I Chunks of action smaller than Planck's constant

[gerald V]

TL;DR Summary

The force law for electric charges allows to observe actions smaller than Plancks constant

The force two static electric charges exert on each other fulfills (velocity of light set unity)
\begin{equation}
F r^2 = N_1 N_2 \alpha \hbar \;,
\end{equation}
where $F$ is the force, $r$ is the mutual distance, $\alpha$ is the dimensionless fine structure constant, $\hbar$ is Planck's (reduced) constant and $N_1 , N_2$ are whole numbers, respectively. Both sides of the equation have the dimension of $action$.

If one holds one of the $N$ at the value unity and lets the other jump by plus/minus unity, the absolute value of the l.h.s.\ changes by an amount of $\alpha \hbar \approx \frac {\hbar}{137}$.
This means by measuring the change of force at any given distance one can immediately observe a jump of action significantly smaller than Planck's constant.

So what about the quantization of action in units of $\hbar$? Where is it valid and where is it not?
Is there any phase space physics behind the above? In particular, do $Fr$ (has dimension of $momentum$) and $r$ span something like phase space? Are these conjugate quantities?
If not, why is there quantization (in smaller chunks, though)?

Thank you very much in advance.

 

[renormalize]

So what about the quantization of action in units of $\hbar$? Where is it valid and where is it not?
Is there any phase space physics behind the above? In particular, do $Fr$ (has dimension of $momentum$) and $r$ span something like phase space? Are these conjugate quantities?
If not, why is there quantization (in smaller chunks, though)?

The notion that action is quantized is an artifact of the "old" quantum theory from before 1925. This is discussed by @A. Neumaier (a frequent contributor to Physics Forums) in his first answer here: https://physics.stackexchange.com/q...-sense-if-any-is-action-a-physical-observable, from which I quote:
"The action of a system along a fixed dynamically-allowed path depends on an assumed initial time and final time, and it goes to zero as these times approach each other - this holds even when it is an operator. Hence its eigenvalues are continuous in time and must go to zero when the time interval tends to zero. This is incompatible with a spectrum consisting of integral or half integral multiples of ℏ."

 

[PeterDonis]

The force two static electric charges exert on each other fulfills (velocity of light set unity)
\begin{equation}
F r^2 = N_1 N_2 \alpha \hbar \;,
\end{equation}
where $F$ is the force, $r$ is the mutual distance, $\alpha$ is the dimensionless fine structure constant, $\hbar$ is Planck's (reduced) constant and $N_1 , N_2$ are whole numbers, respectively. Both sides of the equation have the dimension of $action$.

What does this equation have to do with quantum mechanics? It happens to have $\hbar$ in it because of how you chose to define your units, but that doesn't make it a quantum equation. Action is not an inherently quantum quantity; classical physics can be formulated in terms of action too. Your equation is a simple example of such a formulation.