note that the action of a system along a fixed dynamically allowed path still depends on an assumed initial and final time, and goes to zero as these times approach each other. This implies that the action cannot be bounded below by $\hbar$.

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