# In what sense (if any) is Action a physical observable?

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?

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).

That Planck"s constant $$\hbar$$ 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 $$\hbar$$.

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. The action of a system along a fixed dynamically allowed path depends on an assumed initial and final time, and 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 $$\hbar$$.

• The value of the action between two time slices aka Hamilton's function is an observable on the phase space. – Prof. Legolasov Dec 5 '20 at 8:09
• @Prof.Legolasov: How can it be observed? – Arnold Neumaier Dec 6 '20 at 15:15
• it is a function on the phase space, therefore it can be observed by observing the coordinate and momentum and plugging them into the model-specific formula for the Hamilton-Jacobi function. In QM it becomes an operator that acts on the Hilbert space. – Prof. Legolasov Dec 12 '20 at 7:18
• @Prof.Legolasov: But it is nonlocal in time, hence its computation requires the complete path in phase space, of which only a small initial section is observable at any given time. – Arnold Neumaier Dec 14 '20 at 9:54
• Arnold, can you answer the post below? It suggests that there is an error in your argument. – Christian Jan 26 at 6:28

In the language of the OP, action is a functional, since it is an integral of the Lagrangian... but over an arbitrary path. In other words, it is an abstract mathematical object, which has no counterpart in the real world.

This functional is then minimized in respect to all possible trajectories. In quantum mechanical terms the action along the optimal trajectory corresponds to the phase of a wave function, which is measurable (although defined up to a constant), e.g., in the experiments on Aharonov-Bohm effect, and any other interference experimente. The fact was recognized long before the advent of the path integrals - Landau&Livshitz derive quasiclassical approximation is an eiconal expansion of the phase of the wave function, which they openly call action.

In simple terms: is Planck's elementary quantum of action h-bar, or is it zero? Or again: are photons elementary or divisible?

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.

Planck 1899.

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.

• Hello Motion Mountain, your answer does not really answer the question. Your argumentation is not really physical and seems to be an opinion rather than a scientific truth. – AlmostClueless Dec 5 '20 at 8:31
• @AlmostClueless The improved answer answers the question. The answer may "seem" to be an opinion. But it is not. Arguments are "physical" when they are in agreement with experiment. The answer agrees with experiment. The (common) mistake in the other answer is now explained. – Motion Mountain Jan 3 at 12:03
• An integral over time depends continuously on the initial and final time. 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 multiples of time. Thus this answer is bogus. Action is not angular momentum! – Arnold Neumaier Jan 26 at 17:48
• My criticism is not about measurement. Born's rule says that eigenvalues are measured, and the eigenvalues of the action operator is a continuous function of the initial time and the final time. Hence the action operator cannot have a spectrum consisting of integral or halfintegral multiples of $\hbar$. Thus there is no theoretical reason why the action should be bounded away from zero. Note that in quantum optics one measures angular momentum, not action, and the angular momentum spectrum is that of a compact group, hence quantized. But this is special to angular momentum. – Arnold Neumaier Jan 29 at 10:02
• A photon counter does not measure action but photon number. – Arnold Neumaier Feb 1 at 15:53