In simple terms: is Planck's elementary quantum of action h-bar, or is it zero? Or again: are photons elementary or divisible?
You can decide yourself which answer you like most comparing the arguments below, in this answer, and those in the other answer. The other answer is wrong because it is contradicted by thousands of experiments every single day, and because it contradicts the uncertainty relation. This discussion is a pretty study on the power of prejudices among physicists. Please ask your colleagues and professors to vote! Let us see how this statistics evolves over time. On 22 December 2020, the other, wrong answer has +10 votes, and this, correct one has -5 votes. On 7 February 2021: wrong one +12, correct one -3.
To increase the number of negative votes of this answer, here is a statement that is even more crazy, at first sight:
*Action is quantized in multiples of h-bar.*
Also this statement is almost never made in lectures or books, and thus sounds unfamiliar or even wrong. But it is correct nevertheless. Literature is found at the end. Please add a negative vote to this answer if you do not believe the statement. But let me go back to the original question.
The lines quoted in the question are from my free physics book website www.motionmountain.net, but I discovered this question only in December 2020, several years after it was posed and posted. There are at least two questions here.
First, is action an observable? The answer is yes, of course - even though it is not an observable that is used regularly. The operator is found in many textbooks. The operator is self-adjoint and linear. The quantity can be measured and even has a classical analog. Therefore, action is a physical observable. Indeed, the operator is not local in time - but that is the essence of action: it is an integral over time. (Also 'energy change', for example, is not local in time, and still is an observable.)
Action is the integral of kinetic energy minus potential energy over time. This is a quantity measurable in experiments. Action is the most fundamental quantity in physics. For example, the 10-volume Landau-Lifshitz textbook on physics starts with action on the very first page.
Another argument: The principle of least action implies that action can be measured. Otherwise we could not check whether action is indeed minimized. A further argument: also Schwinger's quantum action principle implies that action W is an observable. His principle uses expressions such as <psi_1|W|psi_2>, W being the action operator. There is no question that action is an observable and measurable. An additional argument: there is also a canonically conjugated observable to action: angle.
The best way to imagine action as an observable is to think about an action movie. A large action value is for movies with a "lot of action". This is a good approximation: if many things, people, cars, planes move, and do so rapidly, if they explode, burn, etc., then the action value (integral over time of kinetic minus potential energy) is large.
Action is observer invariant: action is a scalar. These are properties that only apply to observables. Please add a negative vote to this answer if you believe that action is not an observable.
The second question: is action bounded from below? Obviously, the statement is about a single measurement. The lower bound is the reason for the expression "quantum of action". I just mention that the statement that action is bounded from below is made by Planck, Bohr, Dirac, Sommerfeld, Sackur, Maslov, and many others, as listed in the references given below. Bohr spent quite some time explaining that $\hbar$ is the smallest observable action in nature.
The counter-example to the challenge proposed by Arnold Neumaier is wrong, despite the many votes (and despite him being a very good and very professional physicist): when the time difference delta t gets very small, the measured action value of a moving particle does not go to zero.
Here is the error in Arnold's argument: To determine the action of a free particle (the simplest example), you need to measure its energy at two different points in time. Even though, classically, action W is given by energy E times delta t, in nature and in quantum theory the action value W remains finite when delta t gets very small, because of the uncertainty relation: the energy difference increases when delta t gets smaller. (Measuring energy requires time, that time must be shorter than delta t.) That is quantum theory. For small delta t, h-bar is at work: the uncertainty relation - of energy and time in the case given - prevents that the measured action goes to zero when delta t goes to zero.
Here are two additional experimental examples showing how the challenge fails. First, detecting a spin 1/2 flip requires an action $\hbar$. There is no way to detect a spin flip with a small amount of action. In nature, there is just no way to detect 1/10 of a spin flip.
Second. Every photon detection requires an action $\hbar$. In nature, there is just no way to detect a 1/2 or a 1/100 of a photon. Ask anybody who does quantum optics experiments. There is no way to detect something smaller than a photon. This is the origin of the word "photon"! This is the reason that Planck called it "elementary quantum of action" - to contrast it with a "divisible" quantum of action. Photons cannot be divided! They are elementary.
Einstein never used the word photon - he preferred the term "light quantum".
There is no half photon. The smallest effect of an electromagnetic field is hbar. An electromagnetic field cannot be made to have a smaller effect. This is the radical result that nature tells us. The electromagnetic field cannot be split at will in really tiny amounts. There is a minimum chunk, an "atom" of light. And this smallest chunk is described by h-bar.
If you think that actions smaller that $\hbar$ can be measured, then do so. So far, nobody on the planet was successful.
Many people have tried to do experiments who measure actions smaller than h-bar. The famous discussion match between Einstein and Bohr can be seen as a continuous attempt by Einstein to measure actions below h-bar. But nature does not allow this. Even Einstein, who in the beginning was adamant that this was possible, later changed his mind, and explained the photoelectric effect with it.
When Planck discovered that black body radiation required quantization of action of the electromagnetic field, he did not know that photons had angular momentum. He just found that action was quantized, and called the corresponding quantity the "elementary quantum of action". He found that without quantization of action, he could not explain the spectrum of black body radiation. (He did not say or deduce from experiment that angular momentum was quantized. He said and deduced from experiment that action was quantized.) That was the birth of quantum theory.
Also the photoelectric effects shows that action in quantized. You can find the description in any physics book. If action were not quantized in minimum chunks, the effect would not exist.
Please add a negative vote to this answer if you believe that action is not bounded from below.
One can put it this way: to falsify the statement that "action is bounded from below in a single experiment" you just have to perform a single experiment - a real one or a thought experiment - that yields a value as small as desired, say $\hbar/10$, or smaller. In other words, you just have to find a single counter-example to the lower bound on action, or, in other words, to the quantum of action. That is a simple challenge. Enjoy it.
If you succeed in the challenge, or anybody else succeeds, let me know. I will publicly eat one of my T-shirts - which have the opposite statement printed on it, as can be seen on the page with the T-shirt image - in that case. But you will have a further advantage, much bigger: you will earn a trip to Stockholm and receive a big amount of money.
However, the probability to succeed is as low as the probability that you or anybody else succeeds in moving masses or energy faster than the speed of light. Both c and h-bar are invariant limits in nature. That is why the SI units are based on them.
In short, the quote that started the discussion remains correct, despite the authority and the votes behind the wrong answer. (I want to be clear: Arnold Neumaier almost always can be trusted; he is a serious and professional physicist.) But we always have to be careful when we do physics. We sometimes err, despite our best intentions.
Conclusion: The whole thread is bizarre. In 1899, quantum theory was born with Planck's discovery that action is quantized. In the years between 2017 and December 2020 this fundamental observation about nature is put into question on physics stackexchange. Not only is the quantization put into question, but also the fact that action is observable. Not only is the fact that action is an observable put into question, but it is also denied. Worse, other readers even agree. This is a very unfortunate chain of errors and prejudices. The properties of action imply momentum and energy conservation, among others. If action were not observable, energy and momentum would not be either. To say it drastically: if action were not observable, then nothing in nature would be.
Above all: if action were not quantized in multiples of hbar, photons would not exist. Atoms would not be stable - and would not exist, because electrons would fall into the nucleus. Physics books are full of many other examples telling what would happen if hbar would not be the minimum action.
Many people, me included, have counted photons, atoms and electrons. This is done every day throughout the world. Photons can be counted precisely because hbar is the minimum action that can be measured in any experiment. A honest physicist will never state that photons can be split, or that photons do not exist, or that atoms do not exist: that would be lies. hbar is the minimum action in nature.
Modern literature on the topic:
Levy-Leblond, Quantiques - an excellent quantum mechanics introduction, by the way.
J. Schwinger: his book on quantum theory explores the action operator in detail.
Lorenzo J Curtis, A 21st century perspective as a primer to introductory physics, European Journal of Physics 32 (2011) 1259–1274. This paper makes - independently - the same point as the quote in the question post, in section 5.2.
Lorenzo J. Curtis and David G. Ellis, Use of the Einstein–Brillouin–Keller action quantization, American Journal of Physics 72 (2004) 1521.
V. Hushwater, Quantum Mechanics from the Quantization of the Action Variable, Fortschr. Phys. 46 (1998) 6-8, 863-871.
M. N. Sergeenko, Quantization of the classical action and eigenvalue problem.
S Boughn, Wherefore Quantum Mechanics?
Older authors on the topic:
Maslov's book from 1972 (in French translation) and his papers on the Maslov index, which adds the final details to EBK quantization.
Einstein, Brillouin, Keller - with their three papers that define EBK quantization, which explicitely starts with the quantization of action.
Bronshtein, in his paper on the physics cube.
Sackur, Die universelle Bedeutung des sog. elementaren Wirkungsquantums, Annalen der Physik, 345 (1913) 67-86. He writes on the first page:
"Zu diesem Resultat gelangte ich mittels der Sommerfeldschen Hypothese, daß jede in der Natur ausgeübte Wirkung ein ganzzahliges Vielfache des elementaren Wirkungsquantums h ist."
Sommerfeld, Das Plancksche Wirkungsquantum und seine allgemeine Bedeutung für die Molekülphysik, Physikalische Zeitschr. 12 (1911) 1057-1069.
Bohr, 1911 and 1913.
Comments about the other answer, sentence by sentence.
The book propagates a myth. MM: wrong, see literature.
Experiments measure angular momentum - not action, though these have the same units. MM: wrong; typical photon counters do not measure angular momentum, neither does the photoelectric effect, nor most other experiments. In fact, hbar was discovered before quantum mechanical spin was used or known.
One find that angular momentum in any particular (unit length) direction appears in multiples of ℏ/2, due to the fact that its components generate the compact Lie group SO(3) or its double cover U(2). MM: correct, but not part of the topic.
That Planck"s constant ℏ is called the ''quantum of action'' is solely due to historical reasons. It does not imply that the action is quantized or that its minimal value is ℏ. MM: wrong, it is, as Planck showed analyzing black body radiation.
Indeed, 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. MM: true; but action can be measured: Integrate energy over time (for a free particle).
The action of a system along a fixed dynamically allowed path depends on an assumed initial and final time, MM: correct.
and goes to zero as these times approach each other. MM: wrong, as shown above. This statement contradicts Heisenberg's uncertainty relation.
This holds even when it is an operator. MM: wrong: the action operator has a discrete spectrum. See the cited literature.
Hence its eigenvalues are continuous in time and must go to zero when the time interval tends to zero. MM: Wrong, as this contradicts the uncertainty relation.
This is incompatible with a spectrum consisting of integral or half integral multiples of ℏ. MM: Wrong. See the cited literature.