The Many Dimensions of Oskar Klein

The idea of parallel dimensions in physics has a long history dating back to Bernhard Riemann’s famous 1954 lecture on the foundations of geometry that he gave as a requirement to attain a teaching position at the University of Göttingen.  Riemann laid out a program of study that included physics problems solved in multiple dimensions, but it was Rudolph Lipschitz twenty years later who first composed a rigorous view of physics as trajectories in many dimensions.  Nonetheless, the three spatial dimensions we enjoy in our daily lives remained the only true physical space until Hermann Minkowski re-expressed Einstein’s theory of relativity in 4-dimensional space time.  Even so, Minkowski’s time dimension was not on an equal footing with the three spatial dimensions—the four dimensions were entwined, but time had a different characteristic, what is known as pseudo-Riemannian metric.  It is this pseudo-metric that allows space-time distances to be negative as easily as positive.

In 1919 Theodore Kaluza of the University of Königsberg in Prussia extended Einstein’s theory of gravitation to a fifth spatial dimension, and physics had its first true parallel dimension.  It was more than just an exercise in mathematics—adding a fifth dimension to relativistic dynamics adds new degrees of freedom that allow the dynamical 5-dimensional theory to include more than merely relativistic massive particles and the electric field they generate.  In addition to electro-magnetism, something akin to Einstein’s field equation of gravitation emerges.  Here was a five-dimensional theory that seemed to unify E&M with gravity—a first unified theory of physics.  Einstein, to whom Kaluza communicated his theory, was intrigued but hesitant to forward Kaluza’s paper for publication.  It seemed too good to be true.  But Einstein finally sent it to be published in the proceedings of the Prussian Academy of Sciences [Kaluza, 1921]. He later launched his own effort to explore such unified field theories more deeply.

Yet Kaluza’s theory was fully classical—if a fifth dimension can be called that—because it made no connection to the rapidly developing field of quantum mechanics. The person who took the step to make five-dimensional space-time into a quantum field theory was Oskar Klein.

Oskar Klein (1894 – 1977)

Oskar Klein was a Swedish physicist who was in the “second wave” of quantum physicists just a few years behind the titans Heisenberg and Schrödinger and Pauli.  He began as a student in physical chemistry working in Stockholm under the famous Arrhenius.  It was arranged for him to work in France and Germany in 1914, but he was caught in Paris at the onset of World War I.  Returning to Sweden, he enlisted in military service from 1915 to 1916 and then joined Arrhenius’ group at the Nobel Institute where he met Hendrick Kramers—Bohr’s direct assistant at Copenhagen at that time.  At Kramer’s invitation, Klein traveled to Copenhagen and worked for a year with Kramers and Bohr before returning to defend his doctoral thesis in 1921 in the field of physical chemistry.  Klein’s work with Bohr had opened his eyes to the possibilities of quantum theory, and he shifted his research interest away from physical chemistry.  Unfortunately, there were no positions at that time in such a new field, so Klein accepted a position as assistant professor at the University of Michigan in Ann Arbor where he stayed from 1923 to 1925. 

Oskar Klein in the late 1920’s

The Fifth Dimension

In an odd twist of fate, this isolation of Klein from the mainstream quantum theory being pursued in Europe freed him of the bandwagon effect and allowed him to range freely on topics of his own devising and certainly in directions all his own.  Unaware of Kaluza’s previous work, Klein expanded Minkowski’s space-time from four to five spatial dimensions, just as Kaluza had done, but now with a quantum interpretation.  This was not just an incremental step but had far-ranging consequences in the history of physics.

Klein found a way to keep the fifth dimension Euclidean in its metric properties while rolling itself up compactly into a cylinder with the radius of the Planck length—something inconceivably small.  This compact fifth dimension made the manifold into something akin to an infinitesimal string.  He published a short note in Nature magazine in 1926 on the possibility of identifying the electric charge within the 5-dimensional theory [Klein, 2916a]. He then returned to Sweden to take up a position at the University of Lund.  This odd string-like feature of 5-dimensional space-time was picked up by Einstein and others in their search for unified field theories of physics, but the topic soon drifted from the lime light where it lay dormant for nearly fifty years until the first forays were made into string theory. String theory resurrected the Kaluza-Klein theory which has bourgeoned into the vast topic of String Theory today, including Superstrings that occur in 10+1 dimensions at the frontiers of physics. 

Dirac Electrons without the Spin: Klein-Gordon Equation

Once back in Europe, Klein reengaged with the mainstream trends in the rapidly developing quantum theory and in 1926 developed a relativistic quantum theory of the electron [Klein, 1926b].  Around the same time Walter Gordon also proposed this equation, which is now called the “Klein-Gordon Equation”.  The equation was a classic wave equation that was second order in both space and time.  This was the most natural form for a wave equation for quantum particles and Schrödinger himself had started with this form.  But Schrödinger had quickly realized that the second-order time term in the equation did not capture the correct structure of the hydrogen atom, which led him to express the time-dependent term in first order and non-relativistically—which is today’s “Schrödinger Equation”.  The problem was in the spin of the electron.  The electron is a spin-1/2 particle, a Fermion, which has special transformation properties.  It was Dirac a few years later who discovered how to express the relativistic wave equation for the electron—not by promoting the time-dependent term to second order, but by demoting the space-dependent term to first order.  The first-order expression for both the space and time derivatives goes hand in hand with the Pauli spin matrices for the electron, and the Dirac Equation is the appropriate relativistically-correct wave equation for the electron.

Klein’s relativistic quantum wave equation does turn out to be the relevant form for a spin-less particle like the pion, but the pion decays by the strong nuclear force and the Klein-Gordon equation is not a practical description.  However, the Higgs boson also is a spin-zero particle, and the Klein-Gordon expression does have relevance for this fundamental exchange particle.

Klein Tunneling

In those early days of the late 1920’s, the nature of the nucleus was still a mystery, especially the problem of nuclear radioactivity where a neutron could convert to a proton with the emission of an electron.  Some suggested that the neutron was somehow a proton that had captured an electron in a potential barrier.  Klein showed that this was impossible, that the electrons would be highly relativistic—something known as a Dirac electron—and they would tunnel with perfect probability through any potential barrier [Klein, 1929].  Therefore, Klein concluded, no nucleon or nucleus could bind an electron. 

This phenomenon of unity transmission through a barrier became known as Klein tunneling. The relativistic electron transmits perfectly through an arbitrary potential barrier—independent of its width or height. This is unlike light that transmits through a dielectric slab in resonances that depend on the thickness of the slab—also known as a Fabry-Perot interferometer. The Dirac electron can have any energy, and the potential barrier can have any width, yet the electron will tunnel with 100% probability. How can this happen?

The answer has to do with the dispersion (velocity versus momentum) of the Dirac electron. As the momentum changes in a potential the speed of the Dirac electron stays constant. In the potential barrier, the moment flips sign, but the speed remains unchanged. This is equivalent to the effects of negative refractive index in optics. If a photon travels through a material with negative refractive index, its momentum is flipped, but its speed remains unchanged. From Fermat’s principle, it is speed which determines how a particle like a photon refracts, so if there is no speed change, then there is no reflection.

For the case of Dirac electrons in a potential with field F, speed v and transverse momentum py, the transmission coefficient is given by

If the transverse momentum is zero, then the transmission is perfect. A visual schematic of the role of dispersion and potentials for Dirac electrons undergoing Klein tunneling is shown in the next figure.

Dispersion of Dirac electrons at a potential step. Reprinted from

In this case, even if the transverse momentum is not strictly zero, there can still be perfect transmission. It is simply a matter of matching speeds.

Graphene became famous over the past decade because its electron dispersion relation is just like a relativistic Dirac electron with a Dirac point between conduction and valence bands. Evidence for Klein tunneling in graphene systems has been growing, but clean demonstrations have remained difficult to observe.

Now, published in the Dec. 2020 issue of Science magazine—almost a century after Klein first proposed it—an experimental group at the University of California at Berkeley reports a beautiful experimental demonstration of Klein tunneling—not from a nucleus, but in an acoustic honeycomb sounding board the size of a small table—making an experimental analogy between acoustics and Dirac electrons that bears out Klein’s theory.

The accoustic Klein tunneling sounding board at Berkeley. Reprinted from

In this special sounding board, it is not electrons but phonons—acoustic vibrations—that have a Dirac point. Furthermore, by changing the honeycomb pattern, the bands can be shifted, just like in a p-n-p junction, to produce a potential barrier. The Berkeley group, led by Xiang Zhang (now president of Hong Kong University), fabricated the sounding board that is about a half-meter in length, and demonstrated dramatic Klein tunneling.

It is amazing how long it can take between the time a theory is first proposed and the time a clean experimental demonstration is first performed.  Nearly 90 years has elapsed since Klein first derived the phenomenon. Performing the experiment with actual relativistic electrons was prohibitive, but bringing the Dirac electron analog into the solid state has allowed the effect to be demonstrated easily.


[1921] Kaluza, Theodor (1921). “Zum Unitätsproblem in der Physik”. Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.): 966–972

[1926a] Klein, O. (1926). “The Atomicity of Electricity as a Quantum Theory Law”. Nature 118: 516-516.

[1926b] Klein, O. (1926). “Quantentheorie und fünfdimensionale Relativitätstheorie”. Zeitschrift für Physik. 37 (12): 895

[1929] Klein, O. (1929). “Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac”. Zeitschrift für Physik. 53 (3–4): 157

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.


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.”