Action of classical mechanics is bounded from below by the Planck constant

Question

I was wondering about the dimensions of the Planck constant ($h$) and the dimensions of the action, which are obviously same. Then a train of thought led me to conclude that uncertainty principle can be stated as "minimum action required to make an observation is $hbar/2$".
Also, principle of least action is the governing law for classical dynamics and uncertainty principle is a consequence of commutation relations which are stated as postulate in theory of quantum mechanics.
Now, my question is that can this postulate of quantum mechanics be interpreted/proved as setting the limit for action i.e. $|S| ge kh$ or the vice-versa i.e. $|S| ge kh implies Delta{x}Delta{p} ge hbar/2$, where $k$ is some positive real number.
I have not seen this been interpreted like this. So this could be a trivial interpretation. If this is the case please refer me to the relevant reference. If this is not the case then this could have consequences for interpretation of action, uncertainty principle and QM in general, so please help prove or disprove this relation.
These are some of the observations that an answer may address:

Principle of least action leads to Feynman's path integral formulation of quantum mechanics.
Principle of least action leads to classical Hamiltonian mechanics which can be expressed through Poisson brackets notation which directly corresponds to commutators in QM.
Classically, for SHM kinetic energy and potential energy are the same over a time period so casually it may seem that the action taken over a time period is 0 but considered more carefully the Lagrangian is a function of q and $dot{q}$ which follow uncertainty principle so the integral can be non-zero.

Edit
Looking at the lack of responses, I have to ask - is this a question worth investigating further? If yes, can someone point to some authority in foundations of quantum physics? I am a amateur physicist at the present, working alone and do not know how to proceed.

Andrew · Answer

Bohr-Sommerfeld quantization is based on a relationship like the one you are discussing
begin{equation}
oint vec{p} cdot d vec{q} = n h, n in mathbf{N}
end{equation}
where the integral is over a closed path in phase space, $n$ is an integer, and $h$ is Planck's constant. The left had side has units of action.
The modern point of view is that this relationship arises as a consequence of the WKB approximation (up to a small correction where $nrightarrow n+1/2$). However, note that this approach only applies to bound systems.
In general there is no bound on the action in quantum mechanics. For example, in the path integral approach, there is a contribution from all possible paths weighted by $e^{i S/hbar}$, where $S$ is the action of the path, and there is no bound on the action.

Action of classical mechanics is bounded from below by the Planck constant

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