Nature has a fascinating property, discovered by Max Planck: action is quantized. Or, in simple terms: change occurs in smallest steps.

●  Summary: action is quantized
●  History and experiments
●  How to forget action quantization
●  The silence of Feynman - and others
●  More fallacies and false counter-arguments
●  How to use action quantization
●  Conclusion: bets and future tests
●  Literature and citations
●  Acknowledgements and declarations

Summary: action is quantized

Action is quantized in multiples of the quantum of action h-bar. This is a statement about measured action values.

In 1899, Max Planck discovered, in his measurements on light, the existence of what he later called the `elementary quantum of action'. An `elementary quantum' is an indivisible smallest entity. Planck's discovery and his choice of terms gave `quantum' theory its modern name. (The term `quantum' was introduced by Galileo, who used it to name something that is indivisible. In particular, Galileo used the term in place of the greek term `atom'.)

Today we use h-bar, with a value of about 10-34 Js, to describe this fundamental property about nature, and call h-bar `Planck's elementary quantum of action'. The discovery was extended by Wilson, Bohr, Sommerfeld, Ishiwara, Einstein, de Broglie, Heisenberg, Schrödinger, Born, Jordan, Pauli, Dirac, Brillouin, Keller, Maslov, and many others into textbook quantum theory.

In nature, measured action values are found to be quantized. This means, first of all, that measurements never find a value smaller than h-bar. There is no exception to this statement, in any experiment ever performed. In nature, there is an indivisible quantum of action.

Quantization of action also means that measurements only find action values that are integer multiples of h-bar. There is no exception to this statement, in any experiment ever performed. In nature, action consists of quanta.

Action measures change. The quantization of action thus implies that in nature, change occurs in small steps. This strange property is the reason for all the fascinating and sometimes counter-intuitive properties of quantum theory.

Until today, no physics paper and no physics book ever denied action quantization. Nevertheless, false statements on the issue are found regularly on the internet, including several Wikipedia articles. Even worse, action quantization is being denied by some professional physicists. Action quantization thus resembles the Corona virus: in both cases, there are people who deny its existence. It is shown briefly below how this mistaken pre-quantum belief about action returned, despite over 120 successful years of quantum theory. A few fascinating aspects of quantum theory are also presented. For more, see volume IV of the Motion Mountain Physics Textbook: The Quantum of Change. pdf


History and experiments

Planck discovered that action is quantized while studying thermal radiation. He called the smallest action unit the `elementary quantum of action´. In this way, Planck introduced the term `quantum' into physics. As mentioned, he took the term `quanta' from Galileo, who had already used it for the smallest, indivisible chunks of matter. Planck also used the term `elementary' to stress that every action, every change in nature, is made up of those smallest quanta (Planck, 1906, 1907). He even checked that the value of the quantum of action and the number of action quanta are relativistically invariant.

Planck found that light consists of `light quanta', today called `photons'. Planck also found that h-bar is a fundamental constant of nature.

No experiment every detected half a photon, or some other fraction of a photon. Photon counters are used all over the world, all the time. h-bar is the smallest possible action value. This is valid for light and, as it turned out, for every other physical system. This includes every type of matter and every type of radiation.

No experiment using any quantum particle – e.g., during electron interference, during spin experiments, or in molecular interference – ever found an action value smaller than h-bar.

Several quantization methods – i.e., calculations about quantum effects ‐ started from the quantization of action. Bohr used it, and so did Wilson, Sommerfeld, Ishiwara and many others. Most famous is the Einstein-Brillouin-Keller quantization. All these methods start from action quantization and deduce the quantum properties of atoms to full precision,

Without action quantization, atoms would not exist. Action quantization determines the energy levels of electrons in atoms. In simple terms, action quantization determines the size and the electron clouds of atoms.

Pauli knew that there is a quantum of action. He knew this even when the discovery of spin 1/2 surprised everybody in the community of quantum physicists.

Dirac, in the 1970s, confirmed in a letter to Gardner that there is a smallest action value.

Action quantization also describes every interaction: every interaction between physical systems is an exchange of an integer number of action quanta.

Action has the same unit as angular momentum. Both observables are quantized.

The quantum of action is so fundamental that it is used in the definition of the SI units. The quantum of action is indivisible. The quantum of action is a relativistic invariant, and it is a limit of nature.


How to forget action quantization

In many languages, `Planck's elementary quantum of action' was shortened to `Planck's constant'.

The next step was to avoid using action in the teaching of physics. Despite Landau, Lifshitz, Hilbert, Einstein, Taylor and many other heroes, action got into disuse.

Even the Nobel committee made a mistake: Planck got the prize "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta." In reality, Planck discovered action quanta, not energy quanta.

A further step, decades later, was to claim, erroneously, that action is not a quantum observable. The next mistake is the claim that action cannot be measured. Both claims are wrong - the correspondence principle shows this.

Then followed the claim that action is "not quantized". This claim is in contrast with every experiment. It is wrong.

The final step is to claim that Planck and Einstein were mistaken. This statement may make certain people feel good, but in this case, it is wrong.


The silence of Feynman - and others

Feynman started his career with the path integral formulation of quantum theory. He assumed that particles take classical paths, and then averaged over those paths.

His formulation implies the quantization of action, but he never makes it explicit. The closest is his statement that a particle is a rotating arrow. This has led people to think that any partial rotation can be measured, however small it might be. The truth: only full circles of the arrow motion of a particle can be detected in experiments. (Likewise, Feynman diagrams do not imply that quantum particles follow paths.)

In simple words: Feynman avoided the issue. The avoidance became standard. Many people who wrote popular books about fundamental physics in the past decades have avoided the issue as well.

Fortunately, so far, there is not one published physics paper or physics book stating that action is not quantized. In contrast, the literature section below lists a selection of textbooks and published papers, including recent ones, stating clearly that action is quantized.


More fallacies and false counter-arguments

"Action is not a quantum observable." This statement is wrong; there is a linear and self-adjoint operator for action. (In fact, there are several operators that yield action in the classical limit.) Many textbooks present one or several of these operators. Among them is the textbook by Schwinger.

"Action needs two measurements, not one." True. This often the case in nature; also displacement measurements, energy change measurements, speed measurements and many measurements of other observables require two measurements. Nothing particular can be deduced from this fact.

"Action is not quantized - Planck's result was part of the old quantum theory and was changed later on." Wrong statement, in contrast with observation. Planck's result was not changed later at all, but confirmed by thousands of researchers, without any exception. Planck's heirs do not need to give back his Nobel prize medal. In fact, the "old" quantum theory has been proven to be fully correct, to be equivalent to the "new" quantum theory, and to be very useful in calculations and teaching (see literature).

"Action is not quantized" is claimed in Wikipedia, Quora and several physics forums. However, experiments, starting with those of Planck himself, prove the opposite. Nowadays such experiments are performed every day in laboratories around the world.

"Action is not quantized - Mach-Zehnder interferometers prove it." Wrong, the detectors in these interferometers measure discrete photons, and prove that action is quantized. No device ever observed fractions of a photon.

"Action is not quantized - the action integral is continuous in its starting and final states." Those states obey the uncertainty relations. Taking the uncertainty in to account leads to quantization of the action integral, as shown by Wilson, Sommerfeld, Ishiwara, Einstein, Brillouin, Keller and many others who used the method.

"Straight particle motion over a short time interval yields action values smaller than the quantum of action." No: the uncertainty relation prevents this.

"Action is not quantized for free or unbound states." This statement is wrong, for the same reasons just mentioned.

"Action is not quantized because I can imagine/think that it is not." The second half of the statement is correct. It is even valid for every quantum observable. But it does not change the result that measured, observed action values are always quantized.

It must be stressed that the statement "action is quantized" refers to the measured values of the action. In quantum theory, this is a statement on the eigenvalues of the action operator.

In short, quantum theory owes its name to the quantization of action. Nevertheless, Wikipedia, Quora, and various physics forums are spreading disinformation about the issue. Many years of doublespeak and of careful avoidance of checks with experiments are needed to deny action quantization. In every experiment and in every observation, action is quantized.


How to use action quantization

Action is useful! Action is the quantity that measures how much is happening in a physical system. For example: action is high in action movies; physical action is high in explosions and fast motion.

Action is the most fundamental quantity in physics - and in nature! This is rarely told, but an open secret among physicists.

Action measures the amount of change occurring in a physical system. Every motion occurs in a way that minimizes change.

Action is a relativistic invariant. Action – i.e., change – is minimized in every example of motion. This applies to elementary particles, chemistry and all other quantum motion, to everyday motion, to gravitational waves, to black hole motion and to the rest general relativity.

Have a look at Edwin Taylor's "A call to action" and his web pages on the topic. This is physics teaching at its best.

As Planck wrote in 1907 (my own translation): "To every change in nature corresponds a certain number of quanta of action that is independent of the choice of reference system." This statement can be used to determine the properties of atoms, the scattering of X-rays, the heat capacity of solids, and every other measurable observable of the quantum world.

In nature, change is not continuous. It occurs in smallest steps. This is the revolutionary aspect of quantum theory. (More can be said: Whenever one tries to measure a system or a process using action values below h-bar, probabilities arise. Nevertheless, the limit h-bar cannot be overcome.)

The literature section below lists some papers and books showing how action quantization leads to Schrödinger's equation and to Dirac's equation.


Conclusion: bets and future tests

In science, every statement must be checked continuously, again and again. A sweeping statement like Planck's "action is quantized" must be checked with particular care. If you have a counterargument or notice an issue missing above, just send a note. I will add it to this page.

The above results imply two simple predictions about experiments: (1) Nobody will ever measure an action value that is not a multiple of h-bar. (2) Nobody will ever measure an action value that is smaller than h-bar. These are safe bets.

The chance to falsify these predictions is as low as the chance to observe energy or matter moving faster than light. Fractions of photons are never found. Atoms exist and, in the ground state, all atoms of an element are of equal size.

The quantization of action, i.e., the quantization of change, is one of the deepest and most fascinating properties of nature.

Certain people claim that nature's fundamental limits h-bar, c, c^4/4G or any of their combinations can be overcome. However, this is not the case in nature. In fact, the existence of these three limits – also called collectively the Planck limits – is an essential part of the world around us. The limits determine quantum theory, determine special relativity and determine general relativity. Most importantly, quantum theory follows from the existence of a smallest, indivisible, elementary quantum of action h-bar.

The Planck limits are so essential that most measurement units are directly defined using them. (One measurement unit, the second, is defined indirectly, because G is too hard to measure with high precision.) Most famously, the kilogram is defined using the indivisibility of h-bar, i.e., using the quantization of action.


Literature and citations

You can type "quantization of action" into Google Books. You can also search for "quantum of action" or related concepts, even in other languages. Numerous textbooks come up. You can do the same searches in scientific publications using Google Scholar. (Of course, you can also search for statements denying action quantization. You will not find anything.) A few examples of textbooks and papers follow.

M.N. Sergeenko, General solution of the Schrödinger equation (2022), arXiv:2201.02199. He writes: "The [action] quantization condition (20) [...] reproduces the exact energy spectra for all known solvable 2TP problems in QM." See also his previous papers on the topic.

A. Khrennikov, Is the Devil in h?, Entropy 23 (2021) 632. doi: 10.3390/e23050632. Khrennikov explains in detail Bohr's insistence on an "indivisible quantum of action".

S. Boughn, Wherefore Quantum Mechanics?, (2019) arXiv:1910.08069. He writes "Duane (1923), Breit (1923) and Compton (1923) applied the quantization of action to the interaction of x-ray photons with an infinite, periodic crystal lattice and were able to obtain Bragg’s law of diffraction" and "The analyses of Duane et al. provide seminal illustrations of a direct path from the quantization of action to the wave behavior of particles and photons." And "... the Heisenberg indeterminacy relation is a direct consequence of the quantum of action." And "... the quantization of action that is primal in quantum physics but absent in classical physics."

H. Capellmann, The Development of Elementary Quantum Theory (2017). He explains how Born used the quantization of action to develop modern quantum theory.

A Zagoskin, Quantum Mechanics: A Complete Introduction: Teach Yourself (2015). This 420-page textbook contains the statement: "... the celebrated Planck constant h is nothing else but the quantum of action - the smallest change of action one can ever get." He also has an exercise for the reader on the topic. Asking how to continue the sentence "The Planck constant is called the quantum of action, because" the correct answers are: "action is a multiple of h" and "h is the smallest value action can have".

M. Bartelmann, B. Feuerbacher, T. Krüger, D. Lüst, A. Rebhan and A. Wipf, Theoretische Physik 3 - Quantenmechanik, Springer (2013). This common physics textbook states on page 15: "Das sogenannte Wirkungsintegral S in der Quantisierungsregel (1.43) ist also immer ein Vielfaches einer kleinsten Wirkung h. Dies erklärt den Namen Wirkungsquantum für die Naturkonstante h."

L.J. Curtis, A 21st century perspective as a primer to introductory physics, European Journal of Physics 32 (2011) 1259–1274, section 5.2. He writes "(1) Action exists only as integer multiples of an indivisible basic unit h-bar/2." and "(10) The ‘mechanical action’ (orbital angular momentum) is quantized in multiples of 2(h-bar/2), and is associated with the ‘parity’ or handedness of an atomic state."

M. Bucher, Rise and fall of the old quantum theory (2008), arXiv:0802.1366. Explains the limits of the "old" quantum theory, and why it is still applicable. One of many recent papers on the topic.

E. Zeidler, Quantum Field Theory I - Basics in Mathematics and Physics (2006). He writes: "According to Planck, the smallest amount of action in nature is equal to h = 6.260 0755 · 10−34Js (1.1) where 1 Joule = 1 kg · m2/s2. We also introduce hbar = h/2π. The universal constant h is the famous Planck quantum of action (or the Planck constant). Observe that the action of typical processes in daily life has the magnitude of 1 Js. Therefore, the Planck constant is tiny. Nevertheless, the quantization of action has enormous consequences."

S. Ivanov, The Sources of the Quantum Mechanics: Fundamentals for Chemists, Springer (2006). This textbook states: "The main particularity of the phenomena of the microscopic world consists of character discreteness, appearing in the existence of an indivisible quantum of action h."

L.J. Curtis and D.G. Ellis, Use of the Einstein–Brillouin–Keller action quantization, American Journal of Physics 72 (2004) 1521. A pretty article on how to use action quantization for teaching students.

J. Schwinger, B. Englert, Quantum Mechanics, Springer (2003). This textbook on quantum theory explores the (one type of the) action operator in detail, showing its existence and properties, including the explanation that it is equal the classical action in the limit of large action values.

H. Pietschmann, Quantenmechanik verstehen, Springer (2003). "Planck interpretierte sie als Quantisierung der Wirkung; demnach sollte die Wirkung (die klassische Größe der Dimension Energie x Zeit) immer nur in ganzzahligen Vielfachen dieser Grundeinheit auftreten, ganz ähnlich wie dies in (1.15) von der Ladung gefordert ist."

M.N. Sergeenko, Quantization of the classical action and eigenvalue problem (2002) "As Sommerfeld stated, the method of action-angle variables then provided “a royal road to quantization”. One had only to solve the problem in classical mechanics using action-angle variables, and motion could be immediately quantized by replacing the J’s with integral multiples of Planck’s constant h." and "In conclusion, we have reduced the eigenvalue problem in quantum mechanics to quantization of the classical action."

V. Hushwater, Quantum mechanics from the quantization of the action variable, Fortschr. Phys. 46 (1998) 6-8, 863-871. The paper deduces the time-dependent Schrödinger equation from the quantization of action.

V. Hushwater, A path from the quantization of the action variable to quantum mechanical formalism, Foundations of Physics 28 (1998) 167–184. The paper deduces the time-independent Schrödinger equation from action quantization.

A. Zeilinger, On the interpretation and philosophical foundation of quantum mechanics, in "Vastakohtien todellisuus", Festschrift for K.V. Laurikainen, U. Ketvel et al. (Eds.), Helsinki University Press (1996). In that paper, Zeilinger writes: "... there is a universal smallest action which can be exchanged in a physical process".

A. Messiah, Quantum Mechanics (1995). This well-known textbook states on page 41: "The essential fact is the appearance of discontinuity on this scale, connected with the existence of an indivisible quantum of action."

A. Shimony, Conceptual foundations of quantum, in P. Davies ed., The New Physics (1992). "What quantum mechanics adds to this general principle is the indivisible quantum of action, ..."

J.-M. Levy-Leblond and F. Balibar, Quantique - Rudiments, 1984. This excellent introductory textbook on quantum mechanics shows in the first chapter that action values below h-bar are not observed.

K. Dinstl and P. Fischer, Der Laser - Grundlagen und klinische Anwendung, Springer (1981). The textbook states: "... eine Wirkung ist und daß diese nach Planck nur ein ganzzahliges Vielfaches des Wirkungsquantums h sein kann."

V.P. Maslov, Théorie des perturbations et méthodes asymptotiques (1972). This well-known book and his papers on the Maslov index added the final mathematical details to Einstein-Brillouin-Keller action quantization.

F. Hund, Geschichte der Quantentheorie, BI Hochschultaschenbücher 200/200a (1967). Hund knew all founders of quantum theory personally; he writes about the quantum of action: "erkannte man später die viel umfassendere Rolle, nämlich die Abänderung jeder klassischen Bechreibung".

J.B. Keller, Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems, Ann. Phys. (N.Y.) 4 (1958) 180–188. This is the "K" paper of the EBK action quantization method.

L. de Broglie, The Revolution in Physics: A Non-mathematical Survey of Quanta (1956). Here and elsewhere, he calls h "atome d'action", or "atom of action". He explains the expression by underlining that the action h cannot be divided into smaller parts.

W. Heisenberg, Die Plancksche Entdeckung und die philosophischen Grundfragen des Atomlehre (1958). He writes, for example, "In the last analysis the nonvisualizeable character [unanschauliche Character] of modern atomic physics rests on Planck’s quantum of action – on the existence of a criterion for the notion of atomic smallness in the natural laws."

L. de Broglie, Max Planck und das Wirkungsquantum, Physikalische Blätter (1948). "Indem Planck die Bedeutung seiner Hypothese vertiefte, zögerte er nicht, zu zeigen, daß die Konstante h ein Wirkungsquant darstellt, und daß es letzten Endes die Wirkung und nicht die Energie ist, die gequantelt ist, wie man zuerst hatte vermuten können. Die Plancksche Konstante h stellt deshalb eine Art Wirkungsatom dar, eine unteilbare von der Art, daß sie die kleinste ist, die bei irgend einem physikalischen Phänomen eine Rolle spielt. Und damit schien diese unerwartete Diskontinuität, deren Existenz das Genie Plancks in der Strahlung des schwarzen Körpers enthüllt hatte, sogleich von sehr viel größerer Tragweite und Auswirkung in der ganzen Physik zu sein."

W. Heisenberg, Das Plancksche Wirkungsquantum, 1945.

M. Planck, Zur Geschichte der Auffindung des physikalischen Wirkungsquantums, Naturwissenschaften 31, Nr. 14 (1943) 153–159. He writes on page 158: "Die von der Natur der Oszillatoren unabhängige Konstante a' bezeichnete ich mit h und nannte sie, da sie die Dimension eines Produktes yon Energie und Zeit besitzt, das elementare Wirkungsquantum oder das Wirkungselement, im Gegensatz zum Energieelement hv." And, later on: "Viel aussichtsloser erschien die Aufgabe, den Zahlenwert der zweiten Konstante, h, die zuerst völlig in der Luft hing, experimentell zu prüfen. Daher war es mir eine große Überraschung und Freude, als J. FRANCK und G. HERTZ bei ihren Versuchen über die Erregung einer Spektrallinie durch Elektronenstöße eine Methode zu ihrer Messung fanden, wie man sie sich direkter nicht wünschen kann. Damit war auch der letzte Zweifel an der Realität des Wirkungsquantums verschwunden."

N. Bohr, Atomtheorie und Naturbeschreibung, Springer (1931). The four articles in the book are based on the indivisibility of the quantum of action and continuously underline it. Bohr wrote numerous papers about the "indivisible quantum of action". The expression is found in most of his texts. A few are: N. Bohr, The quantum postulate and the recent development of atomic theory, Suppl. Nat. 121 (1928) 580–590. N. Bohr, Wirkungsquantum und Naturbeschreibung, Naturwissenschaften 17 (1929) 483–486, doi: 10.1007/BF01505680. N. Bohr, The quantum of action and the description of nature, in J. Kalckar, editor, Foundations of Quantum Physics I (1926–1932) Volume 6, Elsevier Amsterdam (1985) pp. 201–217.

A. Eddington, The nature of the physical world (1928). Eddington calls h an "atom of action" (because it cannot be divided further) and has a section with this title.

L. Brillouin, Remarques sur la mécanique ondulatoire, J. Phys. Radium 7 (1926) 353–368. The "B" paper on the EBK method using action quantization. In the summary he states (simplified): "ce qui donne les conditions I = n h (n entier)" and similar statements are found in the text.

A. Einstein, Zum Quantensatz von Sommerfeld und Epstein, Verh. Dtsch. Phys. Ges. 19 (1917) 82–92. This is the starting "E" paper of the EBK action quantization method.

J. Ishiwara, "Universelle Bedeutung des Wirkungsquantums", Proceedings of the Tokyo Mathematico-Physical Society, 2nd Series, 8 (1915) 106–116. doi:10.11429/ptmps1907.8.4_106. English reedition as "The universal meaning of the quantum of action", European Physical Journal H 42 (2017) 523-536. doi:10.1140/epjh/e2017-80041-1.

J. Ishiwara, "Über den Fundamentalsatz der Quantentheorie", Proceedings of the Tokyo Mathematico-Physical Society, 2nd Series, 8 (1915) 318–326. doi:10.11429/ptmps1907.8.10_318.

M. Bronshtein, in his paper on the physics cube. See

O. 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."

A. Sommerfeld, Das Plancksche Wirkungsquantum und seine allgemeine Bedeutung für die Molekülphysik, Physikalische Zeitschrift 12 (1911) 1057-1069.

M. Planck, Zur Dynamik bewegter Systeme (1907). "Nimmt man hinzu den Satz, dass für die Wirkungsgrösse ein ganz bestimmtes Elementarquantum [14] existirt: h = 6,55 ⋅ 10^−27 erg sec, so kann man auch sagen: Einer jeden Veränderung in der Natur entspricht eine bestimmte, von der Wahl des Bezugsystems unabhängige Anzahl von Wirkungselementen."

Max Planck, Vorlesungen über die Theorie der Wärmestrahlung, Verlag Joh. Amb. Barth, Leipzig (1906) page 154. "Und doch wird die Thermodynamik der Strahlung erst dann zum vollständig befriedigenden Abschluß gelangt sein, wenn die Konstante h in ihrer vollen universellen Bedeutung erkannt ist. Ich möchte dieselbe als „elementares Wirkungsquantum" oder als „Wirkungselement" bezeichnen, weil sie von derselben Dimension ist wie diejenige Größe, welcher das Prinzip der kleinsten Wirkung seinen Namen verdankt."

M. Planck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Königlich-Preußischen Akademie der Wissenschaften zu Berlin (1899) 440–480. This text introduces the constant h, and, on the last two pages, the Planck units.


Acknowledgements and declarations

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