Dirac: Quantum Field Theory to Antimatter

Paul Adrian Maurice Dirac (1902 – 1984) was given the moniker of “the strangest man” by Niels Bohr while he was reminiscing about the many great scientists with whom he had worked over the years [1].  It is a moniker that resonates with the innumerable “Dirac stories” that abound in the mythology of the hallways of physics departments around the world.  Dirac was awkward, shy, a loner, rarely said anything, was completely literal, had not the slightest comprehension of art or poetry, nor any clear understanding of human interpersonal interaction.  Dirac was also brilliant, providing the theoretical foundation for the central paradigm of modern physics—quantum field theory.  The discovery of the Higgs boson in 2012, a human achievement that capped nearly a century of scientific endeavor, rests solidly on the theory of quantum fields that permeate space.  The Higgs particle, when it pops into existence at the Large Hadron Collider in Geneva, is a singular quantum excitation of the Higgs field, a field that usually resides in a vacuum state, frothing with quantum fluctuations that imbue all particles—and you and me—with mass.  The Higgs field is Dirac’s legacy.

… all of a sudden he had a new equation with four-dimensional space-time symmetry.

Copenhagen and Bohr

Although Dirac as a young scientist was initially enthralled with relativity theory, he was working under Ralph Fowler (1889 – 1944) in the physics department at Cambridge in 1923 when he had the chance to read advanced proofs of Heisenberg’s matrix mechanics paper.  This chance event launched him on his own trajectory in quantum theory.  After Dirac was awarded his doctorate from Cambridge in 1926, he received a stipend that sent him to work with Niels Bohr (1885 – 1962) in Copenhagen—ground zero of the new physics. During his time there, Dirac became famous for taking long walks across Copenhagen as he played about with things in his mind, performing mental juggling of abstract symbols, envisioning how they would permute and act.  His attention was focused on the electromagnetic field and how it interacted with the quantized states of atoms.  Although the electromagnetic field was the classical field of light, it was also the quantum field of Einstein’s photon, and he wondered how the quantized harmonic oscillators of the electromagnetic field could be generated by quantum wavefunctions acting as operators.  But acting on what?  He decided that, to generate a photon, the wavefunction must operate on a state that had no photons—the ground state of the electromagnetic field known as the vacuum state.

            In late 1926, nearing the end of his stay in Copenhagen with Bohr, Dirac put these thoughts into their appropriate mathematical form and began work on two successive manuscripts.  The first manuscript contained the theoretical details of the non-commuting electromagnetic field operators.  He called the process of generating photons out of the vacuum “second quantization”.  This phrase is a bit of a misnomer, because there is no specific “first quantization” per se, although he was probably thinking of the quantized energy levels of Schrödinger and Heisenberg.  In second quantization, the classical field of electromagnetism is converted to an operator that generates quanta of the associated quantum field out of the vacuum (and also annihilates photons back into the vacuum).  The creation operators can be applied again and again to build up an N-photon state containing N photons that obey Bose-Einstein statistics, as they must, as required by their integer spin, agreeing with Planck’s blackbody radiation. 

            Dirac then went further to show how an interaction of the quantized electromagnetic field with quantized energy levels involved the annihilation and creation of photons as they promoted electrons to higher atomic energy levels, or demoted them through stimulated emission.  Very significantly, Dirac’s new theory explained the spontaneous emission of light from an excited electron level as a direct physical process that creates a photon carrying away the energy as the electron falls to a lower energy level.  Spontaneous emission had been explained first by Einstein more than ten years earlier when he derived the famous A and B coefficients, but Einstein’s arguments were based on the principle of detailed balance, which is a thermodynamic argument.  It is impressive that Einstein’s deep understanding of thermodynamics and statistical mechanics could allow him to derive the necessity of both spontaneous and stimulated emission, but the physical mechanism for these processes was inferred rather than derived. Dirac, in late 1926, had produced the first direct theory of photon exchange with matter.  This was the birth of quantum electrodynamics, known as QED, and the birth of quantum field theory [2].

Fig. 1 Paul Dirac in his early days.

Göttingen and Born

            Dirac’s next stop on his postodctoral fellowship was in Göttingen to work with Max Born (1882 – 1970) and the large group of theoreticians and mathematicians who were like electrons in a cloud orbiting around the nucleus represented by the new quantum theory.  Göttingen was second only to Copenhagen as the Mecca for quantum theorists.  Hilbert was there and von Neumann too, as well as the brash American J. Robert Oppenheimer (1904 – 1967) who was finishing his PhD with Born.  Dirac and Oppenheimer struck up an awkward friendship.  Oppenheimer was considered arrogant by many others in the group, but he was in awe of Dirac who arrived with his manuscript on quantum electrodynamics ready for submission.  Oppenheimer struggled at first to understand Dirac’s new approach to quantizing fields, but he quickly grasped the importance, as did Pascual Jordan (1902 – 1980), who was also in Göttingen.

            Jordan had already worked on ideas very close to Dirac’s on the quantization of fields.  He and Dirac seemed to be going down the same path, independently arriving at very similar conclusions around the same time.  In fact, Jordan was often a step ahead of Dirac, tending to publish just before Dirac, as with non-commuting matrices, transformation theory and the relationship of canonical transformations to second quantization.  However, Dirac’s paper on quantum electrodynamics was a masterpiece in clarity and comprehensiveness, launching a new field in a way that Jordan had not yet achieved with his own work.  But because of the closeness of Jordan’s thinking to Dirac’s, he was able to see immediately how to extend Dirac’s approach.  Within the year, he published a series of papers that established the formalism of quantum electrodynamics as well as quantum field theory.  With Pauli, he systematized the operators for creation and annihilation of photons [3].  With Wigner, he developed second quantization for de Broglie matter waves, defining creation and annihilation operators that obeyed the Pauli exclusion principle of electrons[4].  Jordan was on a roll, forging ahead of Dirac on extensions of quantum electrodynamics and field theory, but Dirac was about to eclipse Jordan once and for all.

St. John’s at Cambridge

            At the end of the Spring semester in 1927, Dirac was offered a position as a fellow of St. John’s College at Cambridge, which he accepted, returning to England to begin his life as a college professor.  During the summer and into the Fall, Dirac returned to his first passion in physics, relativity, which had yet to be successfully incorporated into quantum physics.  Oskar Klein and Walter Gordon had made initial attempts at formulating relativistic quantum theory, but they could not correctly incorporate the spin properties of the electron, and their wave equation had the bad habit of producing negative probabilities.  Probabilities went negative because the Klein-Gordon equation had two time derivatives instead of one.  The reason it had two (while the non-relativistic Schrödinger equation has only one) is because space-time symmetry required the double space derivative of the Schrödinger equation to be paired with a double time derivative.  Dirac, with creative insight, realized that the problem could be flipped by requiring the single time derivative to be paired with a single space derivative.  The problem was that a single space derivative did not seem to make any sense [5].

St. John’s College at Cambridge

            As Dirac puzzled how to get an equation with only single derivatives, he was playing around with Pauli spin matrices and hit on a simple identity that related the spin matrices to the electron momentum.  At first he could not get the identity to apply to four-dimensional relativistic momenta using the usual 2×2 spin matrices.  Then he realized that four-dimensional space-time could be captured if he expanded Pauli’s 2×2 spin matrices to 4×4 spin matrices, and all of a sudden he had a new equation with four-dimensional space-time symmetry with single derivatives on space and time.  As a test of his new equation, he calculated fine details of the experimentally-measured hydrogen spectrum, known as the fine structure, which had resisted theoretical explanation, and he derived answers in close agreement with experiment.  He also showed that the electron had spin-1/2, and he calculated its magnetic moment.  He finished his manuscript at the end of the Fall semester in 1927, and the paper was published in early 1928[6].  His relativistic quantum wave equation was an instant sensation, becoming known for all time as “the Dirac Equation”.  He had succeeded at finding a correct and long-sought relativistic quantum theory where many before had failed.  It was a crowning achievment, placing Dirac firmly in the firmament of the quantum theorists.

Fig. 1 The relativistic Dirac equation. The wavefunction is a four-component spinor. The gamma-del product is a 4×4 matrix operator. The time and space derivatives are both first-order operators.

Antimatter

            In the process of ridding the Klein-Gordon equation of negative probability, which Dirac found abhorent, his new equation created an infinite number of negative energy states, which he did not find abhorent.  It is perhaps a matter of taste what one theoriest is willing to accept over another, and for Dirac, negative energies were better than negative probabilities.  Even so, one needed to deal with an infinite number of negative energy states in quantum theory, because they are available to quantum transitions.  In 1929 and 1930, as Dirac was writing his famous textbook on quantum theory, he became intrigued by the similarity between the positive and negative electron states of the vacuum and the energy levels of valence electrons on atoms.  An electron in a state outside a filled electron shell behaves very much like a single-electron atom, like sodium and lithium with their single valence electrons.  Conversely, an atomic shell that has one electron less than a full complement can be described as having a “hole” that behaves “as if” it were a positive particle.  It is like a bubble in water.  As water sinks, the bubble rises to the top of the water level.  For electrons, if all the electrons go one way in an electric field, then the hole goes the opposite direction, like a positive charge. 

            Dirac took this analogy of nearly-filled atomic shells and applied it to the vacuum states of the electron, viewing the filled negative energy states like the filled electron shells of atoms.  If there is a missing electron, a hole in this infinite sea, then it would behave as if it had positive charge.  Initially, Dirac speculated that the “hole” was the proton, and he even wrote a paper on that possibility.  But Oppenheimer pointed out that the idea was inconsistent with observations, especially the inability of the electron and proton to annihilate, and that the ground state of the infinite electron sea must be completely filled. Hermann Weyl further pointed out that the electron-proton theory did not have the correct symmetry, and Dirac had to rethink.  In early 1931 he hit on an audacious solution to the puzzle.  What if the hole in the infinite negative energy sea did not just behave like a positive particle, but actually was a positive particle, a new particle that Dirac dubbed the “anti-electron”?  The anti-electron would have the same mass as the electron, but would have positive charge. He suggested that such particles might be generated in high-energy collisions in vacuum, and he finished his paper with the suggestion that there also could be an anti-proton with the mass of the proton but with negative charge.  In this singular paper, titled “Quantized Singularities of the Electromagnetic Field” published in 1931, Dirac predicted the existence of antimatter.  A year later the positron was discovered by Carl David Anderson at Cal Tech.  Anderson had originally called the particle the positive electron, but a journal editor of the Physical Review changed it to positron, and the new name stuck.

Fig. 3 An electron-positron pair is created by the absorption of a photon (gamma ray). Positrons have negative energy and can be viewed as a hole in a sea of filled electron states. (Momentum conservation is satisfied if a near-by heavy particle takes up the recoil momentum.)

            The prediction and subsequent experimental validation of antmatter stands out in the history of physics in the 20th Century.  In previous centuries, theory was performed mainly in the service of experiment, explaining interesting new observed phenomena either as consequences of known physics, or creating new physics to explain the observations.  Quantum theory, revolutionary as a way of understanding nature, was developed to explain spectroscopic observations of atoms and molecules and gases.  Similarly, the precession of the perihelion of Mercury was a well-known phenomenon when Einstein used his newly developed general relativity to explain it.  As a counter example, Einstein’s prediction of the deflection of light by the Sun was something new that emerged from theory.  This is one reason why Einstein became so famous after Eddington’s expedition to observe the deflection of apparent star locations during the total eclipse.  Einstein had predicted something that had never been seen before.  Dirac’s prediction of the existence of antimatter similarly is a triumph of rational thought, following the mathematical representation of reality to an inevitable conclusion that cannot be ignored, no matter how wild and initially unimaginable it is.  Dirac went on to receive the Nobel prize in Physics in 1933, sharing the prize that year with Schrödinger (Heisenberg won it the previous year in 1932).


[1] Framelo, “The Strangest Man: The Hidden Life of Paul Dirac” (Basic Books, 2011)

[2] Dirac, P. A. M. (1927). “The quantum theory of the emission and absorption of radiation.” Proceedings of the Royal Society of London Series A114(767): 243-265.;  Dirac, P. A. M. (1927). “The quantum theory of dispersion.” Proceedings of the Royal Society of London Series A114(769): 710-728.

[3] Jordan, P. and W. Pauli, Jr. (1928). “To quantum electrodynamics of free charge fields.” Zeitschrift Fur Physik 47(3-4): 151-173.

[4] Jordan, P. and E. Wigner (1928). “About the Pauli’s equivalence prohibited.” Zeitschrift Fur Physik 47(9-10): 631-651.

[5] This is because two space derivatives measure the curvative of the wavefunction which is related to the kinetic energy of the electron.

[6] Dirac, P. A. M. (1928). “The quantum theory of the electron.” Proceedings of the Royal Society of London Series A 117(778): 610-624.;  Dirac, P. A. M. (1928). “The quantum theory of the electron – Part II.” Proceedings of the Royal Society of London Series A118(779): 351-361.

Physics and the Zen of Motorcycle Maintenance

When I arrived at Berkeley in 1981 to start graduate school in physics, the single action I took that secured my future as a physicist, more than spending scores of sleepless nights studying quantum mechanics by Schiff or electromagnetism by Jackson —was buying a motorcycle!  Why motorcycle maintenance should be the Tao of Physics was beyond me at the time—but Zen is transcendent.

1978_GS550_blk_leftside_520

The Quantum Sadistics

In my first semester of grad school I made two close friends, Keith Swenson and Kent Owen, as we stayed up all night working on impossible problem sets and hand-grading a thousand midterms for an introductory physics class that we were TAs for.  The camaraderie was made tighter when Keith and Kent bought motorcycles and I quickly followed suit, buying my first wheels –– a 1972 Suzuki GT550.    It was an old bike, but in good shape and ready to ride, so the three of us began touring around the San Francisco Bay Area together on weekend rides.  We went out to Mt. Tam, or up to Vallejo, or around the North and South Bay.  Kent thought this was a very cool way for physics grads to spend their time and he came up with a name for our gang –– the “Quantum Sadistics”!  He even made a logo for our “colors” that was an eye shedding a tear drop shaped like the dagger of a quantum raising operator.

At the end of the first year, Keith left the program, not sure he was the right material for a physics degree, and moved to San Diego to head up the software arm of a start-up company that he had founder’s shares in.  Kent and I continued at Berkeley, but soon got too busy to keep up the weekend rides.  My Suzuki was my only set of wheels, so I tooled around with it, keeping it running when it really didn’t want to go any further.  I had to pull its head and dive deep into it to adjust the rockers.  It stayed together enough for a trip all the way down Highway 1 to San Diego to visit Keith and back, and a trip all the way up Highway 1 to Seattle to visit my grandparents and back, having ridden the full length of the Pacific Coast from Tijuana to Vancouver.  Motorcycle maintenance was always part of the process.

Andrew Lange

After a few semesters as a TA for the large lecture courses in physics, it was time to try something real and I noticed a job opening posted on a bulletin board.  It was for a temporary research position in Prof. Paul Richard’s group.  I had TA-ed for him once, but knew nothing of his research, and the interview wasn’t even with him, but with a graduate student named Andrew Lange.  I met with Andrew in a ground-floor lab on the south side of Birge Hall.  He was soft-spoken and congenial, with round architect glasses, fine sandy hair and had about him a hint of something exotic.  He was encouraging in his reactions to my answers.  Then he asked if I had a motorcycle.  I wasn’t sure if he already knew, or whether it was a test of some kind, so I said that I did.  “Do you work on it?”, he asked.  I remember my response.  “Not really,” I said.  In my mind I was no mechanic.  Adjusting the overhead rockers was nothing too difficult.  It wasn’t like I had pulled the pistons.

“It’s important to work on your motorcycle.”

For some reason, he didn’t seem to like my answer.  He probed further.  “Do you change the tires or the oil?”.  I admitted that I did, and on further questioning, he slowly dragged out my story of pulling the head and adjusting the cams.  He seemed to relax, like he had gotten to the bottom of something.  He then gave me some advice, focusing on me with a strange intensity and stressing very carefully, “It’s important to work on your motorcycle.”

I got the job and joined Paul Richards research group.  It was a heady time.  Andrew was designing a rocket-borne far-infrared spectrometer that would launch on a sounding rocket from Nagoya, Japan.  The spectrometer was to make the most detailed measurements ever of the cosmic microwave background (CMB) radiation during a five-minute free fall at the edge of space, before plunging into the Pacific Ocean.  But the spectrometer was missing a set of key optical elements known as far-infrared dichroic beam splitters.  Without these beam splitters, the spectrometer was just a small chunk of machined aluminum.  It became my job to create these beam splitters.  The problem was that no one knew how to do it.  So with Andrew’s help, I scanned the literature, and we settled on a design related to results from the Ulrich group in Germany.

Our spectral range was different than previous cases, so I created a new methodology using small mylar sheets, patterned with photolithography, evaporating thin films of aluminum on both sides of the mylar.  My first photomasks were made using an amazingly archaic technology known as rubylith that had been used in the 70’s to fabricate low-level integrated circuits.  Andrew showed me how to cut the fine strips of red plastic tape at a large scale that was then photo-reduced for contract printing.  I modeled the beam splitters with equivalent circuits to predict the bandpass spectra, and learned about Kramers-Kronig transforms to explain an additional phase shift that appeared in the interferometric tests of the devices.  These were among the first metamaterials ever created (although this was before that word existed), with an engineered magnetic response for millimeter waves.  I fabricated the devices in the silicon fab on the top floor of the electrical engineering building on the Berkeley campus.  It was one of the first university-based VLSI fabs in the country, with high-class clean rooms and us in bunny suits.  But I was doing everything but silicon, modifying all their carefully controlled processes in the photolithography bay.  I made and characterized a full set of 5 of these high-tech beam splitters–right before I was ejected from the lab and banned.  My processes were incompatible with the VLSI activities of the rest of the students.  Fortunately, I had completed the devices, with a little extra material to spare.

I rode my motorcycle with Andrew and his friends around the Bay Area and up to Napa and the wine country.  One memorable weekend Paul had all his grad students come up to his property in Mendocino County to log trees.  Of course, we rode up on our bikes.  Paul’s land was high on a coastal mountain next to the small winery owned by Charles Kittel (the famous Kittel of “Solid State Physics”).  The weekend was rustic.  The long-abandoned hippie-shack on the property was uninhabitable so we roughed it.  After two days of hauling and stacking logs, I took a long way home riding along dark roads under tall redwoods.

Andrew moved his operation to the University of Nagoya, Japan, six months before the launch date.  The spectrometer checked out perfectly.  As launch day approached, it was mounted into the nose cone of the sounding rocket, continuing to pass all calibration tests.  On the day of launch, we held our breath back in Berkeley.  There was a 12 hour time difference, then we received the report.  The launch was textbook perfect, but at the critical moment when the explosive nose-cone bolts were supposed to blow, they failed.  The cone stayed firmly in place, and the spectrometer telemetered back perfect measurements of the inside of the rocket all the way down until it crashed into the Pacific, and the last 9 months of my life sank into the depths of the Marianas Trench.  I read the writing on the thin aluminum wall, and the following week I was interviewing for a new job up at Lawrence Berkeley Laboratory, the DOE national lab high on the hill overlooking the Berkeley campus.

Eugene Haller

The  instrument I used in Paul Richard’s lab to characterize my state-of-the-art dichroic beamsplitters was a far-infrared Fourier-transform spectrometer that Paul had built using a section of 1-foot-diameter glass sewer pipe.  Bob McMurray, a graduate student working with Prof. Eugene Haller on the hill, was a routine user of this makeshift spectrometer, and I had been looking over Bob’s shoulder at the interesting data he was taking on shallow defect centers in semiconductors.   The work sounded fascinating, and as Andrew’s Japanese sounding rocket settled deeper into the ocean floor, I arranged to meet with Eugene Haller in his office at LBL.

I was always clueless about interviews.  I never thought about them ahead of time, and never knew what I needed to say.  On the other hand, I always had a clear idea of what I wanted to accomplish.  I think this gave me a certain solid confidence that may have come through.  So I had no idea what Eugene was getting at as we began the discussion.  He asked me some questions about my project with Paul, which I am sure I answered with lots of details about Kramers-Kronig and the like.  Then came the question strangely reminiscent of when I first met Andrew Lange:  Did I work on my car?  Actually, I didn’t have a car, I had a motorcycle, and said so.  Well then, did I work on my motorcycle?  He had that same strange intensity that Andrew had when he asked me roughly the same question.  He looked like a prosecuting attorney waiting for the suspect to incriminate himself.  Once again, I described pulling the head and adjusting the rockers and cams.

Eugene leaned back in his chair and relaxed.  He began talking in the future tense about the project I would be working on.  It was a new project for the new Center for Advanced Materials at LBL, for which he was the new director.  The science revolved around semiconductors and especially a promising new material known as GaAs.  He never actually said I had the job … all of a sudden it just seemed to be assumed.  When the interview was over, he simply asked me to give him an answer in a few days if I would come up and join his group.

I didn’t know it at the time, by Eugene had a beautiful vintage Talbot roadster that was his baby.  One of his loves was working on his car.  He was a real motor head and knew everything about the mechanics.  He was also an avid short-wave radio enthusiast and knew as much about vacuum tubes as he did about transistors.  Working on cars (or motorcycles) was a guaranteed ticket into his group.  At a recent gathering of his former students and colleagues for his memorial, similar stories circulated about that question:  Did you work on your car?  The answer to this one question mattered more than any answer you gave about physics.

I joined Eugene Haller’s research group at LBL in March of 1984 and received my PhD on topics of semiconductor physics in 1988.  My association with his group opened the door to a post-doc position at AT&T Bell Labs and then to a faculty position at Purdue University where I currently work on the physics of oncology in medicine and have launched two biotech companies—all triggered by the simple purchase of a motorcycle.

Andrew Lange’s career was particularly stellar.  He joined the faculty of Cal Tech, and I was amazed to read in Science magazine in 2004 or 2005, in a section called “Nobel Watch”, that he was a candidate for the Nobel Prize for his work on BoomerAng that had launched and monitored a high-altitude balloon as it circled the South Pole taking unprecedented data on the CMB that constrained the amount of dark matter in the universe.  Around that same time I invited Paul Richards to Purdue to give our weekly physics colloquium to talk about his own work on MAXIMA. There was definitely a buzz going around that the BoomerAng and MAXIMA collaborations were being talked about in Nobel circles. The next year, the Nobel Prize of 2006 was indeed awarded for work on the Cosmic Microwave Background, but to Mather and Smoot for their earlier work on the COBE satellite.

Then, in January 2010, I was shocked to read in the New York Times that Andrew, that vibrant sharp-eyed brilliant physicist, was found lifeless in a hotel room, dead from asphyxiation.  The police ruled it a suicide.  Apparently few had known of his life-long struggle with depression, and it had finally overwhelmed him.  Perhaps he had sold his motorcycle by then.  But I wonder—if he had pulled out his wrenches and gotten to work on its engine, whether he might have been enveloped by the zen of motorcycle maintenance and the crisis would have passed him by.  As Andrew had told me so many years ago, and I wish I could have reminded him, “It’s important to work on your motorcycle.”

Physicists in Revolution: 1848

The opening episode of Victoria on Masterpiece Theatre (PBS) this season finds the queen confronting widespread unrest among her subjects who are pressing for more freedoms and more say in government. Louis-Phillipe, former King of France, has been deposed in the February Revolution of 1848 in Paris and his presence at the Royal Palace does not help the situation.

In 1848 a wave of spontaneous revolution swept across Europe.  It was not a single revolution of many parts, but many separate revolutions with similar goals.  Two essential disruptions of life occurred in the early 1800’s.  The first was the partitioning of Europe at the Congress of Vienna from 1814 to 1815, presided over by Prince Metternich of Austria, that had carved up Napoleon’s conquests and sought to establish a stable order based on the old ideal of absolute monarchy.  In the process, nationalities were separated or suppressed.  The second was the industrialization of Europe in the early 1800’s that created economic upheaval, with masses of working poor fleeing effective serfdom in the fields and flocking to the cities.  Wages fell, food became scarce, legions of the poor and starving bloomed.  Because of these influences, European society had become unstable, supercooled beyond a phase transition and waiting for a seed or catalyst to crystalize the continent into a new state of matter. 

When the wave came, physicists across Europe were caught in the upheaval.  Some were caught up in the fervor and turned their attention to national service, some lost their standing and their positions during the inevitable reactionary backlash, others got the opportunities of their careers.  It was difficult for anyone to be untouched by the 1848 revolutions, and physicist were no exception.

The Spontaneous Fire of Revolution

The extraodinary wave of revolution was sparked by a small rebellion in Sicily in January 1848 that sought to overturn the ruling Bourbons.  It was a small rebellion of little direct consequence to Europe, but it succeeded in establishing a liberal democracy in an independent state that stood as a symbol of what could be achieved by a determined populace.  The people of Paris took notice, and in the sudden and unanticipated February Revolution, the French constitutional monarchy under Louis-Phillipe was overthrown in a few days and replaced by the French Second Republic.  The shock of Louis-Phillipe’s fall reverberated across Europe, feared by those in power and welcomed by those who sought a new world order.  Nationalism, liberalism, socialism and communism were on the rise, and the opportunity to change the world seemed to have arrived.  The Five Days of Milan in Italy, the March Revolution of the German states, the Polish rebellion against Prussia, and the Young Irelander Rebellion in Ireland were all consequences of the unstable conditions and the unprecidented opportunities for the people to enact change.  None of these uprisings were coordinated by any central group.  It was a spontaneous consequence of similar preconditions that existed across nearly all the states of Europe.

Arago and the February Revolution in Paris

The French were no newcomers to street rebellions.  Paris had a history of armed conflict between citizens manning barricades and the superior forces of the powers at be.  The unforgettable scene in Les Misérables of Marius at the barricade and Jean Valjean’s rescue through the sewers of Paris was based on the 1832 June Rebellion in Paris.  Yet this event was merely an echo of the much larger rebellion of 1830 that had toppled the unpopular monarchy of Charles X, followed by the ascension of the Bourgeois Monarch Louis Phillipe at the start of the July Monarchy.  Eighteen years later, Louis Phillipe was still on the throne and the masses were ready again for a change.  Alexis de Tocqueville saw the change coming and remarked, “We are sleeping together in a volcano. … A wind of revolution blows, the storm is on the horizon.”  The storm would sweep up a generation of participants, including the French physicist Francois Arago (1786 – 1853).

Lamartine in front of the Town Hall of Paris on 25 February 1848 (Image by Henri Félix Emmanuel Philippoteaux in public domain).

Arago is one of the under-appreciated French physicists of the 1800’s.  This may be because so many of his peers have become icons in the history of physics: Fourier, Fresnel, Poisson, Laplace, Malus, Biot and Foucault.  The one place where his name appears—the Spot of Arago—was not exclusively his discovery, but rather was an experimental demonstration of an effect derived by Poisson using Fresnel’s new theory of diffraction.  Poisson derived the phenomenon as a means to show the absurdity of Fresnel’s undulatory theory of light, but Arago’s experimental demonstration turned the tables on Poisson and the emissionists (followers of Newton’s particulate theory of light).  Yet Arago played a role behind the scenes as a supporter and motivator of some of the most important discoveries in optics.  In particular, it was Arago’s encouragement and support of the (at that time) unknown Fresnel, that helped establish the Fresnel theory of diffraction and the wave nature of light.  Together, Arago and Fresnel established the transverse nature of the light wave, and Arago is also the little-known discoverer of optical rotation.  As a young scientist, he attempted to measure the drift of the ether, which was a null experiment that foreshadowed the epochal experiments of Michelson and Morley 80 years later.  In his later years, Arago proposed the methodology for measuring the speed of light in both stationary and moving materials, which became the basis for the important measurements of the speed of light by Fizeau and Foucault (who also attempted to measure ether drift).

In addition to his duties as the director of the National Observatory and as the perpetual secretary of the Academie des Sciences (replacing Fourier), he entered politics in 1830 when he was elected as a member of the chamber of deputies.  At the fall of Louis-Phillipe in the February Revolution of 1848, he was appointed as a member of the steering committee of the newly formed government of the French Second Republic, and he was named head of the Marine and Colonies as well as the head of the Department of War.  Although he was a staunch republican and supporter of the people, his position put him in direct conflict with the later stages of the revolutions of 1848. 

The population of Paris became disenchanted with the conservative trends in the Second Republic.  In June of 1848 barricades were again erected in the streets of Paris, this time in opposition to the Republic.  Forces were drawn up on both sides, although many of the Republican forces defected to the insurgents, and attempts were made to mediate the conflict.  At the barricade on the rue Soufflot near the Pantheon, Arago himself approached the barricades to implore defenders to disperse.  It is a measure of the respect Arago held with the people when they replied, “Monsieur Arago, we are full of respect for you, but you have no right to reproach us.  You have never been hungry.  You don’t know what poverty is.” [1] When Arago finally withdrew, he feared that death and carnage were inevitable.  They came at noon on June 23 when the barricade at Porte Saint-Denis was attacked by the National Guards.  This started a general onslaught of all the barricades by Republican forces that left 1,500 workers dead in the streets and more than 11,000 arrested.  Arago resigned from the steering committee on June 24, although he continued to work in the government until the coup d’Etat by Louis Napolean, the nephew of Napoleon Bonaparte, in 1852 when he became Napoleon III, Emperor of the Second French Empire. Louis Napoleon demanded that all government workers take an oath of allegiance to him, but Arago refused.  Yet such was the respect that Arago commanded that Louis Napoleon let him continue unmolested as the astronomer of the Bureau des Longitudes.

Riemann and Jacobi and the March Revolution in Berlin

The February Revolution of Paris was followed a month later by the March Revolutions of the German States.  The center of the German-speaking world at that time was Vienna, and a demonstration by students broke out in Vienna on March 13. Emperor Ferdinand, following the advice of Metternich, called out the army who fired on the crowd, killing several protestors.  Throngs rallied to the protest and arms were distributed, readying for a fight.  Rather than risk unreserved bloodshed, the emperor dismissed Metternich who went into exile to London (following closely the footsteps of the French Louis-Phillipe).  Within the week, the revolutionary fervor had spread to Berlin where a student uprising marched on the royal palace of King Frederick Wilhelm IV on March 18.  They were met by 20,000 troops. 

The March 1848 revolution in Berlin (Image in the public domain).

Not all university students were liberals and revolutionaries, and there were numerous student groups that formed to support the King.  One of the students in one of these loyalist groups was a shy mathematician who joined a loyalist student militia to protect the King.  Bernhard Riemann (1826 – 1866) had come to the University of Berlin after spending a short time in the Mathematics department at the University in Göttingen.  Despite the presence of Gauss there, the mathematics department was not considered strong (this would change dramatically in about 50 years when Göttingen became the center of German mathematics with the arrival of Felix Klein, Karl Schwarzschild and Hermann Minkowski).  At Berlin, Riemann attended lectures by Steiner, Jacobi, Dirichlet and Eisenstein. 

On the night of the uprising, a nervous Riemann found himself among a group of students, few more than 20 years old, guarding the quarters of the King, not knowing what would unfold.  They spent a sleepless night that dawned on the chaos and carnage at the barricades at Alexander Platz with hundreds of citizens dead.  King Wilhelm was caught off guard by the events, and he assured the citizens that he would reorganize the government and yield to the demonstrator’s demands for parliamentary elections, a constitution, and freedom of the press.  Two days later the king attended a mass funeral for the fallen, attended by his generals and ministers who wore the german revolutionary tricolor of black, red and gold.  This ploy worked, and the unrest in Berlin died away before the king was forced to abdicate.  This must have relieved Riemann immensely, because this entire episode was entirely outside his usual meek and mild character.  Yet the character of all the unrelated 1848 revolutions had one thing in common: a sharp division among the populace between the liberals and the conservatives.  As Riemann had elected to join with the loyalists, one of his professors picked the other side.

Carl Gustav Jacob Jacobi (1804 – 1851) had been born in Potsdam and had obtained his first faculty position at the University of Königsberg where he was soon ranked among the top mathematicians in Europe.  However, in his early thirties he was stricken with diabetes, and the harsh winters of Königsberg became to difficult to bear.  He returned to the milder climate of Berlin to a faculty position at the university when the wave of revolution swept over the city.  Jacobi was a liberal thinker and was caught up in the movement, attending meetings at the Constitution Club.  Once the danger to Wilhelm IV had passed, the reactionary forces took their revenge, and Jacobi’s teaching stipend was suspended.  When he threatened to move to the University of Vienna, the royalists relented, so Jacobi too was able to weather the storm. 

The surprising footnote to this story is that Jacobi delivered lectures on a course on the application of differential equations to mechanics in the winter semester of 1847 – 1848 right in the midst of the political turmoil.  His participation in the extraordinary political events of that time apparently did not hamper him from giving one of the most extraordinary sets of lectures in mathematical physics.  Jacobi’s lectures of 1848 were the greatest advance in mathematical physics since Euler had reinterpreted Newton a hundred years earlier.  This is where Jacobi expanded on the work of Hamilton, establishing what is today called the Hamilton-Jacobi theory of dynamics.  He also derived and proved, using Liouville’s theorem of 1838, that the volume of phase space was an invariant in a conservative dynamical system [2].  It is tempting to imagine Jacobi returning home late at night, after rousing discussions of revolution at the Constitution Club, to set to work on his own revolutionary theories in physics.

Doppler and the Hungarian Revolution

Among all the states of Europe, the revolutions of 1848 posed the greatest threat to the Austrian Empire, which was a beaurocratic state entangling scores of diverse nationalities sprawled across the largest state of Europe.  The Austrian Empire was the remnant of the Holy Roman Empire that had succumbed to the Napoleonic invasion.  The lands that were controlled by Austria, after Metternich engineered the Congress of Vienna, included Poles, Ukranians, Romanians, Germans, Czechs, Slovaks, Hungarians, Slovenes, Serbs, Albanians and more.  Holding this diverse array of peoples together was already a challenge, and the revolutions of 1848 carried with them strong feelings of nationalism.  The revolutions spreading across Europe were the perfect catalyst to set off the Hungarian Revolution that grew into a war for independence, and the fierce fighting across Hungary could not be avoided even by cloistered physicists.

Christian Doppler (1803 – 1853) had moved in 1847 from Prague (where he had proposed what came to be called the Doppler effect in 1842 to the Royal Bohemian Society of Sciences) to the Academy of Mines and Forests in Schemnitz (modern Banská Štiavnica in Slovakia, but then part of the Kingdom of Hungary) with more pay and less work.  His health had been failing, and the strenuous duties at Prague had taken their toll.  If the goal of this move to an obscure school far from the center of Austrian power had been to lead a peaceful life, Doppler’s plans were sorely upset.

The news of the protests in Vienna arrived in Schemnitz on the 17th of March, and student demonstrations commenced immediately.  Amidst the uncertainty, Doppler requested a leave of absence from the summer semester and returned to Vienna.  It is not clear why he went there, whether to be near the center of excitement, or to take advantage of the free time to pursue his own researches.  While in Vienna he read a treatise before the Academy on galvano-electric effects.  He returned to Schemnitz in the Fall to relative peace, until the 12th of December, when the Hungarians rejected to acknowledge the new Emperor Franz Josef in Vienna, replacing his Uncle Ferdinand who was forced to abdicate, and the Hungarian war for independence began.

Görgey’s troops crossing the Sturec pass. Their ability to evade the Austrian pursuit was legendary (Image by Keiss Károly in the public domain).

One of Doppler’s former students from his days in Prague was appointed to command the newly formed Hungarian army.  General Arthur Görgey (1818 – 1916) moved to take possession of the northern mining towns (present day Slovakia) and occupied Schemnitz.  When Görgey learned that his old teacher was in the town he sent word to Doppler to meet him at his headquarters.  Meeting with a revolutionary and rebel could have marked Doppler as a traitor in Vienna, but he decided to meet him anyway, taking along one of his colleagues as a “witness” that the discussion were purely academic.  This meeting opens an interesting unsolved question in the history of physics. 

Around this time Doppler was interested in the dynamical properties of the pendulum for cases when the suspension wire was exceptionally long.  Experiments on such extreme pendula could provide insight into changes in gravity with height as well as the effects of the motion of the Earth.  For instance, Coriolis had published his paper on forces in rotating frames many years earlier in 1835.  Because Schemnitz was a mining town, there was ample access to deep mine shafts in which to set up a pendulum with a very long wire.  This is where the story becomes murky.  Within the family of Doppler’s descendants there are stories of Doppler setting up such an experiment, and even a night time visit to the Doppler house by Görgey.  The pendulum was thought to be one of the topics discussed by Doppler and Görgey at their first meeting, and Görgey (from his life as a scientist prior to becoming a revolution general) had arrived to help with the experiment [3]

This story is significant for two reasons.  First, it would be astounding to think of General Görgey taking a break from the revolution to do some physics for fun.  Görgey has not been graced by history with a benevolent reputation.  He was known as a hard and sometimes vicious leader, and towards the end of the short-lived Hungarian Revolution he displaced the President Kossuth to become the dictator of Hungary.  The second reason, which is important for the history of physics, is that if Doppler had performed this experiment in 1848, it would have preceded the famous experiment by Foucault by more than two years.  However, the paper published by Doppler around this time on the dynamics of the pendulum did not mention the experiment, and it remains an open question in the history of physics whether Doppler may have had priority over Foucault.

The Austrian Imperial Army laid siege to Schemnitz and commenced a short bombardment that displaced Görgey and his troops from the town.  Even as Schemnitz was being liberated, a letter arrived informing Doppler that his old mentor Stampfer at the University of Vienna was retiring and that he had been chosen to be his replacement.  The March Revolution had led to the abdication of the previous Austrian emperor and his replacement by the more liberal-minded Franz Josef who was interested in restructuring the educational system in the Austrian empire.  On the advice of Doppler’s supporters who were in the new government, the Institute of Physics was formed and Doppler was named as its first director.  He arrived in the spring of 1850 to take up his new post.

The Legacy of 1848

Despite the early successes and optimism of the revolutions of 1848, reactionary forces were quick to reverse many of the advances made for universal suffrage, constitutional government, freedom of the press, and freedom of expression.  In most cases, monarchs either retained power or soon returned.  Even the reviled Metternich returned to Vienna from exile in London in 1851.  Yet as is so often the case, once a door has been opened it is difficult to shut it again.  The pressure for reforms continued long after the revolutions faded away, and by 1870 many of the specific demands of the people had been instituted by most of the European states.  Russia was an exception, which may explain why the inevitable Russian Revolution half a century later was so severe.            

The revolutions of 1848 cannot be said to have had a long-lasting impact on the progress of physics, although they certainly had a direct impact on the lives of selected physicists.  The most lasting effect of the revolutions on science was the restructuring of educational systems, not only in Austria, but in many of the European states.  This was perhaps one of the first times when the social and economic benefits of science education to the national welfare was understood and implemented across Europe, although a similar recognition had occurred earlier during the French Revolution, for instance leading to the founding of the Ecole Polytechnique.  The most important, though subtle, effect of the revolutions of 1848 on society was the shift away from autocratic rule to democracy, and the freeing of expression and thought from rigid bounds.  The coming revolution in physics at the turn of the next century may have been helped a little by the revolutionary spirit that still echoed from 1848.


[1] pg. 201, Mike Rapport, “1848: Year of Revolution” (Basic Books, 2008)

[2] D. D. Nolte, The Tangled Tale of Phase Space, Chap. 6 in Galileo Unbound (Oxford University Press, 2018)

[3] Schuster, P. Moving the stars : Christian Doppler, his life, his works and principle, and the world after. Pöllauberg, Austria, Living Edition. (2005)



Chandrasekhar’s Limit

Arthur Eddington was the complete package—an observationalist with the mathematical and theoretical skills to understand Einstein’s general theory, and the ability to construct the theory of the internal structure of stars.  He was Zeus in Olympus among astrophysicists.  He always had the last word, and he stood with Einstein firmly opposed to the Schwarzschild singularity.  In 1924 he published a theoretical paper in which he derived a new coordinate frame (now known as Eddington-Finkelstein coordinates) in which the singularity at the Schwarzschild radius is removed.  At the time, he took this to mean that the singularity did not exist and that gravitational cut off was not possible [1].  It would seem that the possibility of dark stars (black holes) had been put to rest.  Both Eddington and Einstein said so!  But just as they were writing the obituary of black holes, a strange new form of matter was emerging from astronomical observations that would challenge the views of these giants.

Something wonderful, but also a little scary, happened when Chandrasekhar included the relativistic effects in his calculation.

White Dwarf

Binary star systems have always held a certain fascination for astronomers.  If your field of study is the (mostly) immutable stars, then the stars that do move provide some excitement.  The attraction of binaries is the same thing that makes them important astrophysically—they are dynamic.  While many double stars are observed in the night sky (a few had been noted by Galileo), some of these are just coincidental alignments of near and far stars.  However, William Herschel began cataloging binary stars in 1779 and became convinced in 1802 that at least some of them must be gravitationally bound to each other.  He carefully measured the positions of binary stars over many years and confirmed that these stars showed relative changes in position, proving that they were gravitational bound binary star systems [2].  The first orbit of a binary star was computed in 1827 by Félix Savary for the orbit of Xi Ursae Majoris.  Finding the orbit of a binary star system provides a treasure trove of useful information about the pair of stars.  Not only can the masses of the stars be determined, but their radii and densities also can be estimated.  Furthermore, by combining this information with the distance to the binaries, it was possible to develop a relationship between mass and luminosity for all stars, even single stars.  Therefore, binaries became a form of measuring stick for crucial stellar properties.

Comparison of Earth to a white dwarf star with a mass equal to the Sun. They have comparable radii but radically different densities.

One of the binary star systems that Hershel discovered was the pair known as 40 Eridani B/C, which he observed on January 31 in 1783.  Of this pair, 40 Eridani B was very dim compared to its companion.  More than a century later, in 1910 when spectrographs were first being used routinely on large telescopes, the spectrum of 40 Eridani B was found to be of an unusual white spectral class.  In the same year, the low luminosity companion of Sirius, known as Sirius B, which shared the same unusual white spectral class, was evaluated in terms of its size and mass and was found to be exceptionally small and dense [3].  In fact, it was too small and too dense to be believed at first, because the densities were beyond any known or even conceivable matter.  The mass of Sirius B is around the mass of the Sun, but its radius is comparable to the radius of the Earth, making the density of the white star about ten thousand times denser than the core of the Sun.  Eddington at first felt the same way about white dwarfs that he felt about black holes, but he was eventually swayed by the astrophysical evidence.  By 1922 many of these small white stars had been discovered, called white dwarfs, and their incredibly large densities had been firmly established.  In his famous book on stellar structure [4], he noted the strange paradox:  As a star cools, its pressure must decrease, as all gases must do as they cool, and the star would shrink, yet the pressure required to balance the force of gravity to stabilize the star against continued shrinkage must increase as the star gets smaller.  How can pressure decrease and yet increase at the same time?  In 1926, on the eve of the birth of quantum mechanics, Eddington could conceive of no mechanism that could resolve this paradox.  So he noted it as an open problem in his book and sent it to press.

Subrahmanyan Chandrasekhar

Three years after the publication of Eddington’s book, an eager and excited nineteen-year-old graduate of the University in Madras India boarded a steamer bound for England.  Subrahmanyan Chandrasekhar (1910—1995) had been accepted for graduate studies at Cambridge University.  The voyage in 1930 took eighteen days via the Suez Canal, and he needed something to do to pass the time.  He had with him Eddington’s book, which he carried like a bible, and he also had a copy of a breakthrough article written by R. H. Fowler that applied the new theory of quantum mechanics to the problem of dense matter composed of ions and electrons [5].  Fowler showed how the Pauli exclusion principle for electrons, that obeyed Fermi-Dirac statistics, created an energetic sea of electrons in their lowest energy state, called electron degeneracy.  This degeneracy was a fundamental quantum property of matter, and carried with it an intrinsic pressure unrelated to thermal properties.  Chandrasekhar realized that this was a pressure mechanism that could balance the force of gravity in a cooling star and might resolve Eddington’s paradox of the white dwarfs.  As the steamer moved ever closer to England, Chandrasekhar derived the new balance between gravitational pressure and electron degeneracy pressure and found the radius of the white dwarf as a function of its mass.  The critical step in Chandrasekhar’s theory, conceived alone on the steamer at sea with access to just a handful of books and papers, was the inclusion of special relativity with the quantum physics.  This was necessary, because the densities were so high and the electrons were so energetic, that they attained speeds approaching the speed of light. 

Something wonderful, but also a little scary, happened when Chandrasekhar included the relativistic effects in his calculation.  He discovered that electron degeneracy pressure could balance the force of gravity if the mass of the white dwarf were smaller than about 1.4 times the mass of the Sun.  But if the dwarf was more massive than this, then even the electron degeneracy pressure would be insufficient to fight gravity, and the star would continue to collapse.  To what?  Schwarzschild’s singularity was one possibility.  Chandrasekhar wrote up two papers on his calculations, and when he arrived in England, he showed them to Fowler, who was to be his advisor at Cambridge.  Fowler was genuinely enthusiastic about  the first paper, on the derivation of the relativistic electron degeneracy pressure, and it was submitted for publication.  The second paper, on the maximum sustainable mass for a white dwarf, which reared the ugly head of Schwarzschild’s singularity, made Fowler uncomfortable, and he sat on the paper, unwilling to give his approval for publication in the leading British astrophysical journal.  Chandrasekhar grew annoyed, and in frustration sent it, without Fowler’s approval, to an American journal, where “The Maximum Mass of Ideal White Dwarfs” was published in 1931 [6].  This paper, written in eighteen days on a steamer at sea, established what became known as the Chandrasekhar limit, for which Chandrasekhar would win the 1983 Nobel Prize in Physics, but not before he was forced to fight major battles for its acceptance.

The Chandrasekhar limit expressed in terms of the Planck Mass and the mass of a proton. The limit is approximately 1.4 times the mass of the Sun. White dwarfs with masses larger than the limit cannot balance gravitational collapse by relativistic electron degeneracy.

Chandrasekhar versus Eddington

Initially there was almost no response to Chandrasekhar’s paper.  Frankly, few astronomers had the theoretical training needed to understand the physics.  Eddington was one exception, which was why he held such stature in the community.  The big question therefore was:  Was Chandrasekhar’s theory correct?  During the three years to obtain his PhD, Chandrasekhar met frequently with Eddington, who was also at Cambridge, and with colleagues outside the university, and they all encouraged Chandrasekhar to tackle the more difficult problem to combine internal stellar structure with his theory.  This could not be done with pen and paper, but required numerical calculation.  Eddington was in possession of an early electromagnetic calculator, and he loaned it to Chandrasekhar to do the calculations.  After many months of tedious work, Chandrasekhar was finally ready to confirm his theory at the 1934 meeting of the British Astrophysical Society. 

The young Chandrasekhar stood up and gave his results in an impeccable presentation before an auditorium crowded with his peers.  But as he left the stage, he was shocked when Eddington himself rose to give the next presentation.  Eddington proceeded to criticize and reject Chandrasekhar’s careful work, proposing instead a garbled mash-up of quantum theory and relativity that would eliminate Chandrasekhar’s limit and hence prevent collapse to the Schwarzschild singularity.  Chandrasekhar sat mortified in the audience.  After the session, many of his friends and colleagues came up to him to give their condolences—if Eddington, the leader of the field and one of the few astronomers who understood Einstein’s theories, said that Chandrasekhar was wrong, then that was that.  Badly wounded, Chandrasekhar was faced with a dire choice.  Should he fight against the reputation of Eddington, fight for the truth of his theory?  But he was at the beginning of his career and could ill afford to pit himself against the giant.  So he turned his back on the problem of stellar death, and applied his talents to the problem of stellar evolution. 

Chandrasekhar went on to have an illustrious career, spent mostly at the University of Chicago (far from Cambridge), and he did eventually return to his limit as it became clear that Eddington was wrong.  In fact, many at the time already suspected Eddington was wrong and were seeking for the answer to the next question: If white dwarfs cannot support themselves under gravity and must collapse, what do they collapse to?  In Pasadena at the California Institute of Technology, an astrophysicist named Fritz Zwicky thought he knew the answer.

Fritz Zwicky’s Neutron Star

Fritz Zwicky (1898—1874) was an irritating and badly flawed genius.  What made him so irritating was that he knew he was a genius and never let anyone forget it.  What made him badly flawed was that he never cared much for weight of evidence.  It was the ideas that mattered—let lesser minds do the tedious work of filling in the cracks.  And what made him a genius was that he was often right!  Zwicky pushed the envelope—he loved extremes.  The more extreme a theory was, the more likely he was to favor it—like his proposal for dark matter.  Most of his colleagues considered him to be a buffoon and borderline crackpot.  He was tolerated by no one—no one except his steadfast collaborator of many years Ernst Baade (until they nearly came to blows on the eve of World War II).  Baade was a German physicist trained at Göttingen and recently arrived at Cal Tech.  He was exceptionally well informed on the latest advances in a broad range of fields.  Where Zwicky made intuitive leaps, often unsupported by evidence, Baade would provide the context.  Baade was a walking Wikipedia for Zwicky, and together they changed the face of astrophysics.

Zwicky and Baade submitted an abstract to the American Physical Society Meeting in 1933, which Kip Thorne has called “…one of the most prescient documents in the history of physics and astronomy” [7].  In the abstract, Zwicky and Baade introduced, for the first time, the existence of supernovae as a separate class of nova and estimated the total energy output of these cataclysmic events, including the possibility that they are the source of some cosmic rays.  They introduced the idea of a neutron star, a star composed purely of neutrons, only a year after Chadwick discovered the neutron’s existence, and they strongly suggested that a supernova is produced by the transformation of a star into a neutron star.  A neutron star would have a mass similar to that of the Sun, but would have a radius of only tens of kilometers.  If the mass density of white dwarfs was hard to swallow, the density of a neutron star was billion times greater!  It would take nearly thirty years before each of the assertions made in this short abstract were proven true, but Zwicky certainly had a clear view, tempered by Baade, of where the field of astrophysics was headed.  But no one listened to Zwicky.  He was too aggressive and backed up his wild assertions with too little substance.  Therefore, neutron stars simmered on the back burner until more substantial physicists could address their properties more seriously.

Two substantial physicists who had the talent and skills that Zwicky lacked were Lev Landau in Moscow and Robert Oppenheimer at Berkeley.  Landau derived the properties of a neutron star in 1937 and published the results to great fanfare.  He was not aware of Zwicky’s work, and he called them neutron cores, because he hypothesized that they might reside at the core of ordinary stars like the Sun.  Oppenheimer, working with a Canadian graduate student George Volkoff at Berkeley, showed that Landau’s idea about stellar cores was not correct, but that the general idea of a neutron core, or rather neutron star, was correct [8].  Once Oppenheimer was interested in neutron stars, he kept going and asked the same question about neutron stars that Chandrasekhar had asked about white dwarfs:  Is there a maximum size for neutron stars beyond which they must collapse?  The answer to this question used the same quantum mechanical degeneracy pressure (now provided by neutrons rather than electrons) and gravitational compaction as the problem of white dwarfs, but it required detailed understanding of nuclear forces, which in 1938 were only beginning to be understood.  However, Oppenheimer knew enough to make a good estimate of the nuclear binding contribution to the total internal pressure and came to a similar conclusion for neutron stars as Chandrasekhar had made for white dwarfs.  There was indeed a maximum mass of a neutron star, a Chandrasekhar-type limit of about three solar masses.  Beyond this mass, even the degeneracy pressure of neutrons could not support gravitational pressure, and the neutron star must collapse.  In Oppenheimer’s mind it was clear what it must collapse to—a black hole (known as gravitational cut-off at that time). This was to lead Oppenheimer and John Wheeler to their famous confrontation over the existence of black holes, which Oppenheimer won, but Wheeler took possession of the battle field [9].

Derivation of the Relativistic Chandrasekhar Limit

White dwarfs are created from the balance between gravitational compression and the degeneracy pressure of electrons caused by the Pauli exclusion principle. When a star collapses gravitationally, the matter becomes so dense that the electrons begin to fill up quantum states until all the lowest-energy states are filled and no more electrons can be added. This results in a balance that stabilizes the gravitational collapse, and the result is a white dwarf with a mass density a million times larger than the Sun.

If the electrons remained non-relativistic, then there would be no upper limit for the size of a star that would form a white dwarf. However, because electrons become relativistic at high enough compaction, if the initial star is too massive, the electron degeneracy pressure becomes limited relativistically and cannot keep the matter from compacting more, and even the white dwarf will collapse (to a neutron star or a black hole). The largest mass that can be supported by a white dwarf is known as the Chandrasekhar limit.

A simplified derivation of the Chandrasekhar limit begins by defining the total energy as the kinetic energy of the degenerate Fermi electron gas plus the gravitational potential energy

The kinetic energy of the degenerate Fermi gas has the relativistic expression


where the Fermi k-vector can be expressed as a function of the radius of the white dwarf and the total number of electrons in the star, as

If the star is composed of pure hydrogen, then the mass of the star is expressed in terms of the total number of electrons and the mass of the proton

The total energy of the white dwarf is minimized by taking its derivative with respect to the radius of the star

When the derivative is set to zero, the term in brackets becomes

This is solved for the radius for which the electron degeneracy pressure stabilizes the gravitational pressure

This is the relativistic radius-mass expression for the size of the stabilized white dwarf as a function of the mass (or total number of electrons). One of the astonishing results of this calculation is the merging of astronomically large numbers (the mass of stars) with both relativity and quantum physics. The radius of the white dwarf is actually expressed as a multiple of the Compton wavelength of the electron!

The expression in the square root becomes smaller as the size of the star increases, and there is an upper bound to the mass of the star beyond which the argument in the square root goes negative. This upper bound is the Chandrasekhar limit defined when the argument equals zero

This gives the final expression for the Chandrasekhar limit (expressed in terms of the Planck mass)

This expression is only approximate, but it does contain the essential physics and magnitude. This limit is on the order of a solar mass. A more realistic numerical calculation yields a limiting mass of about 1.4 times the mass of the Sun. For white dwarfs larger than this value, the electron degeneracy is insufficient to support the gravitational pressure, and the star will collapse to a neutron star or a black hole.


[1] The fact that Eddington coordinates removed the singularity at the Schwarzschild radius was first pointed out by Lemaitre in 1933.  A local observer passing through the Schwarzschild radius would experience no divergence in local properties, even though a distant observer would see that in-falling observer becoming length contracted and time dilated. This point of view of an in-falling observer was explained in 1958 by Finkelstein, who also pointed out that the Schwarzschild radius is an event horizon.

[2] William Herschel (1803), Account of the Changes That Have Happened, during the Last Twenty-Five Years, in the Relative Situation of Double-Stars; With an Investigation of the Cause to Which They Are Owing, Philosophical Transactions of the Royal Society of London 93, pp. 339–382 (Motion of binary stars)

[3] Boss, L. (1910). Preliminary General Catalogue of 6188 stars for the epoch 1900. Carnegie Institution of Washington. (Mass and radius of Sirius B)

[4] Eddington, A. S. (1927). Stars and Atoms. Clarendon Press. LCCN 27015694.

[5] Fowler, R. H. (1926). “On dense matter”. Monthly Notices of the Royal Astronomical Society 87: 114. Bibcode:1926MNRAS..87..114F. (Quantum mechanics of degenerate matter).

[6] Chandrasekhar, S. (1931). “The Maximum Mass of Ideal White Dwarfs”. The Astrophysical Journal 74: 81. Bibcode:1931ApJ….74…81C. doi:10.1086/143324. (Mass limit of white dwarfs).

[7] Kip Thorne (1994) Black Holes & Time Warps: Einstein’s Outrageous Legacy (Norton). pg. 174

[8] Oppenheimer was aware of Zwicky’s proposal because he had a joint appointment between Berkeley and Cal Tech.

[9] See Chapter 7, “The Lens of Gravity” in Galileo Unbound: A Path Across Life, the Universe and Everything (Oxford University Press, 2018).



George Green’s Theorem

For a thirty-year old miller’s son with only one year of formal education, George Green had a strange hobby—he read papers in mathematics journals, mostly from France.  This was his escape from a dreary life running a flour mill on the outskirts of Nottingham, England, in 1823.  The tall wind mill owned by his father required 24-hour attention, with farmers depositing their grain at all hours and the mechanisms and sails needing constant upkeep.  During his one year in school when he was eight years old he had become fascinated by maths, and he had nurtured this interest after leaving school one year later, stealing away to the top floor of the mill to pore over books he scavenged, devouring and exhausting all that English mathematics had to offer.  By the time he was thirty, his father’s business had become highly successful, providing George with enough wages to become a paying member of the private Nottingham Subscription Library with access to the Transactions of the Royal Society as well to foreign journals.  This simple event changed his life and changed the larger world of mathematics.

Green’s windmill in Sneinton, England.

French Analysis in England

George Green was born in Nottinghamshire, England.  No record of his birth exists, but he was baptized in 1793, which may be assumed to be the year of his birth.  His father was a baker in Nottingham, but the food riots of 1800 forced him to move outside of the city to the town of Sneinton, where he bought a house and built an industrial-scale windmill to grind flour for his business.  He prospered enough to send his eight-year old son to Robert Goodacre’s Academy located on Upper Parliament Street in Nottingham.  Green was exceptionally bright, and after one year in school he had absorbed most of what the Academy could teach him, including a smattering of Latin and Greek as well as French along with what simple math that was offered.  Once he was nine, his schooling was over, and he took up the responsibility of helping his father run the mill, which he did faithfully, though unenthusiastically, for the next 20 years.  As the milling business expanded, his father hired a mill manager that took part of the burden off George.  The manager had a daughter Jane Smith, and in 1824 she had her first child with Green.  Six more children were born to the couple over the following fifteen years, though they never married.

Without adopting any microscopic picture of how electric or magnetic fields are produced or how they are transmitted through space, Green could still derive rigorous properties that are independent of any details of the microscopic model.

            During the 20 years after leaving Goodacre’s Academy, Green never gave up learning what he could, teaching himself to read French readily as well as mastering English mathematics.  The 1700’s and early 1800’s had been a relatively stagnant period for English mathematics.  After the priority dispute between Newton and Leibniz over the invention of the calculus, English mathematics had become isolated from continental advances.  This was part snobbery, but also part handicap as the English school struggled with Newton’s awkward fluxions while the continental mathematicians worked with Leibniz’ more fruitful differential notation.  The French mathematicians in the early 1800’s were especially productive, including works by Lagrange, Laplace and Poisson.

            One block away from where Green lived stood the Free Grammar School overseen by headmaster John Topolis.  Topolis was a Cambridge graduate on a minor mission to update the teaching of mathematics in England, well aware that the advances on the continent were passing England by.  For instance, Topolis translated Laplace’s mathematically advanced Méchaniqe Celéste from French into English.  Topolis was also well aware of the work by the other French mathematicians and maintained an active scholarly output that eventually brought him back to Cambridge as Dean of Queen’s College in 1819 when Green was 26 years old.  There is no record whether Topolis and Green knew each other, but their close proximity and common interests point to a natural acquaintance.  One can speculate that Green may even have sought Topolis out, given his insatiable desire to learn more mathematics, and it is likely that Topolis would have introduced Green to the vibrant French school of mathematics.             

By the time Green joined the Nottingham Subscription Library, he must already have been well trained in basic mathematics, and membership in the library allowed him to request loans of foreign journals (sort of like Interlibrary Loan today).  With his library membership beginning in 1823, Green absorbed the latest advances in differential equations and must have begun forming a new viewpoint of the uses of mathematics in the physical sciences.  This was around the same time that he was beginning his family with Jane as well as continuing to run his fathers mill, so his mathematical hobby was relegated to the dark hours of the night.  Nonetheless, he made steady progress over the next five years as his ideas took rough shape and were refined until finally he took pen to paper, and this uneducated miller’s son began a masterpiece that would change the history of mathematics.

Essay on Mathematical Analysis of Electricity and Magnetism

By 1827 Green’s free-time hobby was about to bear fruit, and he took out a modest advertisement to announce its forthcoming publication.  Because he was an unknown, and unknown to any of the local academics (Topolis had already gone back to Cambridge), he chose vanity publishing and published out of pocket.   An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism was printed in March of 1828, and there were 51 subscribers, mostly from among the members of the Nottingham Subscription Library who bought it at 7 shillings and 6 pence per copy, probably out of curiosity or sympathy rather than interest.  Few, if any, could have recognized that Green’s little essay contained several revolutionary elements.

Fig. 1 Cover page of George Green’s Essay

            The topic of the essay was not remarkable, treating mathematical problems of electricity and magnetism, which was in vogue at that time.  As background, he had read works by Cavendish, Poisson, Arago, Laplace, Fourier, Cauchy and Thomas Young (probably Young’s Course of Lectures on Natural Philosopy and the Mechanical Arts (1807)).  He paid close attention to Laplace’s treatment of celestial mechanics and gravitation which had obvious strong analogs to electrostatics and the Coulomb force because of the common inverse square dependence. 

            One radical contribution in Green’s essay was his introduction of the potential function—one of the first uses of the concept of a potential function in mathematical physics—and he gave it its modern name.  Others had used similar constructions, such as Euler [1], D’Alembert [2], Laplace[3] and Poisson [4], but the use had been implicit rather than explicit.  Green shifted the potential function to the forefront, as a central concept from which one could derive other phenomena.  Another radical contribution from Green was his use of the divergence theorem.  This has tremendous utility, because it relates a volume integral to a surface integral.  It was one of the first examples of how measuring something over a closed surface could determine a property contained within the enclosed volume.  Gauss’ law is the most common example of this, where measuring the electric flux through a closed surface determines the amount of enclosed charge.  Lagrange in 1762 [5] and Gauss in 1813 [6] had used forms of the divergence theorem in the context of gravitation, but Green applied it to electrostatics where it has become known as Gauss’ law and is one of the four Maxwell equations.  Yet another contribution was Green’s use of linear superposition to determine the potential of a continuous charge distribution, integrating the potential of a point charge over a continuous charge distribution.  This was equivalent to defining what is today called a Green’s function, which is a common method to solve partial differential equations.

            A subtle contribution of Green’s Essay, but no less influential, was his adoption of a mathematical approach to a physics problem based on the fundamental properties of the mathematical structure rather than on any underlying physical model.  Without adopting any microscopic picture of how electric or magnetic fields are produced or how they are transmitted through space, he could still derive rigorous properties that are independent of any details of the microscopic model.  For instance, the inverse square law of both electrostatics and gravitation is a fundamental property of the divergence theorem (a mathematical theorem) in three-dimensional space.  There is no need to consider what space is composed of, such as the many differing models of the ether that were being proposed around that time.  He would apply this same fundamental mathematical approach in his later career as a Cambridge mathematician to explain the laws of reflection and refraction of light.

George Green: Cambridge Mathematician

A year after the publication of the Essay, Green’s father died a wealthy man, his milling business having become very successful.  Green inherited the family fortune, and he was finally able to leave the mill and begin devoting his energy to mathematics.  Around the same time he began working on mathematical problems with the support of Sir Edward Bromhead.  Bromhead was a Nottingham peer who had been one of the 51 subscribers to Green’s published Essay.  As a graduate of Cambridge he was friends with Herschel, Babbage and Peacock, and he recognized the mathematical genius in this self-educated miller’s son.  The two men spent two years working together on a pair of publications, after which Bromhead used his influence to open doors at Cambridge.

            In 1832, at the age of 40, George Green enrolled as an undergraduate student in Gonville and Caius College at Cambridge.  Despite his concerns over his lack of preparation, he won the first-year mathematics prize.  In 1838 he graduated as fourth wrangler only two positions behind the future famous mathematician James Joseph Sylvester (1814 – 1897).  Based on his work he was elected as a fellow of the Cambridge Philosophical Society in 1840.  Green had finally become what he had dreamed of being for his entire life—a professional mathematician.

            Green’s later papers continued the analytical dynamics trend he had established in his Essay by applying mathematical principles to the reflection and refraction of light. Cauchy had built microscopic models of the vibrating ether to explain and derive the Fresnel reflection and transmission coefficients, attempting to understand the structure of ether.  But Green developed a mathematical theory that was independent of microscopic models of the ether.  He believed that microscopic models could shift and change as newer models refined the details of older ones.  If a theory depended on the microscopic interactions among the model constituents, then it too would need to change with the times.  By developing a theory based on analytical dynamics, founded on fundamental principles such as minimization principles and geometry, then one could construct a theory that could stand the test of time, even as the microscopic understanding changed.  This approach to mathematical physics was prescient, foreshadowing the geometrization of physics in the late 1800’s that would lead ultimately to Einsteins theory of General Relativity.

Green’s Theorem and Greens Function

Green died in 1841 at the age of 49, and his Essay was mostly forgotten.  Ten years later a young William Thomson (later Lord Kelvin) was graduating from Cambridge and about to travel to Paris to meet with the leading mathematicians of the age.  As he was preparing for the trip, he stumbled across a mention of Green’s Essay but could find no copy in the Cambridge archives.  Fortunately, one of the professors had a copy that he lent Thomson.  When Thomson showed the work to Liouville and Sturm it caused a sensation, and Thomson later had the Essay republished in Crelle’s journal, finally bringing the work and Green’s name into the mainstream.

            In physics and mathematics it is common to name theorems or laws in honor of a leading figure, even if the they had little to do with the exact form of the theorem.  This sometimes has the effect of obscuring the historical origins of the theorem.  A classic example of this is the naming of Liouville’s theorem on the conservation of phase space volume after Liouville, who never knew of phase space, but who had published a small theorem in pure mathematics in 1838, unrelated to mechanics, that inspired Jacobi and later Boltzmann to derive the form of Liouville’s theorem that we use today.  The same is true of Green’s Theorem and Green’s Function.  The form of the theorem known as Green’s theorem was first presented by Cauchy [7] in 1846 and later proved by Riemann [8] in 1851.  The equation is named in honor of Green who was one of the early mathematicians to show how to relate an integral of a function over one manifold to an integral of the same function over a manifold whose dimension differed by one.  This property is a consequence of the Generalized Stokes Theorem, of which the Kelvin-Stokes Theorem, the Divergence Theorem and Green’s Theorem are special cases.

Fig. 2 Green’s theorem and its relationship with the Kelvin-Stokes theorem, the Divergence theorem and the Generalized Stokes theorem (expressed in differential forms)

            Similarly, the use of Green’s function for the solution of partial differential equations was inspired by Green’s use of the superposition of point potentials integrated over a continuous charge distribution.  The Green’s function came into more general use in the late 1800’s and entered the mainstream of physics in the mid 1900’s [9].

Fig. 3 The application of Green’s function so solve a linear operator problem, and an example applied to Poisson’s equation.

[1] L. Euler, Novi Commentarii Acad. Sci. Petropolitanae , 6 (1761)

[2] J. d’Alembert, “Opuscules mathématiques” , 1 , Paris (1761)

[3] P.S. Laplace, Hist. Acad. Sci. Paris (1782)

[4] S.D. Poisson, “Remarques sur une équation qui se présente dans la théorie des attractions des sphéroïdes” Nouveau Bull. Soc. Philomathique de Paris , 3 (1813) pp. 388–392

[5] Lagrange (1762) “Nouvelles recherches sur la nature et la propagation du son” (New researches on the nature and propagation of sound), Miscellanea Taurinensia (also known as: Mélanges de Turin ), 2: 11 – 172

[6] C. F. Gauss (1813) “Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata,” Commentationes societatis regiae scientiarium Gottingensis recentiores, 2: 355–378

[7] Augustin Cauchy: A. Cauchy (1846) “Sur les intégrales qui s’étendent à tous les points d’une courbe fermée” (On integrals that extend over all of the points of a closed curve), Comptes rendus, 23: 251–255.

[8] Bernhard Riemann (1851) Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse (Basis for a general theory of functions of a variable complex quantity), (Göttingen, (Germany): Adalbert Rente, 1867

[9] Schwinger, Julian (1993). “The Greening of quantum Field Theory: George and I”: 10283. arXiv:hep-ph/9310283

The Wonderful World of Hamiltonian Maps

Hamiltonian systems are freaks of nature.  Unlike the everyday world we experience that is full of dissipation and inefficiency, Hamiltonian systems live in a world free of loss.  Despite how rare this situation is for us, this unnatural state happens commonly in two extremes: orbital mechanics and quantum mechanics.  In the case of orbital mechanics, dissipation does exist, most commonly in tidal effects, but effects of dissipation in the orbits of moons and planets takes eons to accumulate, making these systems effectively free of dissipation on shorter time scales.  Quantum mechanics is strictly free of dissipation, but there is a strong caveat: ALL quantum states need to be included in the quantum description.  This includes the coupling of discrete quantum states to their environment.  Although it is possible to isolate quantum systems to a large degree, it is never possible to isolate them completely, and they do interact with the quantum states of their environment, if even just the black-body radiation from their container, and even if that container is cooled to milliKelvins.  Such interactions involve so many degrees of freedom, that it all behaves like dissipation.  The origin of quantum decoherence, which poses such a challenge for practical quantum computers, is the entanglement of quantum systems with their environment.

Liouville’s theorem plays a central role in the explanation of the entropy and ergodic properties of ideal gases, as well as in Hamiltonian chaos.

Liouville’s Theorem and Phase Space

A middle ground of practically ideal Hamiltonian mechanics can be found in the dynamics of ideal gases. This is the arena where Maxwell and Boltzmann first developed their theories of statistical mechanics using Hamiltonian physics to describe the large numbers of particles.  Boltzmann applied a result he learned from Jacobi’s Principle of the Last Multiplier to show that a volume of phase space is conserved despite the large number of degrees of freedom and the large number of collisions that take place.  This was the first derivation of what is today known as Liouville’s theorem.

Close-up of the Lozi Map with B = -1 and C = 0.5.

In 1838 Joseph Liouville, a pure mathematician, was interested in classes of solutions of differential equations.  In a short paper, he showed that for one class of differential equation one could define a property that remained invariant under the time evolution of the system.  This purely mathematical paper by Liouville was expanded upon by Jacobi, who was a major commentator on Hamilton’s new theory of dynamics, contributing much of the mathematical structure that we associate today with Hamiltonian mechanics.  Jacobi recognized that Hamilton’s equations were of the same class as the ones studied by Liouville, and the conserved property was a product of differentials.  In the mid-1800’s the language of multidimensional spaces had yet to be invented, so Jacobi did not recognize the conserved quantity as a volume element, nor the space within which the dynamics occurred as a space.  Boltzmann recognized both, and he was the first to establish the principle of conservation of phase space volume. He named this principle after Liouville, even though it was actually Boltzmann himself who found its natural place within the physics of Hamiltonian systems [1].

Liouville’s theorem plays a central role in the explanation of the entropy of ideal gases, as well as in Hamiltonian chaos.  In a system with numerous degrees of freedom, a small volume of initial conditions is stretched and folded by the dynamical equations as the system evolves.  The stretching and folding is like what happens to dough in a bakers hands.  The volume of the dough never changes, but after a long time, a small spot of food coloring will eventually be as close to any part of the dough as you wish.  This analogy is part of the motivation for ergodic systems, and this kind of mixing is characteristic of Hamiltonian systems, in which trajectories can diffuse throughout the phase space volume … usually.

Interestingly, when the number of degrees of freedom are not so large, there is a middle ground of Hamiltonian systems for which some initial conditions can lead to chaotic trajectories, while other initial conditions can produce completely regular behavior.  For the right kind of systems, the regular behavior can hem in the irregular behavior, restricting it to finite regions.  This was a major finding of the KAM theory [2], named after Kolmogorov, Arnold and Moser, which helped explain the regions of regular motion separating regions of chaotic motion as illustrated in Chirikov’s Standard Map.

Discrete Maps

Hamilton’s equations are ordinary continuous differential equations that define a Hamiltonian flow in phase space.  These equations can be solved using standard techniques, such as Runge-Kutta.  However, a much simpler approach for exploring Hamiltonian chaos uses discrete maps that represent the Poincaré first-return map, also known as the Poincaré section.  Testing that a discrete map satisfies Liouville’s theorem is as simple as checking that the determinant of the Floquet matrix is equal to unity.  When the dynamics are represented in a Poincaré plane, these maps are called area-preserving maps.

There are many famous examples of area-preserving maps in the plane.  The Chirikov Standard Map is one of the best known and is often used to illustrate KAM theory.  It is a discrete representation of a kicked rotater, while a kicked harmonic oscillator leads to the Web Map.  The Henon Map was developed to explain the orbits of stars in galaxies.  The Lozi Map is a version of the Henon map that is more accessible analytically.  And the Cat Map was devised by Vladimir Arnold to illustrate what is today called Arnold Diffusion.  All of these maps display classic signatures of (low-dimensional) Hamiltonian chaos with periodic orbits hemming in regions of chaotic orbits.

Chirikov Standard Map
Kicked rotater
Web Map
Kicked harmonic oscillator
Henon Map
Stellar trajectories in galaxies
Lozi Map
Simplified Henon map
Cat MapArnold Diffusion

Table:  Common examples of area-preserving maps.

Lozi Map

My favorite area-preserving discrete map is the Lozi Map.  I first stumbled on this map at the very back of Steven Strogatz’ wonderful book on nonlinear dynamics [3].  It’s one of the last exercises of the last chapter.  The map is particularly simple, but it leads to rich dynamics, both regular and chaotic.  The map equations are

which is area-preserving when |B| = 1.  The constant C can be varied, but the choice C = 0.5 works nicely, and B = -1 produces a beautiful nested structure, as shown in the figure.

Iterated Lozi map for B = -1 and C = 0.5.  Each color is a distinct trajectory.  Many regular trajectories exist that corral regions of chaotic trajectories.  Trajectories become more chaotic farther away from the center.

Python Code for the Lozi Map

"""
Created on Wed May  2 16:17:27 2018
@author: nolte
"""
import numpy as np
from scipy import integrate
from matplotlib import pyplot as plt

B = -1
C = 0.5

np.random.seed(2)
plt.figure(1)

for eloop in range(0,100):

    xlast = np.random.normal(0,1,1)
    ylast = np.random.normal(0,1,1)

    xnew = np.zeros(shape=(500,))
    ynew = np.zeros(shape=(500,))
    for loop in range(0,500):
        xnew[loop] = 1 + ylast - C*abs(xlast)
        ynew[loop] = B*xlast
        xlast = xnew[loop]
        ylast = ynew[loop]
        
    plt.plot(np.real(xnew),np.real(ynew),'o',ms=1)
    plt.xlim(xmin=-1.25,xmax=2)
    plt.ylim(ymin=-2,ymax=1.25)
        
plt.savefig('Lozi')

References:

[1] D. D. Nolte, “The Tangled Tale of Phase Space”, Chapter 6 in Galileo Unbound: A Path Across Life, the Universe and Everything (Oxford University Press, 2018)

[2] H. S. Dumas, The KAM Story: A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov-Arnold-Moser Theory (World Scientific, 2014)

[3] S. H. Strogatz, Nonlinear Dynamics and Chaos (WestView Press, 1994)

Top 10 Topics of Modern Dynamics

“Modern physics” in the undergraduate physics curriculum has been monopolized, on the one hand, by quantum mechanics, nuclear physics, particle physics and astrophysics. “Classical mechanics”, on the other hand, has been monopolized by Lagrangians and Hamiltonians.  While these are all admittedly interesting, the topics of modern dynamics that monopolize the time and actions of most physics-degree holders, as they work in high-tech start-ups, established technology companies, or on Wall Street, are not to be found.  These are the topics of nonlinear dynamics, chaos theory, complex networks, finance, evolutionary dynamics and neural networks, among others.

Cover

There is a growing awareness that the undergraduate physics curriculum needs to be reinvigorated to make a physics degree relevant to the modern workplace.  To that end, I am listing my top 10 topics of modern dynamics that can form the foundation of a revamped upper-division (junior level) mechanics course.  Virtually all of these topics were once reserved for graduate-student-level courses, but all can be introduced to undergraduates in simple and intuitive ways without the need for advanced math.

1) Phase Space

The key change in perspective for modern dynamics that differentiates it from classical dynamics is the emphasis on the set of all possible trajectories that fill a “space” rather than emphasizing single trajectories defined by given initial conditions.  Rather than study the motion of one rock thrown from a cliff top, modern dynamics studies an infinity of rocks thrown from every possible point and with every possible velocity.  The space that contains this infinity of trajectories is known as phase space (or more generally state space).  The equation of motion in state space becomes the dynamical flow, replacing Newton’s second law as the central mathematical structure of physics.  Modern dynamics studies the properties of phase space rather than the properties of single trajectories, and makes rigorous and unique conclusions about classes of possible motions.  This emphasis on classes of behavior is more general and more universal and more powerful, while also providing a fundamental “visual language” with which to describe the complex physics of complex systems.

2) Metric Space

The Cartesian coordinate plane that we were all taught in high school tends to dominate our thinking, biasing us towards linear flat geometries.  Yet most dynamics do not take place in such simple Cartesian spaces.  A case in point, virtually every real-world dynamics problem has constraints that confine the motion to a surface.  Furthermore, the number of degrees of freedom of a dynamical system usually exceed our common 3-space, expanding to hundreds or even to thousands of dimensions.  The surfaces of constraint are hypersurfaces of high dimensions (known as manifolds) and are almost certainly not flat hyperplanes. This daunting prospect of high-dimensional warped spaces can be surprisingly simplified through the concept of Bernhard Riemann’s “metric space”.  Understanding the geometry of a metric space can be as simple as applying Pythagoras’ Theorem to sets of coordinates.  For instance, the metric tensor can be taught and used without requiring students to know anything of tensor calculus.  At the same time, it provides a useful tool for understanding dynamical patterns in phase space as well as orbits around black holes.

3) Invariants

Introductory physics classes emphasize the conservation of energy, linear momentum and angular momentum as if they are special cases.  Yet there is a grand structure that yields a universal set of conservation laws: integrable Hamiltonian systems.  An integrable system is one for which there are as many invariants of motion as there are degrees of freedom.  Amazingly, these conservation laws can all be captured by a single procedure known as (canonical) transformation to action-angle coordinates.  When expressed in action-angle form, these Hamiltonians take on extremely simple expressions.  They are also the starting point for the study of perturbations when small nonintegrable terms are added to the Hamiltonian.  As the perturbations grow, this provides one doorway to the emergence of chaos.

4) Chaos theory

“Chaos theory” is the more popular title for what is generally called “nonlinear dynamics”.  Nonlinear dynamics takes place in state space when the dynamical flow equations have terms that algebraically are products of variables.  One important distinction between chaos theory and nonlinear dynamics is the occurrence of unpredictability that can emerge in the dynamics when the number of variables is equal to three or higher.  The equations, and the resulting dynamics, are still deterministic, but the trajectories are incredibly sensitive to initial conditions (SIC).  In addition, the dynamical trajectories can relax to a submanifold of the original state space known as a strange attractor that typically is a fractal structure.

5) Synchronization

One of the central paradigms of nonlinear dynamics is the autonomous oscillator.  Unlike the harmonic oscillator that eventually decays due to friction, autonomous oscillators are steady-state oscillators that convert steady energy input into oscillatory behavior.  A prime example is the pendulum clock that converts the steady weight of a hanging mass into a sustained oscillation.  When two autonomous oscillators (that normally oscillator at slightly different frequencies) are coupled weakly together, they can synchronize to the same frequency.   This effect was discovered by Christiaan Huygens when he observed two pendulum clocks hanging next to each other on a wall synchronize the swings of their pendula.  Synchronization is a central paradigm in modern dynamics for several reasons.  First, it demonstrates the emergence of order when a collective behavior emerges from a collection of individual systems (this phenomenon of emergence is one of the fundamental principles of complex system science).  Second, synchronized systems include such critical systems as the beating heart and the thinking brain.  Third, synchronization becomes a useful tool to explore coupled systems that have a large number of linked subsystems, as in networks of nodes.

6) Network Dynamics

Networks have become one of the driving forces of our modern interconnected society.  The structure of networks, the dynamics of nodes in networks, and the dynamic growth of networks are all coming into focus as we live our lives in multiple interconnected webs.  Dynamics on networks include problems like diffusion and the spread of infection and connect with topics of percolation theory and critical phenomenon.  Nonlinear dynamics on networks provide key opportunities and examples to study complex interacting systems.

7) Neural Networks

Perhaps the most enigmatic network is the network of neurons in the brain.  The emergence of intelligence and of sentience is one of the greatest scientific questions.  At a much simpler level, the nonlinear dynamics of small numbers of neurons display the properties of autonomous oscillators and synchronization, while larger sets of neurons become interconnected into dynamic networks.  The dynamics of neurons and of neural networks is a  key topic in modern dynamics.  Not only can the physics of the networks be studied, but neural networks become tools for studying other complex systems.

8) Evolutionary Dynamics

The emergence of life and the evolution of species stands as another of the greatest scientific questions of our day.  Although this topic traditionally is studied by the biological sciences (and mathematical biology), physics has a surprising lot to say on the topic.  The dynamics of evolution can be captured in the same types of nonlinear flows that live in state space.  For instance, population dynamics can be described as a large ensemble of interacting individuals that are born, flourish and die dependent on their environment and on their complicated interactions with other members in their ecosystem.  These types of problems have state spaces of extremely high dimension far beyond what we can visualize.  Yet the emergence of structure and of patterns from the complex dynamics helps to reduce the complexity, as do conceptual metaphors like evolutionary fitness landscapes.

9) Economic Dynamics

A non-negligible fraction of both undergraduate and graduate physics degree holders end up on Wall Street or in related industries.  This is partly because physicists are numerically fluent while also possessing sound intuition.  Therefore, economic dynamics is a potentially valuable addition to the modern dynamics curriculum and easily expressed using the concepts of dynamical flows and state space.  Both microeconomics (business competition, business cycles) and macroeconomics (investment and savings, liquidity and money, inflation, unemployment) can be described and analyzed using mathematical flows that are the central toolkit of modern dynamics.

10) Relativity

Special relativity is a common topic in the current upper-division physics curriculum, while general relativity is viewed as too difficult to expose undergraduates to.  This is mostly an artificial division, because Einstein’s “happiest thought” occurred when he realized that an observer in free fall is in a force-free (inertial) frame.  The equivalence principle, that states that a frame in uniform acceleration is indistinguishable from a stationary frame in a uniform gravitational field, opens a wide door that connects special relativity to general relativity.  In an undergraduate course on modern dynamics, the metric tensor (described above) is introduced in simple terms, providing the foundation to develop Minkowski spacetime, and the next natural extension is to warped spacetime—all at the simple level of linear algebra combined with partial differentiation.  General relativity ties in many of the principles of the modern dynamics curriculum (dynamical flows, state space, metric space, invariants, nonlinear dynamics), and the students can simulate orbits around black holes with ease.  I have been teaching General Relativity to undergraduates for over ten years now, and it is a highlight of the course.

Introduction to Modern Dynamics

For further reading and more details, these top 10 topics of modern dynamics are defined and explored in the undergraduate physics textbook “Introduction to Modern Dynamics: Chaos, Networks, Space and Time” published by Oxford University Press (2015).  This textbook is designed for use in a two-semester junior-level mechanics course.  It introduces the topics of modern dynamics, while still presenting traditional materials that the students need for their physics GREs.