# Paul Dirac’s Delta Function

Physical reality is nothing but a bunch of spikes and pulses—or glitches.  Take any smooth phenomenon, no matter how benign it might seem, and decompose it into an infinitely dense array of infinitesimally transient, infinitely high glitches.  Then the sum of all glitches, weighted appropriately, becomes the phenomenon.  This might be called the “glitch” function—but it is better known as Green’s function in honor of the ex-millwright George Green who taught himself mathematics at night to became one of England’s leading mathematicians of the age.

The δ function is thus merely a convenient notation … we perform operations on the abstract symbols, such as differentiation and integration …

PAM Dirac (1930)

The mathematics behind the “glitch” has a long history that began in the golden era of French analysis with the mathematicians Cauchy and Fourier, was employed by the electrical engineer Heaviside, and ultimately fell into the fertile hands of the quantum physicist, Paul Dirac, after whom it is named.

## Augustin-Louis Cauchy (1815)

The French mathematician and physicist Augustin-Louis Cauchy (1789 – 1857) has lent his name to a wide array of theorems, proofs and laws that are still in use today. In mathematics, he was one of the first to establish “modern” functional analysis and especially complex analysis. In physics he established a rigorous foundation for elasticity theory (including the elastic properties of the so-called luminiferous ether).

In the early days of the 1800’s Cauchy was exploring how integrals could be used to define properties of functions.  In modern terminology we would say that he was defining kernel integrals, where a function is integrated over a kernel to yield some property of the function.

In 1815 Cauchy read before the Academy of Paris a paper with the long title “Theory of wave propagation on a surface of a fluid of indefinite weight”.  The paper was not published until more than ten years later in 1827 by which time it had expanded to 300 pages and contained numerous footnotes.  The thirteenth such footnote was titled “On definite integrals and the principal values of indefinite integrals” and it contained one of the first examples of what would later become known as a generalized distribution.  The integral is a function F(μ) integrated over a kernel

Cauchy lets the scale parameter α be “an infinitely small number”.  The kernel is thus essentially zero for any values of μ “not too close to α”.  Today, we would call the kernel given by

in the limit that α vanishes, “the delta function”.

Cauchy’s approach to the delta function is today one of the most commonly used descriptions of what a delta function is.  It is not enough to simply say that a delta function is an infinitely narrow, infinitely high function whose integral is equal to unity.  It helps to illustrate the behavior of the Cauchy function as α gets progressively smaller, as shown in Fig. 1.

In the limit as α approaches zero, the function grows progressively higher and progressively narrower, but the integral over the function remains unity.

## Joseph Fourier (1822)

The delayed publication of Cauchy’s memoire kept it out of common knowledge, so it can be excused if Joseph Fourier (1768 – 1830) may not have known of it by the time he published his monumental work on heat in 1822.  Perhaps this is why Fourier’s approach to the delta function was also different than Cauchy’s.

Fourier noted that an integral over a sinusoidal function, as the argument of the sinusoidal function went to infinity, became independent of the limits of integration. He showed

when ε << 1/p as p went to infinity. In modern notation, this would be the delta function defined through the “sinc” function

and Fourier noted that integrating this form over another function f(x) yielded the value of the function f(α) evaluated at α, rediscovering the results of Cauchy, but using a sinc(x) function in Fig. 2 instead of the Cauchy function of Fig. 1.

## George Green’s Function (1829)

A history of the delta function cannot be complete without mention of George Green, one of the most remarkable British mathematicians of the 1800’s.  He was a miller’s son who had only one year of education and spent most of his early life tending to his father’s mill.  In his spare time, and to cut the tedium of his work, he read the most up-to-date work of the French mathematicians, reading the papers of Cauchy and Poisson and Fourier, whose work far surpassed the British work at that time.  Unbelievably, he mastered the material and developed new material of his own, that he eventually self published.  This is the mathematical work that introduced the potential function and introduced fundamental solutions to unit sources—what today would be called point charges or delta functions.  These fundamental solutions are equivalent to the modern Green’s function, although they were developed rigorously much later by Courant and Hilbert and by Kirchhoff.

The modern idea of a Green’s function is simply the system response to a unit impulse—like throwing a pebble into a pond to launch expanding ripples or striking a bell.  To obtain the solutions for a general impulse, one integrates over the fundamental solutions weighted by the strength of the impulse.  If the system response to a delta function impulse at x = a, that is, a delta function δ(x-a), is G(x-a), then the response of the system to a distributed force f(x) is given by

where G(x-a) is called the Green’s function.

## Oliver Heaviside (1893)

Oliver Heaviside (1850 – 1925) tended to follow his own path, independently of whatever the mathematicians were doing.  Heaviside took particularly pragmatic approaches based on physical phenomena and how they might behave in an experiment.  This is the context in which he introduced once again the delta function, unaware of the work of Cauchy or Fourier.

Heaviside was an engineer at heart who practiced his art by doing. He was not concerned with rigor, only with what works. This part of his personality may have been forged by his apprenticeship in telegraph technology helped by his uncle Charles Wheatstone (of the Wheatstone bridge). While still a young man, Heaviside tried to tackle Maxwell on his new treatise on electricity and magnetism, but he realized his mathematics were lacking, so he began a project of self education that took several years. The product of those years was his development of an idiosyncratic approach to electronics that may be best described as operator algebra. His algebra contained mis-behaved functions, such as the step function that was later named after him. It could also handle the derivative of the step function, which is yet another way of defining the delta function, though certainly not to the satisfaction of any rigorous mathematician—but it worked. The operator theory could even handle the derivative of the delta function.

Perhaps the most important influence by Heaviside was his connection of the delta function to Fourier integrals. He was one of the first to show that

which states that the Fourier transform of a delta function is a complex sinusoid, and the Fourier transform of a sinusoid is a delta function. Heaviside wrote several influential textbooks on his methods, and by the 1920’s these methods, including the Heaviside function and its derivative, had become standard parts of the engineer’s mathematical toolbox.

Given the work by Cauchy, Fourier, Green and Heaviside, what was left for Paul Dirac to do?

## Paul Dirac (1930)

Paul Dirac (1902 – 1984) was given the moniker “The Strangest Man” by Niels Bohr during his visit to Copenhagen shortly after he had received his PhD.  In part, this was because of Dirac’s internal intensity that could make him seem disconnected from those around him. When he was working on a problem in his head, it was not unusual for him to start walking, and by the time he he became aware of his surroundings again, he would have walked the length of the city of Copenhagen. And his solutions to problems were ingenious, breaking bold new ground where others, some of whom were geniuses themselves, were fumbling in the dark.

Among his many influential works—works that changed how physicists thought of and wrote about quantum systems—was his 1930 textbook on quantum mechanics. This was more than just a textbook, because it invented new methods by unifying the wave mechanics of Schrödinger with the matrix mechanics of Born and Heisenberg.

In particular, there had been a disconnect between bound electron states in a potential and free electron states scattering off of the potential. In the one case the states have a discrete spectrum, i.e. quantized, while in the other case the states have a continuous spectrum. There were standard quantum tools for decomposing discrete states by a projection onto eigenstates in Hilbert space, but an entirely different set of tools for handling the scattering states.

Yet Dirac saw a commonality between the two approaches. Specifically, eigenstate decomposition on the one hand used discrete sums of states, while scattering solutions on the other hand used integration over a continuum of states. In the first format, orthogonality was denoted by a Kronecker delta notation, but there was no equivalent in the continuum case—until Dirac introduced the delta function as a kernel in the integrand. In this way, the form of the equations with sums over states multiplied by Kronecker deltas took on the same form as integrals over states multiplied by the delta function.

In addition to introducing the delta function into the quantum formulas, Dirac also explored many of the properties and rules of the delta function. He was aware that the delta function was not a “proper” function, but by beginning with a simple integral property as a starting axiom, he could derive virtually all of the extended properties of the delta function, including properties of its derivatives.

Mathematicians, of course, were appalled and were quick to point out the insufficiency of the mathematical foundation for Dirac’s delta function, until the French mathematician Laurent Schwartz (1915 – 2002) developed the general theory of distributions in the 1940’s, which finally put the delta function in good standing.

Dirac’s introduction, development and use of the delta function was the first systematic definition of its properties. The earlier work by Cauchy, Fourier, Green and Heaviside had all touched upon the behavior of such “spiked” functions, but they had used it in passing. After Dirac, physicists embraced it as a powerful new tool in their toolbox, despite the lag in its formal acceptance by mathematicians, until the work of Schwartz redeemed it.

By David D. Nolte Feb. 17, 2022

## Bibliography

V. Balakrishnan, “All about the Dirac Delta function(?)”, Resonance, Aug., pg. 48 (2003)

M. G. Katz. “Who Invented Dirac’s Delta Function?”, Semantic Scholar (2010).

J. Lützen, The prehistory of the theory of distributions. Studies in the history of mathematics and physical sciences ; 7 (Springer-Verlag, New York, 1982).

# A Short History of Quantum Entanglement

Despite the many apparent paradoxes posed in physics—the twin and ladder paradoxes of relativity theory, Olber’s paradox of the bright night sky, Loschmitt’s paradox of irreversible statistical fluctuations—these are resolved by a deeper look at the underlying assumptions—the twin paradox is resolved by considering shifts in reference frames, the ladder paradox is resolved by the loss of simultaneity, Olber’s paradox is resolved by a finite age to the universe, and Loschmitt’s paradox is resolved by fluctuation theorems.  In each case, no physical principle is violated, and each paradox is fully explained.

However, there is at least one “true” paradox in physics that defies consistent explanation—quantum entanglement.  Quantum entanglement was first described by Einstein with colleagues Podolsky and Rosen in the famous EPR paper of 1935 as an argument against the completeness of quantum mechanics, and it was given its name by Schrödinger the same year in the paper where he introduced his “cat” as a burlesque consequence of entanglement.

Here is a short history of quantum entanglement [1], from its beginnings in 1935 to the recent 2022 Nobel prize in Physics awarded to John Clauser, Alain Aspect and Anton Zeilinger.

## The EPR Papers of 1935

Einstein can be considered as the father of quantum mechanics, even over Planck, because of his 1905 derivation of the existence of the photon as a discrete carrier of a quantum of energy (see Einstein versus Planck).  Even so, as Heisenberg and Bohr advanced quantum mechanics in the mid 1920’s, emphasizing the underlying non-deterministic outcomes of measurements, and in particular the notion of instantaneous wavefunction collapse, they pushed the theory in directions that Einstein found increasingly disturbing and unacceptable.

This feature is an excerpt from an upcoming book, Interference: The History of Optical Interferometry and the Scientists Who Tamed Light (Oxford University Press, July 2023), by David D. Nolte.

At the invitation-only Solvay Congresses of 1927 and 1930, where all the top physicists met to debate the latest advances, Einstein and Bohr began a running debate that was epic in the history of physics as the two top minds went head-to-head as the onlookers looked on in awe.  Ultimately, Einstein was on the losing end.  Although he was convinced that something was missing in quantum theory, he could not counter all of Bohr’s rejoinders, even as Einstein’s assaults became ever more sophisticated, and he left the field of battle beaten but not convinced.  Several years later he launched his last and ultimate salvo.

At the Institute for Advanced Study in Princeton, New Jersey, in the 1930’s Einstein was working with Nathan Rosen and Boris Podolsky when he envisioned a fundamental paradox in quantum theory that occurred when two widely-separated quantum particles were required to share specific physical properties because of simple conservation theorems like energy and momentum.  Even Bohr and Heisenberg could not deny the principle of conservation of energy and momentum, and Einstein devised a two-particle system for which these conservation principles led to an apparent violation of Heisenberg’s own uncertainty principle.  He left the details to his colleagues, with Podolsky writing up the main arguments.  They published the paper in the Physical Review in March of 1935 with the title “Can Quantum-Mechanical Description of Physical Reality be Considered Complete” [2].  Because of the three names on the paper (Einstein, Podolsky, Rosen), it became known as the EPR paper, and the paradox they presented became known as the EPR paradox.

When Bohr read the paper, he was initially stumped and aghast.  He felt that EPR had shaken the very foundations of the quantum theory that he and his institute had fought so hard to establish.  He also suspected that EPR had made a mistake in their arguments, and he halted all work at his institute in Copenhagen until they could construct a definitive answer.  A few months later, Bohr published a paper in the Physical Review in July of 1935, using the identical title that EPR had used, in which he refuted the EPR paradox [3].  There is not a single equation or figure in the paper, but he used his “awful incantation terminology” to maximum effect, showing that one of the EPR assumptions on the assessment of uncertainties to position and momentum was in error, and he was right.

Einstein was disgusted.  He had hoped that this ultimate argument against the completeness of quantum mechanics would stand the test of time, but Bohr had shot it down within mere months.  Einstein was particularly disappointed with Podolsky, because Podolsky had tried too hard to make the argument specific to position and momentum, leaving a loophole for Bohr to wiggle through, where Einstein had wanted the argument to rest on deeper and more general principles.

Despite Bohr’s victory, Einstein had been correct in his initial formulation of the EPR paradox that showed quantum mechanics did not jibe with common notions of reality.  He and Schrödinger exchanged letters commiserating with each other and encouraging each other in their counter beliefs against Bohr and Heisenberg.  In November of 1935, Schrödinger published a broad, mostly philosophical, paper in Naturwissenschaften [4] in which he amplified the EPR paradox with the use of an absurd—what he called burlesque—consequence of wavefunction collapse that became known as Schrödinger’s Cat.  He also gave the central property of the EPR paradox its name: entanglement.

Ironically, both Einstein’s entanglement paradox and Schrödinger’s Cat, which were formulated originally to be arguments against the validity of quantum theory, have become established quantum tools.  Today, entangled particles are the core workhorses of quantum information systems, and physicists are building larger and larger versions of Schrödinger’s Cat that may eventually merge with the physics of the macroscopic world.

## Bohm and Ahronov Tackle EPR

The physicist David Bohm was a rare political exile from the United States.  He was born in the heart of Pennsylvania in the town of Wilkes-Barre, attended Penn State and then the University of California at Berkeley, where he joined Robert Oppenheimer’s research group.  While there, he became deeply involved in the fight for unions and socialism, activities for which he was called before McCarthy’s Committee on Un-American Activities.  He invoked his right to the fifth amendment for which he was arrested.  Although he was later acquitted, Princeton University fired him from his faculty position, and fearing another arrest, he fled to Brazil where his US passport was confiscated by American authorities.  He had become a physicist without a country.

Despite his personal trials, Bohm remained scientifically productive.  He published his influential textbook on quantum mechanics in the midst of his Senate hearings, and after a particularly stimulating discussion with Einstein shortly before he fled the US, he developed and published an alternative version of quantum theory in 1952 that was fully deterministic—removing Einstein’s “God playing dice”—by creating a hidden-variable theory [5].

Hidden-variable theories of quantum mechanics seek to remove the randomness of quantum measurement by assuming that some deeper element of quantum phenomena—a hidden variable—explains each outcome.  But it is also assumed that these hidden variables are not directly accessible to experiment.  In this sense, the quantum theory of Bohr and Heisenberg was “correct” but not “complete”, because there were things that the theory could not predict or explain.

Bohm’s hidden variable theory, based on a quantum potential, was able to reproduce all the known results of standard quantum theory without invoking the random experimental outcomes that Einstein abhorred.  However, it still contained one crucial element that could not sweep away the EPR paradox—it was nonlocal.

Nonlocality lies at the heart of quantum theory.  In its simplest form, the nonlocal nature of quantum phenomenon says that quantum states span spacetime with space-like separations, meaning that parts of the wavefunction are non-causally connected to other parts of the wavefunction.  Because Einstein was fundamentally committed to causality, the nonlocality of quantum theory was what he found most objectionable, and Bohm’s elegant hidden-variable theory, that removed Einstein’s dreaded randomness, could not remove that last objection of non-causality.

After working in Brazil for several years, Bohm moved to the Technion University in Israel where he began a fruitful collaboration with Yakir Ahronov.  In addition to proposing the Ahronov-Bohm effect, in 1957 they reformulated Podolsky’s version of the EPR paradox that relied on continuous values of position and momentum and replaced it with a much simpler model based on the Stern-Gerlach effect on spins and further to the case of positronium decay into two photons with correlated polarizations.  Bohm and Ahronov reassessed experimental results of positronium decay that had been made by Madame Wu in 1950 at Columbia University and found it in full agreement with standard quantum theory.

## John Bell’s Inequalities

John Stuart Bell had an unusual start for a physicist.  His family was too poor to give him an education appropriate to his skills, so he enrolled in vocational school where he took practical classes that included brick laying.  Working later as a technician in a university lab, he caught the attention of his professors who sponsored him to attend the university.  With a degree in physics, he began working at CERN as an accelerator designer when he again caught the attention of his supervisors who sponsored him to attend graduate school.  He graduated with a PhD and returned to CERN as a card-carrying physicist with all the rights and privileges that entailed.

During his university days, he had been fascinated by the EPR paradox, and he continued thinking about the fundamentals of quantum theory.  On a sabbatical to the Stanford accelerator in 1960 he began putting mathematics to the EPR paradox to see whether any local hidden variable theory could be compatible with quantum mechanics.  His analysis was fully general, so that it could rule out as-yet-unthought-of hidden-variable theories.  The result of this work was a set of inequalities that must be obeyed by any local hidden-variable theory.  Then he made a simple check using the known results of quantum measurement and showed that his inequalities are violated by quantum systems.  This ruled out the possibility of any local hidden variable theory (but not Bohm’s nonlocal hidden-variable theory).  Bell published his analysis in 1964 [6] in an obscure journal that almost no one read…except for a curious graduate student at Columbia University who began digging into the fundamental underpinnings of quantum theory against his supervisor’s advice.

## John Clauser’s Tenacious Pursuit

As a graduate student in astrophysics at Columbia University, John Clauser was supposed to be doing astrophysics.  Instead, he spent his time musing over the fundamentals of quantum theory.  In 1967 Clauser stumbled across Bell’s paper while he was in the library.  The paper caught his imagination, but he also recognized that the inequalities were not experimentally testable, because they required measurements that depended directly on hidden variables, which are not accessible.  He began thinking of ways to construct similar inequalities that could be put to an experimental test, and he wrote about his ideas to Bell, who responded with encouragement.  Clauser wrote up his ideas in an abstract for an upcoming meeting of the American Physical Society, where one of the abstract reviewers was Abner Shimony of Boston University.  Clauser was surprised weeks later when he received a telephone call from Shimony.  Shimony and his graduate student Micheal Horne had been thinking along similar lines, and Shimony proposed to Clauser that they join forces.  They met in Boston where they were met Richard Holt, a graudate student at Harvard who was working on experimental tests of quantum mechanics.  Collectively, they devised a new type of Bell inequality that could be put to experimental test [7].  The result has become known as the CHSH Bell inequality (after Clauser, Horne, Shimony and Holt).

When Clauser took a post-doc position in Berkeley, he began searching for a way to do the experiments to test the CHSH inequality, even though Holt had a head start at Harvard.  Clauser enlisted the help of Charles Townes, who convinced one of the Berkeley faculty to loan Clauser his graduate student, Stuart Freedman, to help.  Clauser and Freedman performed the experiments, using a two-photon optical decay of calcium ions and found a violation of the CHSH inequality by 5 standard deviations, publishing their result in 1972 [8].

## Alain Aspect’s Non-locality

Just as Clauser’s life was changed when he stumbled on Bell’s obscure paper in 1967, the paper had the same effect on the life of French physicist Alain Aspect who stumbled on it in 1975.  Like Clauser, he also sought out Bell for his opinion, meeting with him in Geneva, and Aspect similarly received Bell’s encouragement, this time with the hope to build upon Clauser’s work.

In some respects, the conceptual breakthrough achieved by Clauser had been the CHSH inequality that could be tested experimentally.  The subsequent Clauser Freedman experiments were not a conclusion, but were just the beginning, opening the door to deeper tests.  For instance, in the Clauser-Freedman experiments, the polarizers were static, and the detectors were not widely separated, which allowed the measurements to be time-like separated in spacetime.  Therefore, the fundamental non-local nature of quantum physics had not been tested.

Aspect began a thorough and systematic program, that would take him nearly a decade to complete, to test the CHSH inequality under conditions of non-locality.  He began with a much brighter source of photons produced using laser excitation of the calcium ions.  This allowed him to perform the experiment in 100’s of seconds instead of the hundreds of hours by Clauser.  With such a high data rate, Aspect was able to verify violation of the Bell inequality to 10 standard deviations, published in 1981 [9].

However, the real goal was to change the orientations of the polarizers while the photons were in flight to widely separated detectors [10].  This experiment would allow the detection to be space-like separated in spacetime.  The experiments were performed using fast-switching acoustic-optic modulators, and the Bell inequality was violated to 5 standard deviations [11].  This was the most stringent test yet performed and the first to fully demonstrate the non-local nature of quantum physics.

## Anton Zeilinger: Master of Entanglement

If there is one physicist today whose work encompasses the broadest range of entangled phenomena, it would be the Austrian physicist, Anton Zeilinger.  He began his career in neutron interferometery, but when he was bitten by the entanglement bug in 1976, he switched to quantum photonics because of the superior control that can be exercised using optics over sources and receivers and all the optical manipulations in between.

Working with Daniel Greenberger and Micheal Horne, they took the essential next step past the Bohm two-particle entanglement to consider a 3-particle entangled state that had surprising properties.  While the violation of locality by the two-particle entanglement was observed through the statistical properties of many measurements, the new 3-particle entanglement could show violations on single measurements, further strengthening the arguments for quantum non-locality.  This new state is called the GHZ state (after Greenberger, Horne and Zeilinger) [12].

As the Zeilinger group in Vienna was working towards experimental demonstrations of the GHZ state, Charles Bennett of IBM proposed the possibility for quantum teleportation, using entanglement as a core quantum information resource [13].   Zeilinger realized that his experimental set-up could perform an experimental demonstration of the effect, and in a rapid re-tooling of the experimental apparatus [14], the Zeilinger group was the first to demonstrate quantum teleportation that satisfied the conditions of the Bennett teleportation proposal [15].  An Italian-UK collaboration also made an early demonstration of a related form of teleportation in a paper that was submitted first, but published after Zeilinger’s, due to delays in review [16].  But teleportation was just one of a widening array of quantum applications for entanglement that was pursued by the Zeilinger group over the succeeding 30 years [17], including entanglement swapping, quantum repeaters, and entanglement-based quantum cryptography. Perhaps most striking, he has worked on projects at astronomical observatories that entangle photons coming from cosmic sources.

By David D. Nolte Nov. 26, 2022

## Video Lectures

YouTube Lecture on the History of Quantum Entanglement

Physics Colloquium on the Backstory of the 2023 Nobel Prize in Physics

## Timeline

1935 – Einstein EPR

1935 – Bohr EPR

1935 – Schrödinger: Entanglement and Cat

1950 – Madam Wu positron decay

1952 – David Bohm and Non-local hidden variables

1957 – Bohm and Ahronov version of EPR

1963 – Bell’s inequalities

1967 – Clauser reads Bell’s paper

1967 – Commins experiment with Calcium

1969 – CHSH inequality: measurable with detection inefficiencies

1972 – Clauser and Freedman experiment

1975 – Aspect reads Bell’s paper

1976 – Zeilinger reads Bell’s paper

1981 – Aspect two-photon generation source

1982 – Aspect time variable analyzers

1988 – Parametric down-conversion of EPR pairs (Shih and Alley, Ou and Mandel)

1989 – GHZ state proposed

1993 – Bennett quantum teleportation proposal

1995 – High-intensity down-conversion source of EPR pairs (Kwiat and Zeilinger)

1997 – Zeilinger quantum teleportation experiment

1999 – Observation of the GHZ state

## Bibliography

[1] See the full details in: David D. Nolte, Interference: A History of Interferometry and the Scientists Who Tamed Light (Oxford University Press, July 2023)

[2] A. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Physical Review 47, 0777-0780 (1935).

[3] N. Bohr, Can quantum-mechanical description of physical reality be considered complete? Physical Review 48, 696-702 (1935).

[4] E. Schrödinger, Die gegenwärtige Situation in der Quantenmechanik. Die Naturwissenschaften 23, 807-12; 823-28; 844-49 (1935).

[5] D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables .1. Physical Review 85, 166-179 (1952); D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables .2. Physical Review 85, 180-193 (1952).

[6] J. Bell, On the Einstein-Podolsky-Rosen paradox. Physics 1, 195 (1964).

[7] 1. J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, Proposed experiment to test local hidden-variable theories. Physical Review Letters 23, 880-& (1969).

[8] S. J. Freedman, J. F. Clauser, Experimental test of local hidden-variable theories. Physical Review Letters 28, 938-& (1972).

[9] A. Aspect, P. Grangier, G. Roger, EXPERIMENTAL TESTS OF REALISTIC LOCAL THEORIES VIA BELLS THEOREM. Physical Review Letters 47, 460-463 (1981).

[10]  Alain Aspect, Bell’s Theorem: The Naïve Veiw of an Experimentalit. (2004), hal- 00001079

[11] A. Aspect, J. Dalibard, G. Roger, EXPERIMENTAL TEST OF BELL INEQUALITIES USING TIME-VARYING ANALYZERS. Physical Review Letters 49, 1804-1807 (1982).

[12] D. M. Greenberger, M. A. Horne, A. Zeilinger, in 1988 Fall Workshop on Bells Theorem, Quantum Theory and Conceptions of the Universe. (George Mason Univ, Fairfax, Va, 1988), vol. 37, pp. 69-72.

[13] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels. Physical Review Letters 70, 1895-1899 (1993).

[14]  J. Gea-Banacloche, Optical realizations of quantum teleportation, in Progress in Optics, Vol 46, E. Wolf, Ed. (2004), vol. 46, pp. 311-353.

[15] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger, Experimental quantum teleportation. Nature 390, 575-579 (1997).

[16] D. Boschi, S. Branca, F. De Martini, L. Hardy, S. Popescu, Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-podolsky-Rosen Channels. Phys. Rev. Lett. 80, 1121-1125 (1998).

[17]  A. Zeilinger, Light for the quantum. Entangled photons and their applications: a very personal perspective. Physica Scripta 92, 1-33 (2017).

# A Short History of Quantum Tunneling

Quantum physics is often called “weird” because it does things that are not allowed in classical physics and hence is viewed as non-intuitive or strange.  Perhaps the two “weirdest” aspects of quantum physics are quantum entanglement and quantum tunneling.  Entanglement allows a particle state to extend across wide expanses of space, while tunneling allows a particle to have negative kinetic energy.  Neither of these effects has a classical analog.

Quantum entanglement arose out of the Bohr-Einstein debates at the Solvay Conferences in the 1920’s and 30’s, and it was the subject of a recent Nobel Prize in Physics (2022).  The quantum tunneling story is just as old, but it was recognized much earlier by the Nobel Prize in 1972 when it was awarded to Brian Josephson, Ivar Giaever and Leo Esaki—each of whom was a graduate student when they discovered their respective effects and two of whom got their big idea while attending a lecture class.

Always go to class, you never know what you might miss, and the payoff is sometimes BIG

Ivar Giaever

Of the two effects, tunneling is the more common and the more useful in modern electronic devices (although entanglement is coming up fast with the advent of quantum information science). Here is a short history of quantum tunneling, told through a series of publications that advanced theory and experiments.

## Double-Well Potential: Friedrich Hund (1927)

The first analysis of quantum tunneling was performed by Friedrich Hund (1896 – 1997), a German physicist who studied early in his career with Born in Göttingen and Bohr in Copenhagen.  He published a series of papers in 1927 in Zeitschrift für Physik [1] that solved the newly-proposed Schrödinger equation for the case of the double well potential.  He was particularly interested in the formation of symmetric and anti-symmetric states of the double well that contributed to the binding energy of atoms in molecules.  He derived the first tunneling-frequency expression for a quantum superposition of the symmetric and anti-symmetric states

where f is the coherent oscillation frequency, V is the height of the potential and hν is the quantum energy of the isolated states when the atoms are far apart.  The exponential dependence on the potential height V made the tunnel effect extremely sensitive to the details of the tunnel barrier.

## Electron Emission: Lothar Nordheim and Ralph Fowler (1927 – 1928)

The first to consider quantum tunneling from a bound state to a continuum state was Lothar Nordheim (1899 – 1985), a German physicist who studied under David Hilbert and Max Born at Göttingen and worked with John von Neumann and Eugene Wigner and later with Hans Bethe. In 1927 he solved the problem of a particle in a well that is separated from continuum states by a thin finite barrier [2]. Using the new Schrödinger theory, he found transmission coefficients that were finite valued, caused by quantum tunneling of the particle through the barrier. Nordheim’s use of square potential wells and barriers are now, literally, textbook examples that every student of quantum mechanics solves. (For a quantum simulation of wavefunction tunneling through a square barrier see the companion Quantum Tunneling YouTube video.) Nordheim later escaped the growing nationalism and anti-semitism in Germany in the mid 1930’s to become a visiting professor of physics at Purdue University in the United States, moving to a permanent position at Duke University.

One of the giants of mathematical physics in the UK from the 1920s through the 1930’s was Ralph Fowler (1889 – 1944). Three of his doctoral students went on to win Nobel Prizes (Chandrasekhar, Dirac and Mott) and others came close (Bhabha, Hartree, Lennard-Jones). In 1928 Fowler worked with Nordheim on a more realistic version of Nordheim’s surface electron tunneling that could explain thermionic emission of electrons from metals under strong electric fields. The electric field modified Nordheim’s square potential barrier into a triangular barrier (which they treated using WKB theory) to obtain the tunneling rate [3]. This type of tunnel effect is now known as Fowler-Nordheim tunneling.

## Nuclear Alpha Decay: George Gamow (1928)

George Gamov (1904 – 1968) is one of the icons of mid-twentieth-century physics. He was a substantial physicist who also had a solid sense of humor that allowed him to achieve a level of cultural popularity shared by a few of the larger-than-life physicists of his time, like Richard Feynman and Stephen Hawking. His popular books included One Two Three … Infinity as well as a favorite series of books under the rubric of Mr. Tompkins (Mr. Tompkins in Wonderland and Mr. Tompkins Explores the Atom, among others). He also wrote a history of the early years of quantum theory (Thirty Years that Shook Physics).

In 1928 Gamow was in Göttingen (the Mecca of early quantum theory) with Max Born when he realized that the radioactive decay of Uranium by alpha decay might be explained by quantum tunneling. It was known that nucleons were bound together by some unknown force in what would be an effective binding potential, but that charged alpha particles would also feel a strong electrostatic repulsive potential from a nucleus. Gamow combined these two potentials to create a potential landscape that was qualitatively similar to Nordheim’s original system of 1927, but with a potential barrier that was neither square nor triangular (like the Fowler-Nordheim situation).

Gamow was able to make an accurate approximation that allowed him to express the decay rate in terms of an exponential term

where Zα is the atomic charge of the alpha particle, Z is the nuclear charge of the Uranium decay product and v is the speed of the alpha particle detected in external measurements [4].

The very next day after Gamow submitted his paper, Ronald Gurney and Edward Condon of Princeton University submitted a paper [5] that solved the same problem using virtually the same approach … except missing Gamow’s surprisingly concise analytic expression for the decay rate.

## Molecular Tunneling: George Uhlenbeck (1932)

Because tunneling rates depend inversely on the mass of the particle tunneling through the barrier, electrons are more likely to tunnel through potential barriers than atoms. However, hydrogen is a particularly small atom and is therefore the most amenable to experiencing tunneling.

The first example of atom tunneling is associated with hydrogen in the ammonia molecule NH3. The molecule has a pyramidal structure with the Nitrogen hovering above the plane defined by the three hydrogens. However, an equivalent configuration has the Nitrogen hanging below the hydrogen plane. The energies of these two configurations are the same, but the Nitrogen must tunnel from one side of the hydrogen plane to the other through a barrier. The presence of light-weight hydrogen that can “move out of the way” for the nitrogen makes this barrier very small (infrared energies). When the ammonia is excited into its first vibrational excited state, the molecular wavefunction tunnels through the barrier, splitting the excited level by an energy associated with a wavelength of 1.2 cm which is in the microwave. This tunnel splitting was the first microwave transition observed in spectroscopy and is used in ammonia masers.

One of the earliest papers [6] written on the tunneling of nitrogen in ammonia was published by George Uhlenbeck in 1932. George Uhlenbeck (1900 – 1988) was a Dutch-American theoretical physicist. He played a critical role, with Samuel Goudsmit, in establishing the spin of the electron in 1925. Both Uhlenbeck and Goudsmit were close associates of Paul Ehrenfest at Leiden in the Netherlands. Uhlenbeck is also famous for the Ornstein-Uhlenbeck process which is a generalization of Einstein’s theory of Brownian motion that can treat active transport such as intracellular transport in living cells.

## Solid-State Electron Tunneling: Leo Esaki (1957)

Although the tunneling of electrons in molecular bonds and in the field emission from metals had been established early in the century, direct use of electron tunneling in solid state devices had remained elusive until Leo Esaki (1925 – ) observed electron tunneling in heavily doped Germanium and Silicon semiconductors. Esaki joined an early precursor of Sony electronics in 1956 and was supported to obtain a PhD from the University of Tokyo. In 1957 he was working with heavily-doped p-n junction diodes and discovered a phenomenon known as negative differential resistance where the current through an electronic device actually decreases as the voltage increases.

Because the junction thickness was only about 100 atoms, or about 10 nanometers, he suspected and then proved that the electronic current was tunneling quantum mechanically through the junction. The negative differential resistance was caused by a decrease in available states to the tunneling current as the voltage increased.

Esaki tunnel diodes were the fastest semiconductor devices of the time, and the negative differential resistance of the diode in an external circuit produced high-frequency oscillations. They were used in high-frequency communication systems. They were also radiation hard and hence ideal for the early communications satellites. Esaki was awarded the 1973 Nobel Prize in Physics jointly with Ivar Giaever and Brian Josephson.

## Superconducting Tunneling: Ivar Giaever (1960)

Ivar Giaever (1929 – ) is a Norwegian-American physicist who had just joined the GE research lab in Schenectady New York in 1958 when he read about Esaki’s tunneling experiments. He was enrolled at that time as a graduate student in physics at Rensselaer Polytechnic Institute (RPI) where he was taking a course in solid state physics and learning about superconductivity. Superconductivity is carried by pairs of electrons known as Cooper pairs that spontaneously bind together with a binding energy that produced an “energy gap” in the electron energies of the metal, but no one had ever found a way to directly measure it. The Esaki experiment made him immediately think of the equivalent experiment in which Cooper pairs might tunnel between two superconductors (through a thin oxide layer) and yield a measurement of the energy gap. The idea actually came to him during the class lecture.

The experiments used a junction between aluminum and lead (Al—Al2O3—Pb). At first, the temperature of the system was adjusted so that Al remained a normal metal and Pb was superconducting, and Giaever observed a tunnel current with a threshold related to the gap in Pb. Then the temperature was lowered so that both Al and Pb were superconducting, and a peak in the tunnel current appeared at the voltage associated with the difference in the energy gaps (predicted by Harrison and Bardeen).

## The Josephson Effect: Brian Josephson (1962)

In Giaever’s experiments, the external circuits had been designed to pick up “ordinary” tunnel currents in which individual electrons tunneled through the oxide rather than the Cooper pairs themselves. However, in 1962, Brian Josephson (1940 – ), a physics graduate student at Cambridge, was sitting in a lecture (just like Giaever) on solid state physics given by Phil Anderson (who was on sabbatical there from Bell Labs). During lecture he had the idea to calculate whether it was possible for the Cooper pairs themselves to tunnel through the oxide barrier. Building on theoretical work by Leo Falicov who was at the University of Chicago and later at Berkeley (years later I was lucky to have Leo as my PhD thesis advisor at Berkeley), Josephson found a surprising result that even when the voltage was zero, there would be a supercurrent that tunneled through the junction (now known as the DC Josephson Effect). Furthermore, once a voltage was applied, the supercurrent would oscillate (now known as the AC Josephson Effect). These were strange and non-intuitive results, so he showed Anderson his calculations to see what he thought. By this time Anderson had already been extremely impressed by Josephson (who would often come to the board after one of Anderson’s lectures to show where he had made a mistake). Anderson checked over the theory and agreed with Josephson’s conclusions. Bolstered by this reception, Josephson submitted the theoretical prediction for publication [9].

As soon as Anderson returned to Bell Labs after his sabbatical, he connected with John Rowell who was making tunnel junction experiments, and they revised the external circuit configuration to be most sensitive to the tunneling supercurrent, which they observed in short time and submitted a paper for publication. Since then, the Josephson Effect has become a standard element of ultra-sensitive magnetometers, measurement standards for charge and voltage, far-infrared detectors, and have been used to construct rudimentary qubits and quantum computers.

By David D. Nolte: Nov. 6, 2022

## YouTube Video

YouTube Video of Quantum Tunneling Systems

## References:

[1] F. Hund, Z. Phys. 40, 742 (1927). F. Hund, Z. Phys. 43, 805 (1927).

[5] R. W. Gurney, E. U. Condon, Nature 122, 439 (1928). R. W. Gurney, E. U. Condon, Phys. Rev. 33, 127 (1929).

[6] Dennison, D. M. and G. E. Uhlenbeck. “The two-minima problem and the ammonia molecule.” Physical Review 41(3): 313-321. (1932)

[7] L. Esaki, New Phenomenon in Narrow Germanium Para-Normal-Junctions, Phys. Rev., 109, 603-604 (1958); L. Esaki, (1974). Long journey into tunneling, disintegration, Proc. of the Nature 123, IEEE, 62, 825.

[8] I. Giaever, Energy Gap in Superconductors Measured by Electron Tunneling, Phys. Rev. Letters, 5, 147-148 (1960); I. Giaever, Electron tunneling and superconductivity, Science, 183, 1253 (1974)

[9] B. D. Josephson, Phys. Lett. 1, 251 (1962); B.D. Josephson, The discovery of tunneling supercurrent, Science, 184, 527 (1974).

[10] P. W. Anderson, J. M. Rowell, Phys. Rev. Lett. 10, 230 (1963); Philip W. Anderson, How Josephson discovered his effect, Physics Today 23, 11, 23 (1970)

[11] Eugen Merzbacher, The Early History of Quantum Tunneling, Physics Today 55, 8, 44 (2002)

[12] Razavy, Mohsen. Quantum Theory Of Tunneling, World Scientific Publishing Company, 2003.

# Is There a Quantum Trajectory? The Phase-Space Perspective

At the dawn of quantum theory, Heisenberg, Schrödinger, Bohr and Pauli were embroiled in a dispute over whether trajectories of particles, defined by their positions over time, could exist. The argument against trajectories was based on an apparent paradox: To draw a “line” depicting a trajectory of a particle along a path implies that there is a momentum vector that carries the particle along that path. But a line is a one-dimensional curve through space, and since at any point in time the particle’s position is perfectly localized, then by Heisenberg’s uncertainty principle, it can have no definable momentum to carry it along.

My previous blog shows the way out of this paradox, by assembling wavepackets that are spread in both space and momentum, explicitly obeying the uncertainty principle. This is nothing new to anyone who has taken a quantum course. But the surprising thing is that in some potentials, like a harmonic potential, the wavepacket travels without broadening, just like classical particles on a trajectory. A dramatic demonstration of this can be seen in this YouTube video. But other potentials “break up” the wavepacket, especially potentials that display classical chaos. Because phase space is one of the best tools for studying classical chaos, especially Hamiltonian chaos, it can be enlisted to dig deeper into the question of the quantum trajectory—not just about the existence of a quantum trajectory, but why quantum systems retain a shadow of their classical counterparts.

## Phase Space

Phase space is the state space of Hamiltonian systems. Concepts of phase space were first developed by Boltzmann as he worked on the problem of statistical mechanics. Phase space was later codified by Gibbs for statistical mechanics and by Poincare for orbital mechanics, and it was finally given its name by Paul and Tatiana Ehrenfest (a husband-wife team) in correspondence with the German physicist Paul Hertz (See Chapter 6, “The Tangled Tale of Phase Space”, in Galileo Unbound by D. D. Nolte (Oxford, 2018)).

The stretched-out phase-space functions … are very similar to the stochastic layer that forms in separatrix chaos in classical systems.

The idea of phase space is very simple for classical systems: it is just a plot of the momentum of a particle as a function of its position. For a given initial condition, the trajectory of a particle through its natural configuration space (for instance our 3D world) is traced out as a path through phase space. Because there is one momentum variable per degree of freedom, then the dimensionality of phase space for a particle in 3D is 6D, which is difficult to visualize. But for a one-dimensional dynamical system, like a simple harmonic oscillator (SHO) oscillating in a line, the phase space is just two-dimensional, which is easy to see. The phase-space trajectories of an SHO are simply ellipses, and if the momentum axis is scaled appropriately, the trajectories are circles. The particle trajectory in phase space can be animated just like a trajectory through configuration space as the position and momentum change in time p(x(t)). For the SHO, the point follows the path of a circle going clockwise.

A more interesting phase space is for the simple pendulum, shown in Fig. 2. There are two types of orbits: open and closed. The closed orbits near the origin are like those of a SHO. The open orbits are when the pendulum is spinning around. The dividing line between the open and closed orbits is called a separatrix. Where the separatrix intersects itself is a saddle point. This saddle point is the most important part of the phase space portrait: it is where chaos emerges when perturbations are added.

One route to classical chaos is through what is known as “separatrix chaos”. It is easy to see why saddle points (also known as hyperbolic points) are the source of chaos: as the system trajectory approaches the saddle, it has two options of which directions to go. Any additional degree of freedom in the system (like a harmonic drive) can make the system go one way on one approach, and the other way on another approach, mixing up the trajectories. An example of the stochastic layer of separatrix chaos is shown in Fig. 3 for a damped driven pendulum. The chaotic behavior that originates at the saddle point extends out along the entire separatrix.

The main question about whether or not there is a quantum trajectory depends on how quantum packets behave as they approach a saddle point in phase space. Since packets are spread out, it would be reasonable to assume that parts of the packet will go one way, and parts of the packet will go another. But first, one has to ask: Is a phase-space description of quantum systems even possible?

## Quantum Phase Space: The Wigner Distribution Function

Phase-space portraits are arguably the most powerful tool in the toolbox of classical dynamics, and one would like to retain its uses for quantum systems. However, there is that pesky paradox about quantum trajectories that cannot admit the existence of one-dimensional curves through such a phase space. Furthermore, there is no direct way of taking a wavefunction and simply “finding” its position or momentum to plot points on such a quantum phase space.

The answer was found in 1932 by Eugene Wigner (1902 – 1905), an Hungarian physicist working at Princeton. He realized that it was impossible to construct a quantum probability distribution in phase space that had positive values everywhere. This is a problem, because negative probabilities have no direct interpretation. But Wigner showed that if one relaxed the requirements a bit, so that expectation values computed over some distribution function (that had positive and negative values) gave correct answers that matched experiments, then this distribution function would “stand in” for an actual probability distribution.

The distribution function that Wigner found is called the Wigner distribution function. Given a wavefunction ψ(x), the Wigner distribution is defined as

The Wigner distribution function is the Fourier transform of the convolution of the wavefunction. The pure position dependence of the wavefunction is converted into a spread-out position-momentum function in phase space. For a Gaussian wavefunction ψ(x) with a finite width in space, the W-function in phase space is a two-dimensional Gaussian with finite widths in both space and momentum. In fact, the Δx-Δp product of the W-function is precisely the uncertainty production of the Heisenberg uncertainty relation.

The question of the quantum trajectory from the phase-space perspective becomes whether a Wigner function behaves like a localized “packet” that evolves in phase space in a way analogous to a classical particle, and whether classical chaos is reflected in the behavior of quantum systems.

## The Harmonic Oscillator

The quantum harmonic oscillator is a rare and special case among quantum potentials, because the energy spacings between all successive states are all the same. This makes it possible for a Gaussian wavefunction, which is a superposition of the eigenstates of the harmonic oscillator, to propagate through the potential without broadening. To see an example of this, watch the first example in this YouTube video for a Schrödinger cat state in a two-dimensional harmonic potential. For this very special potential, the Wigner distribution behaves just like a (broadened) particle on an orbit in phase space, executing nice circular orbits.

A comparison of the classical phase-space portrait versus the quantum phase-space portrait is shown in Fig. 5. Where the classical particle is a point on an orbit, the quantum particle is spread out, obeying the Δx-Δp Heisenberg product, but following the same orbit as the classical particle.

However, a significant new feature appears in the Wigner representation in phase space when there is a coherent superposition of two states, known as a “cat” state, after Schrödinger’s cat. This new feature has no classical analog. It is the coherent interference pattern that appears at the zero-point of the harmonic oscillator for the Schrödinger cat state. There is no such thing as “classical” coherence, so this feature is absent in classical phase space portraits.

Two examples of Wigner distributions are shown in Fig. 6 for a statistical (incoherent) mixture of packets and a coherent superposition of packets. The quantum coherence signature is present in the coherent case but not the statistical mixture case. The coherence in the Wigner distribution represents “off-diagonal” terms in the density matrix that leads to interference effects in quantum systems. Quantum computing algorithms depend critically on such coherences that tend to decay rapidly in real-world physical systems, known as decoherence, and it is possible to make statements about decoherence by watching the zero-point interference.

Whereas Gaussian wave packets in the quantum harmonic potential behave nearly like classical systems, and their phase-space portraits are almost identical to the classical phase-space view (except for the quantum coherence), most quantum potentials cause wave packets to disperse. And when saddle points are present in the classical case, then we are back to the question about how quantum packets behave as they approach a saddle point in phase space.

## Quantum Pendulum and Separatrix Chaos

One of the simplest anharmonic oscillators is the simple pendulum. In the classical case, the period diverges if the pendulum gets very close to going vertical. A similar thing happens in the quantum case, but because the motion has strong anharmonicity, an initial wave packet tends to spread dramatically as parts of the wavefunction less vertical stretch away from the part of the wave function that is more nearly vertical. Fig. 7 is a snap-shot about a eighth of a period after the wave packet was launched. The packet has already stretched out along the separatrix. A double-cat-state was used, so there is a second packet that has coherent interference with the first. To see a movie of the time evolution of the wave packet and the orbit in quantum phase space, see the YouTube video.

The simple pendulum does have a saddle point, but it is degenerate because the angle is modulo -2-pi. A simple potential that has a non-degenerate saddle point is a double-well potential.

## Quantum Double-Well and Separatrix Chaos

The symmetric double-well potential has a saddle point at the mid-point between the two well minima. A wave packet approaching the saddle will split into to packets that will follow the individual separatrixes that emerge from the saddle point (the unstable manifolds). This effect is seen most dramatically in the middle pane of Fig. 8. For the full video of the quantum phase-space evolution, see this YouTube video. The stretched-out distribution in phase space is highly analogous to the separatrix chaos seen for the classical system.

## Conclusion

A common statement often made about quantum chaos is that quantum systems tend to suppress chaos, only exhibiting chaos for special types of orbits that produce quantum scars. However, from the phase-space perspective, the opposite may be true. The stretched-out Wigner distribution functions, for critical wave packets that interact with a saddle point, are very similar to the stochastic layer that forms in separatrix chaos in classical systems. In this sense, the phase-space description brings out the similarity between classical chaos and quantum chaos.

By David D. Nolte Sept. 25, 2022

## YouTube Video

YouTube Video of Dynamics in Quantum Phase Space

## References

1. T. Curtright, D. Fairlie, C. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space.  (World Scientific, New Jersey, 2014).

2. J. R. Nagel, A Review and Application of the Finite-Difference Time-Domain Algorithm Applied to the Schrödinger Equation, ACES Journal, Vol. 24, NO. 1, pp. 1-8 (2009)

# Is There a Quantum Trajectory?

Heisenberg’s uncertainty principle is a law of physics – it cannot be violated under any circumstances, no matter how much we may want it to yield or how hard we try to bend it.  Heisenberg, as he developed his ideas after his lone epiphany like a monk on the isolated island of Helgoland off the north coast of Germany in 1925, became a bit of a zealot, like a religious convert, convinced that all we can say about reality is a measurement outcome.  In his view, there was no independent existence of an electron other than what emerged from a measuring apparatus.  Reality, to Heisenberg, was just a list of numbers in a spread sheet—matrix elements.  He took this line of reasoning so far that he stated without exception that there could be no such thing as a trajectory in a quantum system.  When the great battle commenced between Heisenberg’s matrix mechanics against Schrödinger’s wave mechanics, Heisenberg was relentless, denying any reality to Schrödinger’s wavefunction other than as a calculation tool.  He was so strident that even Bohr, who was on Heisenberg’s side in the argument, advised Heisenberg to relent [1].  Eventually a compromise was struck, as Heisenberg’s uncertainty principle allowed Schrödinger’s wave functions to exist within limits—his uncertainty limits.

## Disaster in the Poconos

Yet the idea of an actual trajectory of a quantum particle remained a type of heresy within the close quantum circles.  Years later in 1948, when a young Richard Feynman took the stage at a conference in the Poconos, he almost sabotaged his career in front of Bohr and Dirac—two of the giants who had invented quantum mechanics—by having the audacity to talk about particle trajectories in spacetime diagrams.

Feynman was making his first presentation of a new approach to quantum mechanics that he had developed based on path integrals. The challenge was that his method relied on space-time graphs in which “unphysical” things were allowed to occur.  In fact, unphysical things were required to occur, as part of the sum over many histories of his path integrals.  For instance, a key element in the approach was allowing electrons to travel backwards in time as positrons, or a process in which the electron and positron annihilate into a single photon, and then the photon decays back into an electron-positron pair—a process that is not allowed by mass and energy conservation.  But this is a possible history that must be added to Feynman’s sum.

It all looked like nonsense to the audience, and the talk quickly derailed.  Dirac pestered him with questions that he tried to deflect, but Dirac persisted like a raven.  A question was raised about the Pauli exclusion principle, about whether an orbital could have three electrons instead of the required two, and Feynman said that it could—all histories were possible and had to be summed over—an answer that dismayed the audience.  Finally, as Feynman was drawing another of his space-time graphs showing electrons as lines, Bohr rose to his feet and asked derisively whether Feynman had forgotten Heisenberg’s uncertainty principle that made it impossible to even talk about an electron trajectory.

It was hopeless.  The audience gave up and so did Feynman as the talk just fizzled out.  It was a disaster.  What had been meant to be Feynman’s crowning achievement and his entry to the highest levels of theoretical physics, had been a terrible embarrassment.  He slunk home to Cornell where he sank into one of his depressions.  At the close of the Pocono conference, Oppenheimer, the reigning king of physics, former head of the successful Manhattan Project and newly selected to head the prestigious Institute for Advanced Study at Princeton, had been thoroughly disappointed by Feynman.

But what Bohr and Dirac and Oppenheimer had failed to understand was that as long as the duration of unphysical processes was shorter than the energy differences involved, then it was literally obeying Heisenberg’s uncertainty principle.  Furthermore, Feynman’s trajectories—what became his famous “Feynman Diagrams”—were meant to be merely cartoons—a shorthand way to keep track of lots of different contributions to a scattering process.  The quantum processes certainly took place in space and time, conceptually like a trajectory, but only so far as time durations, and energy differences and locations and momentum changes were all within the bounds of the uncertainty principle.  Feynman had invented a bold new tool for quantum field theory, able to supply deep results quickly.  But no one at the Poconos could see it.

## Coherent States

When Feynman had failed so miserably at the Pocono conference, he had taken the stage after Julian Schwinger, who had dazzled everyone with his perfectly scripted presentation of quantum field theory—the competing theory to Feynman’s.  Schwinger emerged the clear winner of the contest.  At that time, Roy Glauber (1925 – 2018) was a young physicist just taking his PhD from Schwinger at Harvard, and he later received a post-doc position at Princeton’s Institute for Advanced Study where he became part of a miniature revolution in quantum field theory that revolved around—not Schwinger’s difficult mathematics—but Feynman’s diagrammatic method.  So Feynman won in the end.  Glauber then went on to Caltech, where he filled in for Feynman’s lectures when Feynman was off in Brazil playing the bongos.  Glauber eventually returned to Harvard where he was already thinking about the quantum aspects of photons in 1956 when news of the photon correlations in the Hanbury-Brown Twiss (HBT) experiment were published.  Three years later, when the laser was invented, he began developing a theory of photon correlations in laser light that he suspected would be fundamentally different than in natural chaotic light.

Because of his background in quantum field theory, and especially quantum electrodynamics, it was fairly easy to couch the quantum optical properties of coherent light in terms of Dirac’s creation and annihilation operators of the electromagnetic field. Glauber developed a “coherent state” operator that was a minimum uncertainty state of the quantized electromagnetic field, related to the minimum-uncertainty wave functions derived initially by Schrödinger in the late 1920’s. The coherent state represents a laser operating well above the lasing threshold and behaved as “the most classical” wavepacket that can be constructed.  Glauber was awarded the Nobel Prize in Physics in 2005 for his work on such “Glauber states” in quantum optics.

## Quantum Trajectories

Glauber’s coherent states are built up from the natural modes of a harmonic oscillator.  Therefore, it should come as no surprise that these coherent-state wavefunctions in a harmonic potential behave just like classical particles with well-defined trajectories. The quadratic potential matches the quadratic argument of the the Gaussian wavepacket, and the pulses propagate within the potential without broadening, as in Fig. 3, showing a snapshot of two wavepackets propagating in a two-dimensional harmonic potential. This is a somewhat radical situation, because most wavepackets in most potentials (or even in free space) broaden as they propagate. The quadratic potential is a special case that is generally not representative of how quantum systems behave.

To illustrate this special status for the quadratic potential, the wavepackets can be launched in a potential with a quartic perturbation. The quartic potential is anharmonic—the frequency of oscillation depends on the amplitude of oscillation unlike for the harmonic oscillator, where amplitude and frequency are independent. The quartic potential is integrable, like the harmonic oscillator, and there is no avenue for chaos in the classical analog. Nonetheless, wavepackets broaden as they propagate in the quartic potential, eventually spread out into a ring in the configuration space, as in Fig. 4.

A potential with integrability has as many conserved quantities to the motion as there are degrees of freedom. Because the quartic potential is integrable, the quantum wavefunction may spread, but it remains highly regular, as in the “ring” that eventually forms over time. However, integrable potentials are the exception rather than the rule. Most potentials lead to nonintegrable motion that opens the door to chaos.

A classic (and classical) potential that exhibits chaos in a two-dimensional configuration space is the famous Henon-Heiles potential. This has a four-dimensional phase space which admits classical chaos. The potential has a three-fold symmetry which is one reason it is non-integral, since a particle must “decide” which way to go when it approaches a saddle point. In the quantum regime, wavepackets face the same decision, leading to a breakup of the wavepacket on top of a general broadening. This allows the wavefunction eventually to distribute across the entire configuration space, as in Fig. 5.

## Youtube Video

Movies of quantum trajectories can be viewed at my Youtube Channel, Physics Unbound. The answer to the question “Is there a quantum trajectory?” can be seen visually as the movies run—they do exist in a very clear sense under special conditions, especially coherent states in a harmonic oscillator. And the concept of a quantum trajectory also carries over from a classical trajectory in cases when the classical motion is integrable, even in cases when the wavefunction spreads over time. However, for classical systems that display chaotic motion, wavefunctions that begin as coherent states break up into chaotic wavefunctions that fill the accessible configuration space for a given energy. The character of quantum evolution of coherent states—the most classical of quantum wavefunctions—in these cases reflects the underlying character of chaotic motion in the classical analogs. This process can be seen directly watching the movies as a wavepacket approaches a saddle point in the potential and is split. Successive splits of the multiple wavepackets as they interact with the saddle points is what eventually distributes the full wavefunction into its chaotic form.

Therefore, the idea of a “quantum trajectory”, so thoroughly dismissed by Heisenberg, remains a phenomenological guide that can help give insight into the behavior of quantum systems—both integrable and chaotic.

As a side note, the laws of quantum physics obey time-reversal symmetry just as the classical equations do. In the third movie of “A Quantum Ballet“, wavefunctions in a double-well potential are tracked in time as they start from coherent states that break up into chaotic wavefunctions. It is like watching entropy in action as an ordered state devolves into a disordered state. But at the half-way point of the movie, the imaginary part of the wavefunction has its sign flipped, and the dynamics continue. But now the wavefunctions move from disorder into an ordered state, seemingly going against the second law of thermodynamics. Flipping the sign of the imaginary part of the wavefunction at just one instant in time plays the role of a time-reversal operation, and there is no violation of the second law.

By David D. Nolte, Sept. 4, 2022

## YouTube Video

YouTube Video of Quantum Trajectories

## References

[1] See Chapter 8 , On the Quantum Footpath, in Galileo Unbound, D. D. Nolte (Oxford University Press, 2018)

[2] J. R. Nagel, A Review and Application of the Finite-Difference Time-Domain Algorithm Applied to the Schrödinger Equation, ACES Journal, Vol. 24, NO. 1, pp. 1-8 (2009)

# Quantum Chaos and the Cheshire Cat

Alice’s disturbing adventures in Wonderland tumbled upon her like a string of accidents as she wandered a world of chaos.  Rules were never what they seemed and shifted whenever they wanted.  She even met a cat who grinned ear-to-ear and could disappear entirely, or almost entirely, leaving only its grin.

The vanishing Cheshire Cat reminds us of another famous cat—Arnold’s Cat—that introduced the ideas of stretching and folding of phase-space volumes in non-integrable Hamiltonian systems.  But when Arnold’s Cat becomes a Quantum Cat, a central question remains: What happens to the chaotic behavior of the classical system … does it survive the transition to quantum mechanics?  The answer is surprisingly like the grin of the Cheshire Cat—the cat vanishes, but the grin remains.  In the quantum world of the Cheshire Cat, the grin of the classical cat remains even after the rest of the cat vanished.

The Cheshire Cat fades away, leaving only its grin, like a fine filament, as classical chaos fades into quantum, leaving behind a quantum scar.

## The Quantum Mechanics of Classically Chaotic Systems

The simplest Hamiltonian systems are integrable—they have as many constants of the motion as degrees of freedom.  This holds for quantum systems as well as for classical.  There is also a strong correspondence between classical and quantum systems for the integrable cases—literally the Correspondence Principle—that states that quantum systems at high quantum number approach classical behavior.  Even at low quantum numbers, classical resonances are mirrored by quantum eigenfrequencies that can show highly regular spectra.

But integrable systems are rare—surprisingly rare.  Almost no real-world Hamiltonian system is integrable, because the real world warps the ideal.  No spring can displace indefinitely, and no potential is perfectly quadratic.  There are always real-world non-idealities that destroy one constant of the motion or another, opening the door to chaos.

When classical Hamiltonian systems become chaotic, they don’t do it suddenly.  Almost all transitions to chaos in Hamiltonian systems are gradual.  One of the best examples of this is the KAM theory that starts with invariant action integrals that generate invariant tori in phase space.  As nonintegrable perturbations increase, the tori break up slowly into island chains of stability as chaos infiltrates the separatrixes—first as thin filaments of chaos surrounding the islands—then growing in width to take up more and more of phase space.  Even when chaos is fully developed, small islands of stability can remain—the remnants of stable orbits of the unperturbed system.

When the classical becomes quantum, chaos softens.  Quantum wave functions don’t like to be confined—they spread and they tunnel.  The separatrix of classical chaos—that barrier between regions of phase space—cannot constrain the exponential tails of wave functions.  And the origin of chaos itself—the homoclinic point of the separatrix—gets washed out.  Then the regular orbits of the classical system reassert themselves, and they appear, like the vestige of the Cheshire Cat, as a grin.

## The Quantum Circus

The empty stadium is a surprisingly rich dynamical system that has unexpected structure in both the classical and the quantum domain.  Its importance in classical dynamics comes from the fact that its periodic orbits are unstable and its non-periodic orbits are ergodic (filling all available space if given long enough).  The stadium itself is empty so that particles (classical or quantum) are free to propagate between reflections from the perfectly-reflecting walls of the stadium.  The ergodicity comes from the fact that the stadium—like a classic Roman chariot-race stadium, also known as a circus—is not a circle, but has a straight stretch between two half circles.  This simple modification takes the stable orbits of the circle into the unstable orbits of the stadium.

A single classical orbit in a stadium is shown in Fig 1. This is an ergodic orbit that is non-periodic and eventually would fill the entire stadium space. There are other orbits that are nearly periodic, such as one that bounces back and forth vertically between the linear portions, but even this orbit will eventually wander into the circular part of the stadium and then become ergodic. The big quantum-classical question is what happens to these classical orbits when the stadium is shrunk to the nanoscale?

Simulating an evolving quantum wavefunction in free space is surprisingly simple. Given a beginning quantum wavefunction A(x,y,t0), the discrete update equation is

Perfect reflection from the boundaries of the stadium are incorporated through imposing a boundary condition that sends the wavefunction to zero. Simple!

A snap-shot of a wavefunction evolving in the stadium is shown in Fig. 2. To see a movie of the time evolution, see my YouTube episode.

The time average of the wavefunction after a long time has passed is shown in Fig. 3. Other than the horizontal nodal line down the center of the stadium, there is little discernible structure or symmetry. This is also true for the mean squared wavefunction shown in Fig. 4, although there is some structure that may be emerging in the semi-circular regions.

On the other hand, for special initial conditions that have a lot of symmetry, something remarkable happens. Fig. 5 shows several mean-squared results for special initial conditions. There is definite structure in these cases that were given the somewhat ugly name “quantum scars” in the 1980’s by Eric Heller who was one of the first to study this phenomenon [1].

One can superpose highly-symmetric classical trajectories onto the figures, as shown in the bottom row. All of these classical orbits go through a high-symmetry point, such as the center of the stadium (on the left image) and through the focal point of the circular mirrors (in the other two images). The astonishing conclusion of this exercise is that the highly-symmetric periodic classical orbits remain behind as quantum scars—like the Cheshire Cat’s grin—when the system is in the quantum realm. The classical orbits that produce quantum scars have the important property of being periodic but unstable. A slight perturbation from the symmetric trajectory causes it to eventually become ergodic (chaotic). These scars are regions with enhanced probability density, what might be termed “quantum trajectories”, but do not show strong interference patterns.

It is important to make the distinction that it is also possible to construct special wavefunctions that are strictly periodic, such as a wave bouncing perfectly vertically between the straight portions. This leads to large-scale interference patterns that are not the same as the quantum scars.

## Quantum Chaos versus Laser Speckle

In addition to the bouncing-wave cases that do not strictly produce quantum scars, there is another “neutral” phenomenon that produces interference patterns that look a lot like scars, but are simply the random addition of lots of plane waves with the same wavelength [2]. A snapshot in time of one of these superpositions is shown in Fig. 6. To see how the waves add together, see the YouTube channel episode.

By David D. Nolte, Aug. 14, 2022

## YouTube Video

YouTube Video of Quantum Chaos

## References

[1] Heller E J, Bound-state eigenfunctions of classically chaotic hamiltonian-systems – scars of periodic-orbits, Physical Review Letters 53 ,1515 (1984)

[2] Gutzwiller M C, Chaos in classical and quantum mechanics (New York: New York : Springer-Verlag, 1990)

# The Solvay Debates: Einstein versus Bohr

Einstein is the alpha of the quantum. Einstein is also the omega. Although he was the one who established the quantum of energy and matter (see my Blog Einstein vs Planck), Einstein pitted himself in a running debate against Niels Bohr’s emerging interpretation of quantum physics that had, in Einstein’s opinion, severe deficiencies. Between sessions during a series of conferences known as the Solvay Congresses over a period of eight years from 1927 to 1935, Einstein constructed a challenges of increasing sophistication to confront Bohr and his quasi-voodoo attitudes about wave-function collapse. To meet the challenge, Bohr sharpened his arguments and bested Einstein, who ultimately withdrew from the field of battle. Einstein, as quantum physics’ harshest critic, played a pivotal role, almost against his will, establishing the Copenhagen interpretation of quantum physics that rules to this day, and also inventing the principle of entanglement which lies at the core of almost all quantum information technology today.

### Debate Timeline

• Fifth Solvay Congress: 1927 October Brussels: Debate Round 1
• Einstein and ensembles
• Sixth Solvay Congress: 1930 Debate Round 2
• Photon in a box
• Seventh Solvay Congress: 1933
• Einstein absent (visiting the US when Hitler takes power…decides not to return to Germany.)
• Physical Review 1935: Debate Round 3
• EPR paper and Bohr’s response
• Schrödinger’s Cat
• Notable Nobel Prizes
• 1918 Planck
• 1921 Einstein
• 1922 Bohr
• 1932 Heisenberg
• 1933 Dirac and Schrödinger

## The Solvay Conferences

The Solvay congresses were unparalleled scientific meetings of their day.  They were attended by invitation only, and invitations were offered only to the top physicists concerned with the selected topic of each meeting.  The Solvay congresses were held about every three years always in Belgium, supported by the Belgian chemical industrialist Ernest Solvay.  The first meeting, held in 1911, was on the topic of radiation and quanta.

The fifth meeting, held in 1927, was on electrons and photons and focused on the recent rapid advances in quantum theory.  The old quantum guard was invited—Planck, Bohr and Einstein.  The new quantum guard was invited as well—Heisenberg, de Broglie, Schrödinger, Born, Pauli, and Dirac.  Heisenberg and Bohr joined forces to present a united front meant to solidify what later became known as the Copenhagen interpretation of quantum physics.  The basic principles of the interpretation include the wavefunction of Schrödinger, the probabilistic interpretation of Born, the uncertainty principle of Heisenberg, the complementarity principle of Bohr and the collapse of the wavefunction during measurement.  The chief conclusion that Heisenberg and Bohr sought to impress on the assembled attendees was that the theory of quantum processes was complete, meaning that unknown or uncertain  characteristics of measurements could not be attributed to lack of knowledge or understanding, but were fundamental and permanently inaccessible.

Einstein was not convinced with that argument, and he rose to his feet to object after Bohr’s informal presentation of his complementarity principle.  Einstein insisted that uncertainties in measurement were not fundamental, but were caused by incomplete information, that , if known, would accurately account for the measurement results.  Bohr was not prepared for Einstein’s critique and brushed it off, but what ensued in the dining hall and the hallways of the Hotel Metropole in Brussels over the next several days has become one of the most famous scientific debates of the modern era, known as the Bohr-Einstein debate on the meaning of quantum theory.  The debate gently raged night and day through the fifth congress, and was renewed three years later at the 1930 congress.  It finished, in a final flurry of published papers in 1935 that launched some of the central concepts of quantum theory, including the idea of quantum entanglement and, of course, Schrödinger’s cat.

Einstein’s strategy, to refute Bohr, was to construct careful thought experiments that envisioned perfect experiments, without errors, that measured properties of ideal quantum systems.  His aim was to paint Bohr into a corner from which he could not escape, caught by what Einstein assumed was the inconsistency of complementarity.  Einstein’s “thought experiments” used electrons passing through slits, diffracting as required by Schrödinger’s theory, but being detected by classical measurements.  Einstein would present a thought experiment to Bohr, who would then retreat to consider the way around Einstein’s arguments, returning the next hour or the next day with his answer, only to be confronted by yet another clever device of Einstein’s clever imagination that would force Bohr to retreat again.  The spirit of this back and forth encounter between Bohr and Einstein is caught dramatically in the words of Paul Ehrenfest who witnessed the debate first hand, partially mediating between Bohr and Einstein, both of whom he respected deeply.

“Brussels-Solvay was fine!… BOHR towering over everybody.  At first not understood at all … , then  step by step defeating everybody.  Naturally, once again the awful Bohr incantation terminology.  Impossible for anyone else to summarise … (Every night at 1 a.m., Bohr came into my room just to say ONE SINGLE WORD to me, until three a.m.)  It was delightful for me to be present during the conversation between Bohr and Einstein.  Like a game of chess, Einstein all the time with new examples.  In a certain sense a sort of Perpetuum Mobile of the second kind to break the UNCERTAINTY RELATION.  Bohr from out of philosophical smoke clouds constantly searching for the tools to crush one example after the other.  Einstein like a jack-in-the-box; jumping out fresh every morning.  Oh, that was priceless.  But I am almost without reservation pro Bohr and contra Einstein.  His attitude to Bohr is now exacly like the attitude of the defenders of absolute simultaneity towards him …” [1]

The most difficult example that Einstein constructed during the fifth Solvary Congress involved an electron double-slit apparatus that could measure, in principle, the momentum imparted to the slit by the passing electron, as shown in Fig.3.  The electron gun is a point source that emits the electrons in a range of angles that illuminates the two slits.  The slits are small relative to a de Broglie wavelength, so the electron wavefunctions diffract according to Schrödinger’s wave mechanics to illuminate the detection plate.  Because of the interference of the electron waves from the two slits, electrons are detected clustered in intense fringes separated by dark fringes.

So far, everyone was in agreement with these suggested results.  The key next step is the assumption that the electron gun emits only a single electron at a time, so that only one electron is present in the system at any given time.  Furthermore, the screen with the double slit is suspended on a spring, and the position of the screen is measured with complete accuracy by a displacement meter.  When the single electron passes through the entire system, it imparts a momentum kick to the screen, which is measured by the meter.  It is also detected at a specific location on the detection plate.  Knowing the position of the electron detection, and the momentum kick to the screen, provides information about which slit the electron passed through, and gives simultaneous position and momentum values to the electron that have no uncertainty, apparently rebutting the uncertainty principle.

This challenge by Einstein was the culmination of successively more sophisticated examples that he had to pose to combat Bohr, and Bohr was not going to let it pass unanswered.  With ingenious insight, Bohr recognized that the key element in the apparatus was the fact that the screen with the slits must have finite mass if the momentum kick by the electron were to produce a measurable displacement.  But if the screen has finite mass, and hence a finite momentum kick from the electron, then there must be an uncertainty in the position of the slits.  This uncertainty immediately translates into a washout of the interference fringes.  In fact the more information that is obtained about which slit the electron passed through, the more the interference is washed out.  It was a perfect example of Bohr’s own complementarity principle.  The more the apparatus measures particle properties, the less it measures wave properties, and vice versa, in a perfect balance between waves and particles.

Einstein grudgingly admitted defeat at the end of the first round, but he was not defeated.  Three years later he came back armed with more clever thought experiments, ready for the second round in the debate.

## The Sixth Solvay Conference: 1930

At the Solvay Congress of 1930, Einstein was ready with even more difficult challenges.  His ultimate idea was to construct a box containing photons, just like the original black bodies that launched Planck’s quantum hypothesis thirty years before.  The box is attached to a weighing scale so that the weight of the box plus the photons inside can be measured with arbitrarily accuracy. A shutter over a hole in the box is opened for a time T, and a photon is emitted.  Because the photon has energy, it has an equivalent weight (Einstein’s own famous E = mc2), and the mass of the box changes by an amount equal to the photon energy divided by the speed of light squared: m = E/c2.  If the scale has arbitrary accuracy, then the energy of the photon has no uncertainty.  In addition, because the shutter was open for only a time T, the time of emission similarly has no uncertainty.  Therefore, the product of the energy uncertainty and the time uncertainty is much smaller than Planck’s constant, apparently violating Heisenberg’s precious uncertainty principle.

Bohr was stopped in his tracks with this challenge.  Although he sensed immediately that Einstein had missed something (because Bohr had complete confidence in the uncertainty principle), he could not put his finger immediately on what it was.  That evening he wandered from one attendee to another, very unhappy, trying to persuade them and saying that Einstein could not be right because it would be the end of physics.  At the end of the evening, Bohr was no closer to a solution, and Einstein was looking smug.  However, by the next morning Bohr reappeared tired but in high spirits, and he delivered a master stroke.  Where Einstein had used special relaitivity against Bohr, Bohr now used Einstein’s own general relativity against him.

The key insight was that the weight of the box must be measured, and the process of measurement was just as important as the quantum process being measured—this was one of the cornerstones of the Copenhagen interpretation.  So Bohr envisioned a measuring apparatus composed of a spring and a scale with the box suspended in gravity from the spring.  As the photon leaves the box, the weight of the box changes, and so does the deflection of the spring, changing the height of the box.  This change in height, in a gravitational potential, causes the timing of the shutter to change according to the law of gravitational time dilation in general relativity.  By calculating the the general relativistic uncertainty in the time, coupled with the special relativistic uncertainty in the weight of the box, produced a product that was at least as big as Planck’s constant—Heisenberg’s uncertainty principle was saved!

## Entanglement and Schrödinger’s Cat

Einstein ceded the point to Bohr but was not convinced. He still believed that quantum mechanics was not a “complete” theory of quantum physics and he continued to search for the perfect thought experiment that Bohr could not escape. Even today when we have become so familiar with quantum phenomena, the Copenhagen interpretation of quantum mechanics has weird consequences that seem to defy common sense, so it is understandable that Einstein had his reservations.

After the sixth Solvay congress Einstein and Schrödinger exchanged many letters complaining to each other about Bohr’s increasing strangle-hold on the interpretation of quantum mechanics. Egging each other on, they both constructed their own final assault on Bohr. The irony is that the concepts they devised to throw down quantum mechanics have today become cornerstones of the theory. For Einstein, his final salvo was “Entanglement”. For Schrödinger, his final salvo was his “cat”. Today, Entanglement and Schrödinger’s Cat have become enshrined on the alter of quantum interpretation even though their original function was to thwart that interpretation.

The final round of the debate was carried out, not at a Solvay congress, but in the Physical review journal by Einstein [2] and Bohr [3], and in the Naturwissenshaften by Schrödinger [4].

In 1969, Heisenberg looked back on these years and said,

To those of us who participated in the development of atomic theory, the five years following the Solvay Conference in Brussels in 1927 looked so wonderful that we often spoke of them as the golden age of atomic physics. The great obstacles that had occupied all our efforts in the preceding years had been cleared out of the way, the gate to an entirely new field, the quantum mechanics of the atomic shells stood wide open, and fresh fruits seemed ready for the picking. [5]

## References

[1] A. Whitaker, Einstein, Bohr, and the quantum dilemma : from quantum theory to quantum information, 2nd ed. Cambridge University Press, 2006. (pg. 210)

[2] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Physical Review, vol. 47, no. 10, pp. 0777-0780, May (1935)

[3] N. Bohr, “Can quantum-mechanical description of physical reality be considered complete?,” Physical Review, vol. 48, no. 8, pp. 696-702, Oct (1935)

[4] E. Schrodinger, “The current situation in quantum mechanics,” Naturwissenschaften, vol. 23, pp. 807-812, (1935)

[5] W Heisenberg, Physics and beyond : Encounters and conversations (Harper, New York, 1971)

# Timelines in the History and Physics of Dynamics (with links to primary texts)

These timelines in the History of Dynamics are organized along the Chapters in Galileo Unbound (Oxford, 2018). The book is about the physics and history of dynamics including classical and quantum mechanics as well as general relativity and nonlinear dynamics (with a detour down evolutionary dynamics and game theory along the way). The first few chapters focus on Galileo, while the following chapters follow his legacy, as theories of motion became more abstract, eventually to encompass the evolution of species within the same theoretical framework as the orbit of photons around black holes.

## Galileo: A New Scientist

Galileo Galilei was the first modern scientist, launching a new scientific method that superseded, after one and a half millennia, Aristotle’s physics.  Galileo’s career began with his studies of motion at the University of Pisa that were interrupted by his move to the University of Padua and his telescopic discoveries of mountains on the moon and the moons of Jupiter.  Galileo became the first rock star of science, and he used his fame to promote the ideas of Copernicus and the Sun-centered model of the solar system.  But he pushed too far when he lampooned the Pope.  Ironically, Galileo’s conviction for heresy and his sentence to house arrest for the remainder of his life gave him the free time to finally finish his work on the physics of motion, which he published in Two New Sciences in 1638.

1543 Copernicus dies, publishes posthumously De Revolutionibus

1564    Galileo born

1581    Enters University of Pisa

1585    Leaves Pisa without a degree

1586    Invents hydrostatic balance

1588    Receives lecturship in mathematics at Pisa

1592    Chair of mathematics at Univeristy of Padua

1595    Theory of the tides

1595    Invents military and geometric compass

1596    Le Meccaniche and the principle of horizontal inertia

1600    Bruno Giordano burned at the stake

1601    Death of Tycho Brahe

1609    Galileo constructs his first telescope, makes observations of the moon

1610    Galileo discovers 4 moons of Jupiter, Starry Messenger (Sidereus Nuncius), appointed chief philosopher and mathematician of the Duke of Tuscany, moves to Florence, observes Saturn, Venus goes through phases like the moon

1611    Galileo travels to Rome, inducted into the Lyncean Academy, name “telescope” is first used

1611    Scheiner discovers sunspots

1611    Galileo meets Barberini, a cardinal

1611 Johannes Kepler, Dioptrice

1613    Letters on sunspots published by Lincean Academy in Rome

1614    Galileo denounced from the pulpit

1615    (April) Bellarmine writes an essay against Coperinicus

1615    Galileo investigated by the Inquisition

1615    Writes Letter to Christina, but does not publish it

1615    (December) travels to Rome and stays at Tuscan embassy

1616    (January) Francesco Ingoli publishes essay against Copernicus

1616    (March) Decree against copernicanism

1616    Galileo publishes theory of tides, Galileo meets with Pope Paul V, Copernicus’ book is banned, Galileo warned not to support the Coperinican system, Galileo decides not to reply to Ingoli, Galileo proposes eclipses of Jupter’s moons to determine longitude at sea

1618    Three comets appear, Grassi gives a lecture not hostile to Galileo

1618    Galileo, through Mario Guiducci, publishes scathing attack on Grassi

1619    Jesuit Grassi (Sarsi) publishes attack on Galileo concerning 3 comets

1619    Marina Gamba dies, Galileo legitimizes his son Vinczenzio

1619 Kepler’s Laws, Epitome astronomiae Copernicanae.

1623    Barberini becomes Urban VIII, The Assayer published (response to Grassi)

1624    Galileo visits Rome and Urban VIII

1629    Birth of his grandson Galileo

1630    Death of Johanes Kepler

1632    Publication of the Dialogue Concerning the Two Chief World Systems, Galileo is indicted by the Inquisition (68 years old)

1633    (February) Travels to Rome

1633    Convicted, abjurs, house arrest in Rome, then Siena, then home to Arcetri

1638    Blind, publication of Two New Sciences

1642    Galileo dies (77 years old)

## Galileo’s Trajectory

Galileo’s discovery of the law of fall and the parabolic trajectory began with early work on the physics of motion by predecessors like the Oxford Scholars, Tartaglia and the polymath Simon Stevin who dropped lead weights from the leaning tower of Delft three years before Galileo (may have) dropped lead weights from the leaning tower of Pisa.  The story of how Galileo developed his ideas of motion is described in the context of his studies of balls rolling on inclined plane and the surprising accuracy he achieved without access to modern timekeeping.

1583    Galileo Notices isochronism of the pendulum

1588    Receives lecturship in mathematics at Pisa

1589 – 1592  Work on projectile motion in Pisa

1592    Chair of mathematics at Univeristy of Padua

1596    Le Meccaniche and the principle of horizontal inertia

1600    Guidobaldo shares technique of colored ball

1602    Proves isochronism of the pendulum (experimentally)

1604    First experiments on uniformly accelerated motion

1604    Wrote to Scarpi about the law of fall (s ≈ t2)

1607-1608  Identified trajectory as parabolic

1609    Velocity proportional to time

1632    Publication of the Dialogue Concerning the Two Chief World Systems, Galileo is indicted by the Inquisition (68 years old)

1636    Letter to Christina published in Augsburg in Latin and Italian

1638    Blind, publication of Two New Sciences

1641    Invented pendulum clock (in theory)

1642    Dies (77 years old)

## On the Shoulders of Giants

Galileo’s parabolic trajectory launched a new approach to physics that was taken up by a new generation of scientists like Isaac Newton, Robert Hooke and Edmund Halley.  The English Newtonian tradition was adopted by ambitious French iconoclasts who championed Newton over their own Descartes.  Chief among these was Pierre Maupertuis, whose principle of least action was developed by Leonhard Euler and Joseph Lagrange into a rigorous new science of dynamics.  Along the way, Maupertuis became embroiled in a famous dispute that entangled the King of Prussia as well as the volatile Voltaire who was mourning the death of his mistress Emilie du Chatelet, the lone female French physicist of the eighteenth century.

1644    Descartes’ vortex theory of gravitation

1662    Fermat’s principle

1669 – 1690    Huygens expands on Descartes’ vortex theory

1687 Newton’s Principia

1698    Maupertuis born

1729    Maupertuis entered University in Basel.  Studied under Johann Bernoulli

1736    Euler publishes Mechanica sive motus scientia analytice exposita

1737   Maupertuis report on expedition to Lapland.  Earth is oblate.  Attacks Cassini.

1744    Maupertuis Principle of Least Action.  Euler Principle of Least Action.

1745    Maupertuis becomes president of Berlin Academy.  Paris Academy cancels his membership after a campaign against him by Cassini.

1746    Maupertuis principle of Least Action for mass

1751    Samuel König disputes Maupertuis’ priority

1756    Cassini dies.  Maupertuis reinstated in the French Academy

1759    Maupertuis dies

1759    du Chatelet’s French translation of Newton’s Principia published posthumously

1760    Euler 3-body problem (two fixed centers and coplanar third body)

1760-1761 Lagrange, Variational calculus (J. L. Lagrange, “Essai d’une nouvelle méthod pour dEeterminer les maxima et lest minima des formules intégrales indéfinies,” Miscellanea Teurinensia, (1760-1761))

1762    Beginning of the reign of Catherine the Great of Russia

1763    Euler colinear 3-body problem

1765    Euler publishes Theoria motus corporum solidorum on rotational mechanics

1766    Euler returns to St. Petersburg

1766    Lagrange arrives in Berlin

1772    Lagrange equilateral 3-body problem, Essai sur le problème des trois corps, 1772, Oeuvres tome 6

1775    Beginning of the American War of Independence

1776    Adam Smith Wealth of Nations

1781    William Herschel discovers Uranus

1783    Euler dies in St. Petersburg

1787    United States Constitution written

1787    Lagrange moves from Berlin to Paris

1788    Lagrange, Méchanique analytique

1789    Beginning of the French Revolution

1799    Pierre-Simon Laplace Mécanique Céleste (1799-1825)

## Geometry on My Mind

This history of modern geometry focuses on the topics that provided the foundation for the new visualization of physics.  It begins with Carl Gauss and Bernhard Riemann, who redefined geometry and identified the importance of curvature for physics.  Vector spaces, developed by Hermann Grassmann, Giuseppe Peano and David Hilbert, are examples of the kinds of abstract new spaces that are so important for modern physics, such as Hilbert space for quantum mechanics.  Fractal geometry developed by Felix Hausdorff later provided the geometric language needed to solve problems in chaos theory.

1629    Fermat described higher-dim loci

1637    Descarte’s Geometry

1649    van Schooten’s commentary on Descartes Geometry

1694    Leibniz uses word “coordinate” in its modern usage

1697    Johann Bernoulli shortest distance between two points on convex surface

1732    Euler geodesic equations for implicit surfaces

1748    Euler defines modern usage of function

1801    Gauss calculates orbit of Ceres

1807    Fourier analysis (published in 1822(

1807    Gauss arrives in Göttingen

1827    Karl Gauss establishes differential geometry of curved surfaces, Disquisitiones generales circa superficies curvas

1830    Bolyai and Lobachevsky publish on hyperbolic geometry

1834    Jacobi n-fold integrals and volumes of n-dim spheres

1836    Liouville-Sturm theorem

1838    Liouville’s theorem

1841    Jacobi determinants

1843    Arthur Cayley systems of n-variables

1843    Hamilton discovers quaternions

1844    Hermann Grassman n-dim vector spaces, Die Lineale Ausdehnungslehr

1846    Julius Plücker System der Geometrie des Raumes in neuer analytischer Behandlungsweise

1848 Jacobi Vorlesungen über Dynamik

1848    “Vector” coined by Hamilton

1854    Riemann’s habilitation lecture

1861    Riemann n-dim solution of heat conduction

1868    Publication of Riemann’s Habilitation

1869    Christoffel and Lipschitz work on multiple dimensional analysis

1871    Betti refers to the n-ply of numbers as a “space”.

1871    Klein publishes on non-euclidean geometry

1872 Boltzmann distribution

1872    Jordan Essay on the geometry of n-dimensions

1872    Felix Klein’s “Erlangen Programme”

1872    Weierstrass’ Monster

1872    Dedekind cut

1872    Cantor paper on irrational numbers

1872    Cantor meets Dedekind

1872 Lipschitz derives mechanical motion as a geodesic on a manifold

1874    Cantor beginning of set theory

1877    Cantor one-to-one correspondence between the line and n-dimensional space

1881    Gibbs codifies vector analysis

1883    Cantor set and staircase Grundlagen einer allgemeinen Mannigfaltigkeitslehre

1884    Abbott publishes Flatland

1887    Peano vector methods in differential geometry

1890    Peano space filling curve

1891    Hilbert space filling curve

1887    Darboux vol. 2 treats dynamics as a point in d-dimensional space.  Applies concepts of geodesics for trajectories.

1898    Ricci-Curbastro Lesons on the Theory of Surfaces

1902    Lebesgue integral

1904    Hilbert studies integral equations

1904    von Koch snowflake

1906    Frechet thesis on square summable sequences as infinite dimensional space

1908    Schmidt Geometry in a Function Space

1910    Brouwer proof of dimensional invariance

1913    Hilbert space named by Riesz

1914    Hilbert space used by Hausdorff

1915    Sierpinski fractal triangle

1918    Hausdorff non-integer dimensions

1918    Weyl’s book Space, Time, Matter

1918    Fatou and Julia fractals

1920    Banach space

1927    von Neumann axiomatic form of Hilbert Space

1935    Frechet full form of Hilbert Space

1967    Mandelbrot coast of Britain

1982    Mandelbrot’s book The Fractal Geometry of Nature

## The Tangled Tale of Phase Space

Phase space is the central visualization tool used today to study complex systems.  The chapter describes the origins of phase space with the work of Joseph Liouville and Carl Jacobi that was later refined by Ludwig Boltzmann and Rudolf Clausius in their attempts to define and explain the subtle concept of entropy.  The turning point in the history of phase space was when Henri Poincaré used phase space to solve the three-body problem, uncovering chaotic behavior in his quest to answer questions on the stability of the solar system.  Phase space was established as the central paradigm of statistical mechanics by JW Gibbs and Paul Ehrenfest.

1804    Jacobi born (1904 – 1851) in Potsdam

1804    Napoleon I Emperor of France

1806    William Rowan Hamilton born (1805 – 1865)

1807    Thomas Young describes “Energy” in his Course on Natural Philosophy (Vol. 1 and Vol. 2)

1808    Bethoven performs his Fifth Symphony

1809    Joseph Liouville born (1809 – 1882)

1821    Hermann Ludwig Ferdinand von Helmholtz born (1821 – 1894)

1824    Carnot published Reflections on the Motive Power of Fire

1834    Jacobi n-fold integrals and volumes of n-dim spheres

1834-1835       Hamilton publishes his principle (1834, 1835).

1836    Liouville-Sturm theorem

1837    Queen Victoria begins her reign as Queen of England

1838    Liouville develops his theorem on products of n differentials satisfying certain first-order differential equations.  This becomes the classic reference to Liouville’s Theorem.

1847    Helmholtz  Conservation of Energy (force)

1849    Thomson makes first use of “Energy” (From reading Thomas Young’s lecture notes)

1850    Clausius establishes First law of Thermodynamics: Internal energy. Second law:  Heat cannot flow unaided from cold to hot.  Not explicitly stated as first and second laws

1851    Thomson names Clausius’ First and Second laws of Thermodynamics

1852    Thomson describes general dissipation of the universe (“energy” used in title)

1854    Thomson defined absolute temperature.  First mathematical statement of 2nd law.  Restricted to reversible processes

1854    Clausius stated Second Law of Thermodynamics as inequality

1857    Clausius constructs kinetic theory, Mean molecular speeds

1858    Clausius defines mean free path, Molecules have finite size. Clausius assumed that all molecules had the same speed

1860    Maxwell publishes first paper on kinetic theory. Distribution of speeds. Derivation of gas transport properties

1865    Loschmidt size of molecules

1865    Clausius names entropy

1868    Boltzmann adds (Boltzmann) factor to Maxwell distribution

1872    Boltzmann transport equation and H-theorem

1876    Loschmidt reversibility paradox

1877    Boltzmann  S = k logW

1890    Poincare: Recurrence Theorem. Recurrence paradox with Second Law (1893)

1896    Zermelo criticizes Boltzmann

1896    Boltzmann posits direction of time to save his H-theorem

1898    Boltzmann Vorlesungen über Gas Theorie

1905    Boltzmann kinetic theory of matter in Encyklopädie der mathematischen Wissenschaften

1906    Boltzmann dies

1910    Paul Hertz uses “Phase Space” (Phasenraum)

1911    Ehrenfest’s article in Encyklopädie der mathematischen Wissenschaften

1913    A. Rosenthal writes the first paper using the phrase “phasenraum”, combining the work of Boltzmann and Poincaré. “Beweis der Unmöglichkeit ergodischer Gassysteme” (Ann. D. Physik, 42, 796 (1913)

1913    Plancheral, “Beweis der Unmöglichkeit ergodischer mechanischer Systeme” (Ann. D. Physik, 42, 1061 (1913).  Also uses “Phasenraum”.

## The Lens of Gravity

Gravity provided the backdrop for one of the most important paradigm shifts in the history of physics.  Prior to Albert Einstein’s general theory of relativity, trajectories were paths described by geometry.  After the theory of general relativity, trajectories are paths caused by geometry.  This chapter explains how Einstein arrived at his theory of gravity, relying on the space-time geometry of Hermann Minkowski, whose work he had originally harshly criticized.  The confirmation of Einstein’s theory was one of the dramatic high points in 20th century history of physics when Arthur Eddington journeyed to an island off the coast of Africa to observe stellar deflections during a solar eclipse.  If Galileo was the first rock star of physics, then Einstein was the first worldwide rock star of science.

1697    Johann Bernoulli was first to find solution to shortest path between two points on a curved surface (1697).

1728    Euler found the geodesic equation.

1783    The pair 40 Eridani B/C was discovered by William Herschel on 31 January

1783    John Michell explains infalling object would travel faster than speed of light

1796    Laplace describes “dark stars” in Exposition du system du Monde

1827    The first orbit of a binary star computed by Félix Savary for the orbit of Xi Ursae Majoris.

1827    Gauss curvature Theoriem Egregum

1844    Bessel notices periodic displacement of Sirius with period of half a century

1844    The name “geodesic line” is attributed to Liouville.

1845    Buys Ballot used musicians with absolute pitch for the first experimental verification of the Doppler effect

1854    Riemann’s habilitationsschrift

1862    Discovery of Sirius B (a white dwarf)

1868    Darboux suggested motions in n-dimensions

1872    Lipshitz first to apply Riemannian geometry to the principle of least action.

1895    Hilbert arrives in Göttingen

1902    Minkowski arrives in Göttingen

1905    Einstein’s miracle year

1906    Poincaré describes Lorentz transformations as rotations in 4D

1907    Einstein has “happiest thought” in November

1907    Einstein’s relativity review in Jahrbuch

1908    Minkowski’s Space and Time lecture

1908    Einstein appointed to unpaid position at University of Bern

1909    Minkowski dies

1909    Einstein appointed associate professor of theoretical physics at U of Zürich

1910    40 Eridani B was discobered to be of spectral type A (white dwarf)

1910    Size and mass of Sirius B determined (heavy and small)

1911    Laue publishes first textbook on relativity theory

1911    Einstein accepts position at Prague

1911    Einstein goes to the limits of special relativity applied to gravitational fields

1912    Einstein’s two papers establish a scalar field theory of gravitation

1912    Einstein moves from Prague to ETH in Zürich in fall.  Begins collaboration with Grossmann.

1913    Einstein EG paper

1914    Adams publishes spectrum of 40 Eridani B

1915    Sirius B determined to be also a low-luminosity type A white dwarf

1915    Einstein Completes paper

1916    Density of 40 Eridani B by Ernst Öpik

1916    Schwarzschild paper

1916 Einstein’s publishes theory of gravitational waves

1919    Eddington expedition to Principe

1920    Eddington paper on deflection of light by the sun

1922    Willem Luyten coins phrase “white dwarf”

1924    Eddington found a set of coordinates that eliminated the singularity at the Schwarzschild radius

1926    R. H. Fowler publishes paper on degenerate matter and composition of white dwarfs

1931    Chandrasekhar calculated the limit for collapse to white dwarf stars at 1.4MS

1933    Georges Lemaitre states the coordinate singularity was an artefact

1934    Walter Baade and Fritz Zwicky proposed the existence of the neutron star only a year after the discovery of the neutron by Sir James Chadwick.

1939    Oppenheimer and Snyder showed ultimate collapse of a 3MS  “frozen star”

1958    David Finkelstein paper

1965    Antony Hewish and Samuel Okoye discovered “an unusual source of high radio brightness temperature in the Crab Nebula”. This source turned out to be the Crab Nebula neutron star that resulted from the great supernova of 1054.

1967    Jocelyn Bell and Antony Hewish discovered regular radio pulses from CP 1919. This pulsar was later interpreted as an isolated, rotating neutron star.

1967    Wheeler’s “black hole” talk

1974    Joseph Taylor and Russell Hulse discovered the first binary pulsar, PSR B1913+16, which consists of two neutron stars (one seen as a pulsar) orbiting around their center of mass.

2015    LIGO detects gravitational waves on Sept. 14 from the merger of two black holes

2017    LIGO detects the merger of two neutron stars

## On the Quantum Footpath

The concept of the trajectory of a quantum particle almost vanished in the battle between Werner Heisenberg’s matrix mechanics and Erwin Schrödinger’s wave mechanics.  It took Niels Bohr and his complementarity principle of wave-particle duality to cede back some reality to quantum trajectories.  However, Schrödinger and Einstein were not convinced and conceived of quantum entanglement to refute the growing acceptance of the Copenhagen Interpretation of quantum physics.  Schrödinger’s cat was meant to be an absurdity, but ironically it has become a central paradigm of practical quantum computers.  Quantum trajectories took on new meaning when Richard Feynman constructed quantum theory based on the principle of least action, inventing his famous Feynman Diagrams to help explain quantum electrodynamics.

1885    Balmer Theory

1897    J. J. Thomson discovered the electron

1904    Thomson plum pudding model of the atom

1911    Bohr PhD thesis filed. Studies on the electron theory of metals.  Visited England.

1911    Rutherford nuclear model

1911    First Solvay conference

1911    “ultraviolet catastrophe” coined by Ehrenfest

1913    Bohr combined Rutherford’s nuclear atom with Planck’s quantum hypothesis: 1913 Bohr model

1913    Ehrenfest adiabatic hypothesis

1914-1916       Bohr at Manchester with Rutherford

1916    Bohr appointed Chair of Theoretical Physics at University of Copenhagen: a position that was made just for him

1916    Schwarzschild and Epstein introduce action-angle coordinates into quantum theory

1920    Heisenberg enters University of Munich to obtain his doctorate

1920    Bohr’s Correspondence principle: Classical physics for large quantum numbers

1921    Bohr Founded Institute of Theoretical Physics (Copenhagen)

1922-1923       Heisenberg studies with Born, Franck and Hilbert at Göttingen while Sommerfeld is in the US on sabbatical.

1923    Heisenberg Doctorate.  The exam does not go well.  Unable to derive the resolving power of a microscope in response to question by Wien.  Becomes Born’s assistant at Göttingen.

1924    Heisenberg visits Niels Bohr in Copenhagen (and met Einstein?)

1924    Heisenberg Habilitation at Göttingen on anomalous Zeeman

1924 – 1925    Heisenberg worked with Bohr in Copenhagen, returned summer of 1925 to Göttiingen

1924    Pauli exclusion principle and state occupancy

1924    de Broglie hypothesis extended wave-particle duality to matter

1924    Bohr Predicted Halfnium (72)

1924    Kronig’s proposal for electron self spin

1924    Bose (Einstein)

1925    Dirac, reading proof from Heisenberg, recognized the analogy of noncommutativity with Poisson brackets and the correspondence with Hamiltonian mechanics.

1925    Uhlenbeck and Goudschmidt: spin

1926    Born, Heisenberg, Kramers: virtual oscillators at transition frequencies: Matrix mechanics (alternative to Bohr-Kramers-Slater 1924 model of orbits).  Heisenberg was Born’s student at Göttingen.

1926    Schrödinger wave mechanics

1927    de Broglie hypotehsis confirmed by Davisson and Germer

1927    Complementarity by Bohr: wave-particle duality “Evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects.

1927    Heisenberg uncertainty principle (Heisenberg was in Copenhagen 1926 – 1927)

1927    Solvay Conference in Brussels

1928    Heisenberg to University of Leipzig

1928    Dirac relativistic QM equation

1929    de Broglie Nobel Prize

1930    Solvay Conference

1932    Heisenberg Nobel Prize

1932    von Neumann operator algebra

1933    Dirac Lagrangian form of QM (basis of Feynman path integral)

1933    Schrödinger and Dirac Nobel Prize

1935    Einstein, Poldolsky and Rosen EPR paper

1935 Bohr’s response to Einsteins “EPR” paradox

1935    Schrodinger’s cat

1939    Feynman graduates from MIT

1941    Heisenberg (head of German atomic project) visits Bohr in Copenhagen

1942    Feynman PhD at Princeton, “The Principle of Least Action in Quantum Mechanics

1942 – 1945    Manhattan Project, Bethe-Feynman equation for fission yield

1943    Bohr escapes to Sweden in a fishing boat.  Went on to England secretly.

1945    Pauli Nobel Prize

1945    Death of Feynman’s wife Arline (married 4 years)

1945    Fall, Feynman arrives at Cornell ahead of Hans Bethe

1947    Shelter Island conference: Lamb Shift, did Kramer’s give a talk suggesting that infinities could be subtracted?

1947    Fall, Dyson arrives at Cornell

1948    Pocono Manor, Pennsylvania, troubled unveiling of path integral formulation and Feynman diagrams, Schwinger’s master presentation

1948    Feynman and Dirac. Summer drive across the US with Dyson

1949    Dyson joins IAS as a postdoc, trains a cohort of theorists in Feynman’s technique

1949    Karplus and Kroll first g-factor calculation

1950    Feynman moves to Cal Tech

1965    Schwinger, Tomonaga and Feynman Nobel Prize

1967    Hans Bethe Nobel Prize

## From Butterflies to Hurricanes

Half a century after Poincaré first glimpsed chaos in the three-body problem, the great Russian mathematician Andrey Kolmogorov presented a sketch of a theorem that could prove that orbits are stable.  In the hands of Vladimir Arnold and Jürgen Moser, this became the KAM theory of Hamiltonian chaos.  This chapter shows how KAM theory fed into topology in the hands of Stephen Smale and helped launch the new field of chaos theory.  Edward Lorenz discovered chaos in numerical models of atmospheric weather and discovered the eponymous strange attractor.  Mathematical aspects of chaos were further developed by Mitchell Feigenbaum studying bifurcations in the logistic map that describes population dynamics.

1760    Euler 3-body problem (two fixed centers and coplanar third body)

1763    Euler colinear 3-body problem

1772    Lagrange equilateral 3-body problem

1881-1886       Poincare memoires “Sur les courbes de ́finies par une equation differentielle”

1890    Poincare “Sur le probleme des trois corps et les equations de la dynamique”. First-return map, Poincare recurrence theorem, stable and unstable manifolds

1892 – 1899    Poincare New Methods in Celestial Mechanics

1892    Lyapunov The General Problem of the Stability of Motion

1899    Poincare homoclinic trajectory

1913    Birkhoff proves Poincaré’s last geometric theorem, a special case of the three-body problem.

1927    van der Pol and van der Mark

1937    Coarse systems, Andronov and Pontryagin

1938    Morse theory

1942    Hopf bifurcation

1945    Cartwright and Littlewood study the van der Pol equation (Radar during WWII)

1954    Kolmogorov A. N., On conservation of conditionally periodic motions for a small change in Hamilton’s function.

1960    Lorenz: 12 equations

1962    Moser On Invariant Curves of Area-Preserving Mappings of an Annulus.

1963    Lorenz: 3 equations

1964    Arnold diffusion

1965    Smale’s horseshoe

1969    Chirikov standard map

1971    Ruelle-Takens (Ruelle coins phrase “strange attractor”)

1972    “Butterfly Effect” given for Lorenz’ talk (by Philip Merilees)

1975    Gollub-Swinney observe route to turbulence along lines of Ruelle

1975    Yorke coins “chaos theory”

1976    Robert May writes review article of the logistic map

1977    New York conference on bifurcation theory

1987    James Gleick Chaos: Making a New Science

## Darwin in the Clockworks

The preceding timelines related to the central role played by families of trajectories phase space to explain the time evolution of complex systems.  These ideas are extended to explore the history and development of the theory of natural evolution by Charles Darwin.  Darwin had many influences, including ideas from Thomas Malthus in the context of economic dynamics.  After Darwin, the ideas of evolution matured to encompass broad topics in evolutionary dynamics and the emergence of the idea of fitness landscapes and game theory driving the origin of new species.  The rise of genetics with Gregor Mendel supplied a firm foundation for molecular evolution, leading to the moleculer clock of Linus Pauling and the replicator dynamics of Richard Dawkins.

1202    Fibonacci

1766    Thomas Robert Malthus born

1776    Adam Smith The Wealth of Nations

1798    Malthus “An Essay on the Principle of Population

1817    Ricardo Principles of Political Economy and Taxation

1838    Cournot early equilibrium theory in duopoly

1848    John Stuart Mill

1848    Karl Marx Communist Manifesto

1859    Darwin Origin of Species

1867    Karl Marx Das Kapital

1871    Darwin Descent of Man, and Selection in Relation to Sex

1871    Jevons Theory of Political Economy

1871    Menger Principles of Economics

1874    Walrus Éléments d’économie politique pure, or Elements of Pure Economics (1954)

1890    Marshall Principles of Economics

1908    Hardy constant genetic variance

1910    Brouwer fixed point theorem

1910    Alfred J. Lotka autocatylitic chemical reactions

1913    Zermelo determinancy in chess

1922    Fisher dominance ratio

1922    Fisher mutations

1925    Lotka predator-prey in biomathematics

1926    Vita Volterra published same equations independently

1927    JBS Haldane (1892—1964) mutations

1928    von Neumann proves the minimax theorem

1930    Fisher ratio of sexes

1932    Wright Adaptive Landscape

1932    Haldane The Causes of Evolution

1933    Kolmogorov Foundations of the Theory of Probability

1934    Rudolph Carnap The Logical Syntax of Language

1936    John Maynard Keynes, The General Theory of Employment, Interest and Money

1936    Kolmogorov generalized predator-prey systems

1938    Borel symmetric payoff matrix

1942    Sewall Wright    Statistical Genetics and Evolution

1943    McCulloch and Pitts A Logical Calculus of Ideas Immanent in Nervous Activity

1944    von Neumann and Morgenstern Theory of Games and Economic Behavior

1950    Prisoner’s Dilemma simulated at Rand Corportation

1950    John Nash Equilibrium points in n-person games and The Bargaining Problem

1951    John Nash Non-cooperative Games

1952    McKinsey Introduction to the Theory of Games (first textbook)

1953    John Nash Two-Person Cooperative Games

1953    Watson and Crick DNA

1955    Braithwaite’s Theory of Games as a Tool for the Moral Philosopher

1961    Lewontin Evolution and the Theory of Games

1962    Patrick Moran The Statistical Processes of Evolutionary Theory

1962    Linus Pauling molecular clock

1968    Motoo Kimura  neutral theory of molecular evolution

1972    Maynard Smith introduces the evolutionary stable solution (ESS)

1972    Gould and Eldridge Punctuated equilibrium

1973    Maynard Smith and Price The Logic of Animal Conflict

1973    Black Scholes

1977    Eigen and Schuster The Hypercycle

1978    Replicator equation (Taylor and Jonker)

1982    Hopfield network

1982    John Maynard Smith Evolution and the Theory of Games

1984    R. Axelrod The Evolution of Cooperation

## The Measure of Life

This final topic extends the ideas of dynamics into abstract spaces of high dimension to encompass the idea of a trajectory of life.  Health and disease become dynamical systems defined by all the proteins and nucleic acids that comprise the physical self.  Concepts from network theory, autonomous oscillators and synchronization contribute to this viewpoint.  Healthy trajectories are like stable limit cycles in phase space, but disease can knock the system trajectory into dangerous regions of health space, as doctors turn to new developments in personalized medicine try to return the individual to a healthy path.  This is the ultimate generalization of Galileo’s simple parabolic trajectory.

1642    Galileo dies

1656    Huygens invents pendulum clock

1665    Huygens observes “odd kind of sympathy” in synchronized clocks

1673    Huygens publishes Horologium Oscillatorium sive de motu pendulorum

1736    Euler Seven Bridges of Königsberg

1845    Kirchhoff’s circuit laws

1852    Guthrie four color problem

1857    Cayley trees

1858    Hamiltonian cycles

1887    Cajal neural staining microscopy

1913    Michaelis Menten dynamics of enzymes

1924    Berger, Hans: neural oscillations (Berger invented the EEG)

1926    van der Pol dimensioness form of equation

1927    van der Pol periodic forcing

1943    McCulloch and Pits mathematical model of neural nets

1948    Wiener cybernetics

1952    Hodgkin and Huxley action potential model

1952    Turing instability model

1956    Sutherland cyclic AMP

1957    Broadbent and Hammersley bond percolation

1959    Erdös and Renyi random graphs

1962    Cohen EGF discovered

1965    Sebeok coined zoosemiotics

1966    Mesarovich systems biology

1967    Winfree biological rythms and coupled oscillators

1969    Glass Moire patterns in perception

1970    Rodbell G-protein

1971    phrase “strange attractor” coined (Ruelle)

1972    phrase “signal transduction” coined (Rensing)

1975    phrase “chaos theory” coined (Yorke)

1975    Kuramoto transition

1976    Robert May logistic map

1977    Mackey-Glass equation and dynamical disease

1982    Hopfield network

1990    Strogatz and Murillo pulse-coupled oscillators

1997    Tomita systems biology of a cell

1998    Strogatz and Watts Small World network

1999    Barabasi Scale Free networks

# Who Invented the Quantum? Einstein vs. Planck

Albert Einstein defies condensation—it is impossible to condense his approach, his insight, his motivation—into a single word like “genius”.  He was complex, multifaceted, contradictory, revolutionary as well as conservative.  Some of his work was so simple that it is hard to understand why no-one else did it first, even when they were right in the middle of it.  Lorentz and Poincaré spring to mind—they had been circling the ideas of spacetime for decades—but never stepped back to see what the simplest explanation could be.  Einstein did, and his special relativity was simple and beautiful, and the math is just high-school algebra.  On the other hand, parts of his work—like gravitation—are so embroiled in mathematics and the religion of general covariance that it remains opaque to physics neophytes 100 years later and is usually reserved for graduate study.

Yet there is a third thread in Einstein’s work that relies on pure intuition—neither simple nor complicated—but almost impossible to grasp how he made his leap.  This was the case when he proposed the real existence of the photon—the quantum particle of light.  For ten years after this proposal, it was considered by almost everyone to be his greatest blunder. It even came up when Planck was nominating Einstein for membership in the German Academy of Science. Planck said

That he may sometimes have missed the target of his speculations, as for example, in his hypothesis of light quanta, cannot really be held against him.

In this single statement, we have the father of the quantum being criticized by the father of the quantum discontinuity.

## Max Planck’s Discontinuity

In histories of the development of quantum theory, the German physicist Max Planck (1858—1947) is characterized as an unlikely revolutionary.  He was an establishment man, in the stolid German tradition, who was already embedded in his career, in his forties, holding a coveted faculty position at the University of Berlin.  In his research, he was responding to a theoretical challenge issued by Kirchhoff many years ago in 1860 to find the function of temperature and wavelength that described and explained the observed spectrum of radiating bodies.  Planck was not looking for a revolution.  In fact, he was looking for the opposite.  One of his motivations in studying the thermodynamics of electromagnetic radiation was to rebut the statistical theories of Boltzmann.  Planck had never been convinced by the atomistic and discrete approach Boltzmann had used to explain entropy and the second law of thermodynamics.  With the continuum of light radiation he thought he had the perfect system that would show how entropy behaved in a continuous manner, without the need for discrete quantities.

Therefore, Planck’s original intentions were to use blackbody radiation to argue against Boltzmann—to set back the clock.  For this reason, not only was Planck an unlikely revolutionary, he was a counter-revolutionary.  But Planck was a revolutionary because that is what he did, whatever his original intentions were, and he accepted his role as a revolutionary when he had the courage to stand in front of his scientific peers and propose a quantum hypothesis that lay at the heart of physics.

Blackbody radiation, at the end of the nineteenth century, was a topic of keen interest and had been measured with high precision.  This was in part because it was such a “clean” system, having fundamental thermodynamic properties independent of any of the material properties of the black body, unlike the so-called ideal gases, which always showed some dependence on the molecular properties of the gas. The high-precision measurements of blackbody radiation were made possible by new developments in spectrometers at the end of the century, as well as infrared detectors that allowed very precise and repeatable measurements to be made of the spectrum across broad ranges of wavelengths.

In 1893 the German physicist Wilhelm Wien (1864—1928) had used adiabatic expansion arguments to derive what became known as Wien’s Displacement Law that showed a simple linear relationship between the temperature of the blackbody and the peak wavelength.  Later, in 1896, he showed that the high-frequency behavior could be described by an exponential function of temperature and wavelength that required no other properties of the blackbody.  This was approaching the solution of Kirchhoff’s challenge of 1860 seeking a universal function.  However, at lower frequencies Wien’s approximation failed to match the measured spectrum.  In mid-year 1900, Planck was able to define a single functional expression that described the experimentally observed spectrum.  Planck had succeeded in describing black-body radiation, but he had not satisfied Kirchhoff’s second condition—to explain it.

Therefore, to describe the blackbody spectrum, Planck modeled the emitting body as a set of ideal oscillators.  As an expert in the Second Law, Planck derived the functional form for the radiation spectrum, from which he found the entropy of the oscillators that produced the spectrum.  However, once he had the form for the entropy, he needed to explain why it took that specific form.  In this sense, he was working backwards from a known solution rather than forwards from first principles.  Planck was at an impasse.  He struggled but failed to find any continuum theory that could work.

Then Planck turned to Boltzmann’s statistical theory of entropy, the same theory that he had previously avoided and had hoped to discredit.  He described this as “an act of despair … I was ready to sacrifice any of my previous convictions about physics.”  In Boltzmann’s expression for entropy, it was necessary to “count” possible configurations of states.  But counting can only be done if the states are discrete.  Therefore, he lumped the energies of the oscillators into discrete ranges, or bins, that he called “quanta”.  The size of the bins was proportional to the frequency of the oscillator, and the proportionality constant had the units of Maupertuis’ quantity of action, so Planck called it the “quantum of action”. Finally, based on this quantum hypothesis, Planck derived the functional form of black-body radiation.

Planck presented his findings at a meeting of the German Physical Society in Berlin on November 15, 1900, introducing the word quantum (plural quanta) into physics from the Latin word that means quantity [1].  It was a casual meeting, and while the attendees knew they were seeing an intriguing new physical theory, there was no sense of a revolution.  But Planck himself was aware that he had created something fundamentally new.  The radiation law of cavities depended on only two physical properties—the temperature and the wavelength—and on two constants—Boltzmann’s constant kB and a new constant that later became known as Planck’s constant h = ΔE/f = 6.6×10-34 J-sec.  By combining these two constants with other fundamental constants, such as the speed of light, Planck was able to establish accurate values for long-sought constants of nature, like Avogadro’s number and the charge of the electron.

Although Planck’s quantum hypothesis in 1900 explained the blackbody radiation spectrum, his specific hypothesis was that it was the interaction of the atoms and the light field that was somehow quantized.  He certainly was not thinking in terms of individual quanta of the light field.

## Einstein’s Quantum

When Einstein analyzed the properties of the blackbody radiation in 1905, using his deep insight into statistical mechanics, he was led to the inescapable conclusion that light itself must be quantized in amounts E = hf, where h is Planck’s constant and f is the frequency of the light field.  Although this equation is exactly the same as Planck’s from 1900, the meaning was completely different.  For Planck, this was the discreteness of the interaction of light with matter.  For Einstein, this was the quantum of light energy—whole and indivisible—just as if the light quantum were a particle with particle properties.  For this reason, we can answer the question posed in the title of this Blog—Einstein takes the honor of being the inventor of the quantum.

Einstein’s clarity of vision is a marvel to behold even to this day.  His special talent was to take simple principles, ones that are almost trivial and beyond reproach, and to derive something profound.  In Special Relativity, he simply assumed the constancy of the speed of light and derived Lorentz’s transformations that had originally been based on obtuse electromagnetic arguments about the electron.  In General Relativity, he assumed that free fall represented an inertial frame, and he concluded that gravity must bend light.  In quantum theory, he assumed that the low-density limit of Planck’s theory had to be consistent with light in thermal equilibrium in thermal equilibrium with the black body container, and he concluded that light itself must be quantized into packets of indivisible energy quanta [2].  One immediate consequence of this conclusion was his simple explanation of the photoelectric effect for which the energy of an electron ejected from a metal by ultraviolet irradiation is a linear function of the frequency of the radiation.  Einstein published his theory of the quanta of light [3] as one of his four famous 1905 articles in Annalen der Physik in his Annus Mirabilis

Einstein’s theory of light quanta was controversial and was slow to be accepted.  It is ironic that in 1914 when Einstein was being considered for a position at the University in Berlin, Planck himself, as he championed Einstein’s case to the faculty, implored his colleagues to accept Einstein despite his ill-conceived theory of light quanta [4].  This comment by Planck goes far to show how Planck, father of the quantum revolution, did not fully grasp, even by 1914, the fundamental nature and consequences of his original quantum hypothesis.  That same year, the American physicist Robert Millikan (1868—1953) performed a precise experimental measurement of the photoelectric effect, with the ostensible intention of proving Einstein wrong, but he accomplished just the opposite—providing clean experimental evidence confirming Einstein’s theory of the photoelectric effect.

## The Stimulated Emission of Light

About a year after Millikan proved that the quantum of energy associated with light absorption was absorbed as a whole quantum of energy that was not divisible, Einstein took a step further in his theory of the light quantum. In 1916 he published a paper in the proceedings of the German Physical Society that explored how light would be in a state of thermodynamic equilibrium when interacting with atoms that had discrete energy levels. Once again he used simple arguments, this time using the principle of detailed balance, to derive a new and unanticipated property of light—stimulated emission!

The stimulated emission of light occurs when an electron is in an excited state of a quantum system, like an atom, and an incident photon stimulates the emission of a second photon that has the same energy and phase as the first photon. If there are many atoms in the excited state, then this process leads to a chain reaction as 1 photon produces 2, and 2 produce 4, and 4 produce 8, etc. This exponential gain in photons with the same energy and phase is the origin of laser radiation. At the time that Einstein proposed this mechanism, lasers were half a century in the future, but he was led to this conclusion by extremely simple arguments about transition rates.

Detailed balance is a principle that states that in thermal equilibrium all fluxes are balanced. In the case of atoms with ground states and excited states, this principle requires that as many transitions occur from the ground state to the excited state as from the excited state to the ground state. The crucial new element that Einstein introduced was to distinguish spontaneous emission from stimulated emission. Just as the probability to absorb a photon must be proportional to the photon density, there must be an equivalent process that de-excites the atom that also must be proportional the photon density. In addition, an electron must be able to spontaneously emit a photon with a rate that is independent of photon density. This leads to distinct coefficients in the transition rate equations that are today called the “Einstein A and B coefficients”. The B coefficients relate to the photon density, while the A coefficient relates to spontaneous emission.

Using the principle of detailed balance together with his A and B coefficients as well as Boltzmann factors describing the number of excited states relative to ground state atoms in equilibrium at a given temperature, Einstein was able to derive an early form of what is today called the Bose-Einstein occupancy function for photons.

## Derivation of the Einstein A and B Coefficients

Detailed balance requires the rate from m to n to be the same as the rate from n to m

where the first term is the spontaneous emission rate from the excited state m to the ground state n, the second term is the stimulated emission rate, and the third term (on the right) is the absorption rate from n to m. The numbers in each state are Nm and Nn, and the density of photons is ρ. The relative numbers in the excited state relative to the ground state is given by the Boltzmann factor

By assuming that the stimulated transition coefficient from n to m is the same as m to n, and inserting the Boltzmann factor yields

The Planck density of photons for ΔE = hf is

which yields the final relation between the spontaneous emission coefficient and the stimulated emission coefficient

The total emission rate is

where the p-bar is the average photon number in the cavity. One of the striking aspects of this derivation is that no assumptions are made about the physical mechanisms that determine the coefficient B. Only arguments of detailed balance are required to arrive at these results.

## Einstein’s Quantum Legacy

Einstein was awarded the Nobel Prize in 1921 for the photoelectric effect, not for the photon nor for any of Einstein’s other theoretical accomplishments.  Even in 1921, the quantum nature of light remained controversial.  It was only in 1923, after the American physicist Arthur Compton (1892—1962) showed that energy and momentum were conserved in the scattering of photons from electrons, that the quantum nature of light began to be accepted.  The very next year, in 1924, the quantum of light was named the “photon” by the American American chemical physicist Gilbert Lewis (1875—1946).

A blog article like this, that attributes the invention of the quantum to Einstein rather than Planck, must say something about the irony of this attribution.  If Einstein is the father of the quantum, he ultimately was led to disinherit his own brain child.  His final and strongest argument against the quantum properties inherent in the Copenhagen Interpretation was his famous EPR paper which, against his expectations, launched the concept of entanglement that underlies the coming generation of quantum computers.

By David D. Nolte, Jan. 13, 2020

## Einstein’s Quantum Timeline

1900 – Planck’s quantum discontinuity for the calculation of the entropy of blackbody radiation.

1905 – Einstein’s “Miracle Year”. Proposes the light quantum.

1911 – First Solvay Conference on the theory of radiation and quanta.

1913 – Bohr’s quantum theory of hydrogen.

1914 – Einstein becomes a member of the German Academy of Science.

1915 – Millikan measurement of the photoelectric effect.

1916 – Einstein proposes stimulated emission.

1921 – Einstein receives Nobel Prize for photoelectric effect and the light quantum. Third Solvay Conference on atoms and electrons.

1927 – Heisenberg’s uncertainty relation. Fifth Solvay International Conference on Electrons and Photons in Brussels. “First” Bohr-Einstein debate on indeterminancy in quantum theory.

1930 – Sixth Solvay Conference on magnetism. “Second” Bohr-Einstein debate.

1935 – Einstein-Podolsky-Rosen (EPR) paper on the completeness of quantum mechanics.

## Selected Einstein Quantum Papers

Einstein, A. (1905). “Generation and conversion of light with regard to a heuristic point of view.” Annalen Der Physik 17(6): 132-148.

Einstein, A. (1907). “Die Plancksche Theorie der Strahlung und die Theorie der spezifischen W ̈arme.” Annalen der Physik 22: 180–190.

Einstein, A. (1909). “On the current state of radiation problems.” Physikalische Zeitschrift 10: 185-193.

Einstein, A. and O. Stern (1913). “An argument for the acceptance of molecular agitation at absolute zero.” Annalen Der Physik 40(3): 551-560.

Einstein, A. (1916). “Strahlungs-Emission un -Absorption nach der Quantentheorie.” Verh. Deutsch. Phys. Ges. 18: 318.

Einstein, A. (1917). “Quantum theory of radiation.” Physikalische Zeitschrift 18: 121-128.

Einstein, A., B. Podolsky and N. Rosen (1935). “Can quantum-mechanical description of physical reality be considered complete?” Physical Review 47(10): 0777-0780.

## Notes

[1] M. Planck, “Elementary quanta of matter and electricity,” Annalen Der Physik, vol. 4, pp. 564-566, Mar 1901.

[2] Klein, M. J. (1964). Einstein’s First Paper on Quanta. The natural philosopher. D. A. Greenberg and D. E. Gershenson. New York, Blaidsdell. 3.

[3] A. Einstein, “Generation and conversion of light with regard to a heuristic point of view,” Annalen Der Physik, vol. 17, pp. 132-148, Jun 1905.

[4] Chap. 2 in “Mind at Light Speed“, by David Nolte (Free Press, 2001)

[5] Einstein, A. (1916). “Strahlungs-Emission un -Absorption nach der Quantentheorie.” Verh. Deutsch. Phys. Ges. 18: 318.

[6] Einstein, A. (1917). “Quantum theory of radiation.” Physikalische Zeitschrift 18: 121-128.

# Science 1916: Schwarzschild, Einstein, Planck, Born, Frobenius et al.

In one of my previous blog posts, as I was searching for Schwarzschild’s original papers on Einstein’s field equations and quantum theory, I obtained a copy of the January 1916 – June 1916 volume of the Proceedings of the Royal Prussian Academy of Sciences through interlibrary loan.  The extremely thick volume arrived at Purdue about a week after I ordered it online.  It arrived from Oberlin College in Ohio that had received it as a gift in 1928 from the library of Professor Friedrich Loofs of the University of Halle in Germany.  Loofs had been the Haskell Lecturer at Oberlin for the 1911-1912 semesters.

As I browsed through the volume looking for Schwarzschild’s papers, I was amused to find a cornucopia of turn-of-the-century science topics recorded in its pages.  There were papers on the overbite and lips of marsupials.  There were papers on forgotten languages.  There were papers on ancient Greek texts.  On the origins of religion.  On the philosophy of abstraction.  Histories of Indian dramas.  Reflections on cancer.  But what I found most amazing was a snapshot of the field of physics and mathematics in 1916, with historic papers by historic scientists who changed how we view the world. Here is a snapshot in time and in space, a period of only six months from a single journal, containing papers from authors that reads like a who’s who of physics.

In 1916 there were three major centers of science in the world with leading science publications: London with the Philosophical Magazine and Proceedings of the Royal Society; Paris with the Comptes Rendus of the Académie des Sciences; and Berlin with the Proceedings of the Royal Prussian Academy of Sciences and Annalen der Physik. In Russia, there were the scientific Journals of St. Petersburg, but the Bolshevik Revolution was brewing that would overwhelm that country for decades.  And in 1916 the academic life of the United States was barely worth noticing except for a few points of light at Yale and Johns Hopkins.

Berlin in 1916 was embroiled in war, but science proceeded relatively unmolested.  The six-month volume of the Proceedings of the Royal Prussian Academy of Sciences contains a number of gems.  Schwarzschild was one of the most prolific contributors, publishing three papers in just this half-year volume, plus his obituary written by Einstein.  But joining Schwarzschild in this volume were Einstein, Planck, Born, Warburg, Frobenious, and Rubens among others—a pantheon of German scientists mostly cut off from the rest of the world at that time, but single-mindedly following their individual threads woven deep into the fabric of the physical world.

## Karl Schwarzschild (1873 – 1916)

Schwarzschild had the unenviable yet effective motivation of his impending death to spur him to complete several projects that he must have known would make his name immortal.  In this six-month volume he published his three most important papers.  The first (pg. 189) was on the exact solution to Einstein’s field equations to general relativity.  The solution was for the restricted case of a point mass, yet the derivation yielded the Schwarzschild radius that later became known as the event horizon of a non-roatating black hole.  The second paper (pg. 424) expanded the general relativity solutions to a spherically symmetric incompressible liquid mass.

The subject, content and success of these two papers was wholly unexpected from this observational astronomer stationed on the Russian Front during WWI calculating trajectories for German bombardments.  He would not have been considered a theoretical physicist but for the importance of his results and the sophistication of his methods.  Within only a year after Einstein published his general theory, based as it was on the complicated tensor calculus of Levi-Civita, Christoffel and Ricci-Curbastro that had taken him years to master, Schwarzschild found a solution that evaded even Einstein.

Schwarzschild’s third and final paper (pg. 548) was on an entirely different topic, still not in his official field of astronomy, that positioned all future theoretical work in quantum physics to be phrased in the language of Hamiltonian dynamics and phase space.  He proved that action-angle coordinates were the only acceptable canonical coordinates to be used when quantizing dynamical systems.  This paper answered a central question that had been nagging Bohr and Einstein and Ehrenfest for years—how to quantize dynamical coordinates.  Despite the simple way that Bohr’s quantized hydrogen atom is taught in modern physics, there was an ambiguity in the quantization conditions even for this simple single-electron atom.  The ambiguity arose from the numerous possible canonical coordinate transformations that were admissible, yet which led to different forms of quantized motion.

Schwarzschild’s doctoral thesis had been a theoretical topic in astrophysics that applied the celestial mechanics theories of Henri Poincaré to binary star systems.  Within Poincaré’s theory were integral invariants that were conserved quantities of the motion.  When a dynamical system had as many constraints as degrees of freedom, then every coordinate had an integral invariant.  In this unexpected last paper from Schwarzschild, he showed how canonical transformation to action-angle coordinates produced a unique representation in terms of action variables (whose dimensions are the same as Planck’s constant).  These action coordinates, with their associated cyclical angle variables, are the only unambiguous representations that can be quantized.  The important points of this paper were amplified a few months later in a publication by Schwarzschild’s friend Paul Epstein (1871 – 1939), solidifying this approach to quantum mechanics.  Paul Ehrenfest (1880 – 1933) continued this work later in 1916 by defining adiabatic invariants whose quantum numbers remain unchanged under slowly varying conditions, and the program started by Schwarzschild was definitively completed by Paul Dirac (1902 – 1984) at the dawn of quantum mechanics in Göttingen in 1925.

## Albert Einstein (1879 – 1955)

In 1916 Einstein was mopping up after publishing his definitive field equations of general relativity the year before.  His interests were still cast wide, not restricted only to this latest project.  In the 1916 Jan. to June volume of the Prussian Academy Einstein published two papers.  Each is remarkably short relative to the other papers in the volume, yet the importance of the papers may stand in inverse proportion to their length.

The first paper (pg. 184) is placed right before Schwarzschild’s first paper on February 3.  The subject of the paper is the expression of Maxwell’s equations in four-dimensional space time.  It is notable and ironic that Einstein mentions Hermann Minkowski (1864 – 1909) in the first sentence of the paper.  When Minkowski proposed his bold structure of spacetime in 1908, Einstein had been one of his harshest critics, writing letters to the editor about the absurdity of thinking of space and time as a single interchangeable coordinate system.  This is ironic, because Einstein today is perhaps best known for the special relativity properties of spacetime, yet he was slow to adopt the spacetime viewpoint. Einstein only came around to spacetime when he realized around 1910 that a general approach to relativity required the mathematical structure of tensor manifolds, and Minkowski had provided just such a manifold—the pseudo-Riemannian manifold of space time.  Einstein subsequently adopted spacetime with a passion and became its greatest champion, calling out Minkowski where possible to give him his due, although he had already died tragically of a burst appendix in 1909.

The importance of Einstein’s paper hinges on his derivation of the electromagnetic field energy density using electromagnetic four vectors.  The energy density is part of the source term for his general relativity field equations.  Any form of energy density can warp spacetime, including electromagnetic field energy.  Furthermore, the Einstein field equations of general relativity are nonlinear as gravitational fields modify space and space modifies electromagnetic fields, producing a coupling between gravity and electromagnetism.  This coupling is implicit in the case of the bending of light by gravity, but Einstein’s paper from 1916 makes the connection explicit.

Einstein’s second paper (pg. 688) is even shorter and hence one of the most daring publications of his career.  Because the field equations of general relativity are nonlinear, they are not easy to solve exactly, and Einstein was exploring approximate solutions under conditions of slow speeds and weak fields.  In this “non-relativistic” limit the metric tensor separates into a Minkowski metric as a background on which a small metric perturbation remains.  This small perturbation has the properties of a wave equation for a disturbance of the gravitational field that propagates at the speed of light.  Hence, in the June 22 issue of the Prussian Academy in 1916, Einstein predicts the existence and the properties of gravitational waves.  Exactly one hundred years later in 2016, the LIGO collaboration announced the detection of gravitational waves generated by the merger of two black holes.

## Max Planck (1858 – 1947)

Max Planck was active as the secretary of the Prussian Academy in 1916 yet was still fully active in his research.  Although he had launched the quantum revolution with his quantum hypothesis of 1900, he was not a major proponent of quantum theory even as late as 1916.  His primary interests lay in thermodynamics and the origins of entropy, following the theoretical approaches of Ludwig Boltzmann (1844 – 1906).  In 1916 he was interested in how to best partition phase space as a way to count states and calculate entropy from first principles.  His paper in the 1916 volume (pg. 653) calculated the entropy for single-atom solids.

## Max Born (1882 – 1970)

Max Born was to be one of the leading champions of the quantum mechanical revolution based at the University of Göttingen in the 1920’s. But in 1916 he was on leave from the University of Berlin working on ranging for artillery.  Yet he still pursued his academic interests, like Schwarzschild.  On pg. 614 in the Proceedings of the Prussian Academy, Born published a paper on anisotropic liquids, such as liquid crystals and the effect of electric fields on them.  It is astonishing to think that so many of the flat-panel displays we have today, whether on our watches or smart phones, are technological descendants of work by Born at the beginning of his career.

## Ferdinand Frobenius (1849 – 1917)

Like Schwarzschild, Frobenius was at the end of his career in 1916 and would pass away one year later, but unlike Schwarzschild, his career had been a long one, receiving his doctorate under Weierstrass and exploring elliptic functions, differential equations, number theory and group theory.  One of the papers that established him in group theory appears in the May 4th issue on page 542 where he explores the series expansion of a group.

## Heinrich Rubens (1865 – 1922)

Max Planck owed his quantum breakthrough in part to the exquisitely accurate experimental measurements made by Heinrich Rubens on black body radiation.  It was only by the precise shape of what came to be called the Planck spectrum that Planck could say with such confidence that his theory of quantized radiation interactions fit Rubens spectrum so perfectly.  In 1916 Rubens was at the University of Berlin, having taken the position vacated by Paul Drude in 1906.  He was a specialist in infrared spectroscopy, and on page 167 of the Proceedings he describes the spectrum of steam and its consequences for the quantum theory.

## Emil Warburg (1946 – 1931)

Emil Warburg’s fame is primarily as the father of Otto Warburg who won the 1931 Nobel prize in physiology.  On page 314 Warburg reports on photochemical processes in BrH gases.     In an obscure and very indirect way, I am an academic descendant of Emil Warburg.  One of his students was Robert Pohl who was a famous early researcher in solid state physics, sometimes called the “father of solid state physics”.  Pohl was at the physics department in Göttingen in the 1920’s along with Born and Franck during the golden age of quantum mechanics.  Robert Pohl’s son, Robert Otto Pohl, was my professor when I was a sophomore at Cornell University in 1978 for the course on introductory electromagnetism using a textbook by the Nobel laureate Edward Purcell, a quirky volume of the Berkeley Series of physics textbooks.  This makes Emil Warburg my professor’s father’s professor.

## Papers in the 1916 Vol. 1 of the Prussian Academy of Sciences

Schulze,  Alt– und Neuindisches

Orth,  Zur Frage nach den Beziehungen des Alkoholismus zur Tuberkulose

Schulze,  Die Erhabunen auf der Lippin- und Wangenschleimhaut der Säugetiere

von Wilamwitz-Moellendorff, Die Samie des Menandros

Engler,  Bericht über das >>Pflanzenreich<<

von Harnack,  Bericht über die Ausgabe der griechischen Kirchenväter der dri ersten Jahrhunderte

Meinecke,  Germanischer und romanischer Geist im Wandel der deutschen Geschichtsauffassung

Rubens und Hettner,  Das langwellige Wasserdampfspektrum und seine Deutung durch die Quantentheorie

Einstein,  Eine neue formale Deutung der Maxwellschen Feldgleichungen der Electrodynamic

Schwarschild,  Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie

Helmreich,  Handschriftliche Verbesserungen zu dem Hippokratesglossar des Galen

Prager,  Über die Periode des veränderlichen Sterns RR Lyrae

Holl,  Die Zeitfolge des ersten origenistischen Streits

Lüders,  Zu den Upanisads. I. Die Samvargavidya

Warburg,  Über den Energieumsatz bei photochemischen Vorgängen in Gasen. VI.

Hellman,  Über die ägyptischen Witterungsangaben im Kalender von Claudius Ptolemaeus

Meyer-Lübke,  Die Diphthonge im Provenzaslischen

Diels,  Über die Schrift Antipocras des Nikolaus von Polen

Müller und Sieg,  Maitrisimit und >>Tocharisch<<

Meyer,  Ein altirischer Heilsegen

Schwarzschild,  Über das Gravitationasfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie

Brauer,  Die Verbreitung der Hyracoiden

Correns,  Untersuchungen über Geschlechtsbestimmung bei Distelarten

Brahn,  Weitere Untersuchungen über Fermente in der Lever von Krebskranken

Erdmann,  Methodologische Konsequenzen aus der Theorie der Abstraktion

Bang,  Studien zur vergleichenden Grammatik der Türksprachen. I.

Frobenius,  Über die  Kompositionsreihe einer Gruppe

Schwarzschild,  Zur Quantenhypothese

Fischer und Bergmann,  Über neue Galloylderivate des Traubenzuckers und ihren Vergleich mit der Chebulinsäure

Schuchhardt,  Der starke Wall und die breite, zuweilen erhöhte Berme bei frügeschichtlichen Burgen in Norddeutschland

Born,  Über anisotrope Flüssigkeiten

Planck,  Über die absolute Entropie einatomiger Körper

Haberlandt,  Blattepidermis und Lichtperzeption

Einstein,  Näherungsweise Integration der Feldgleichungen der Gravitation

Lüders,  Die Saubhikas.  Ein Beitrag zur Gecschichte des indischen Dramas