The secret to fixing the fatal flaws at the heart of quantum theory may be hidden in three obscure books by Jean Écalle in the 80s of the 20th century. But physicists overlooked potentially transformative ideas because the book was full of original mathematical objects and fancy wording. Odd terms like "superseries," "analyzable embryo," "heterogeneous derivative," and "accelerated summation" abound. And in this mathematics, there may be what is needed to overcome a dilemma in physics.
Today, physicists have learned to make amazingly accurate predictions about the subatomic world. However, these predictions, while accurate, are only approximate. If absolute precision is sought, textbook quantum theory collapses, producing infinite answers — answers that many physicists consider mathematical garbage.
But now, by poring over those three textbooks, physicists have discovered a mathematical theory called resurgence. This theory made physicists realize that these infinite answers contain countless treasures that allow them to dig out the finite and flawless answers to any quantum problem from any infinity.
The upturn theory research community, although small, has made steady progress over the years. Prototype versions of this technique have yielded precise results in quantum mechanics, which itself is limited to describing the behavior of particles. The development of more complex mathematical methods has allowed some physicists to delve into quantum field theory and, more recently, string theory. However, this is only the beginning of the grand dreams of practitioners of the recovery theory. Their goal is not only to find a new way to deal with infinity problems in physical theory, but also to better match our finite world in theory and practice.
Possibility of divergence
Quantum field theory forces physicists to confront the problem of infinity. These quantum fields are unimaginably complex systems – transient and coherent fluctuations tumbling in seemingly empty space. In principle, these transient fluctuations can appear at any moment, in any quantity, and at any energy, which challenges physicists who need to interpret endless subatomic interactions in order to understand the precise results of even simple experiments.
In the 40s of the 20th century, Shin'ichirō Tomonaga, Julian Schwinger, and Richard Feynman all developed equivalent methods for deriving finite answers from the infinite complexity of quantum electromagnetic fields. Now best known is Feynman's method, which uses an infinite number of "Feynman diagrams" to represent increasingly complex quantum possibilities. Start with the simplest plot of events (e.g., electrons traveling through space) and calculate some measurable property, such as how well electrons wobble in a magnetic field. Next, the results are added to more complex scenarios, such as electrons briefly releasing during flight and then absorbing a photon. Then add subatomic processes involving two transient fluctuations, then three, and so on, and this widely used mathematical technique is known as perturbation theory.
This calculation yields an infinite "power series":
For electromagnetic fields, the value of x is a key constant (i.e., α) in nature, close to 1/137. This is a relatively small number, suitable for indicating the relative weakness of the force, and raising this tiny number to a greater power causes the terms to shrink rapidly.
Feynman diagrams provide physicists with the coefficients for each term—the value of a—which is the difficult point in calculation. Take the electron's "g-factor," which is a number related to how a particle oscillates in a magnetic field. The simplest Feynman diagram gives the a_0, which is exactly equal to 2. However, if you consider a slightly more complex Feynman diagram, such as the case where the first transient fluctuations occur, you need to calculate the a_1 item, which is where infinity comes in. Asanaga, Schwinger, and Feynman devised a way to make this limited. Their calculated electron g-factor was about 2.002, which matched experimental measurements at the time, proved that quantum field theory made sense, and earned the three of them the 1965 Nobel Prize in Physics.
Their method also ushered in a new era in which physicists had to climb higher and higher Feynman diagrams in order to calculate more A-values. These mountains are steep and grow very fast. In 2017, a physicist completed a 20-year-long work that precisely calculated the g-factor of an electron, which required calculating complex equations from 891 Feynman diagrams. The results reveal only the fifth term of the series.
Feynman diagrams still play a pivotal role in modern physics. A similar but more complex set of calculations for μ (electrons' cousins) made headlines in 2021. One experiment showed that the theoretical prediction differed from the actual measurement to the eighth decimal place. This tiny anomaly represents the best hope, revealing uncharted territory beyond the vast theoretical system built on the basis of Feynman's work.
However, this series of experimental victories obscures the fact that this approach to quantum field theory is fundamentally unworkable.
The decline of Feynman diagrams
Freeman Dyson, another pioneer of physics, was the first physicist to realize that the quantum theory of perturbations could be doomed. It was 1952, and while others celebrated that the first few of the Feynman power series could become small and finite, Dyson began to worry about other parts of the series.
Physicists naively hope that the electromagnetic field treatment described by Feynman diagrams will become what mathematicians call "convergence." In a convergent series, each subsequent term is much smaller than the previous term, and the more terms, the closer the series sum is to a finite number. Conversely, the series may also be "divergent". The divergent series sum is meaningless.
The first few terms of Feynman's summation are indeed shrinking because the α value is small, and Dyson initially concluded that perturbation quantum electromagnetism should converge as a whole.
But later, Dyson combined mathematical and physical reasoning to make a more complex guess about this series. From a mathematical point of view, Dyson knew that when x becomes smaller, the convergent power series converges faster because higher terms (involving powers of x) contract faster.
But when he let x pass zero, everything collapsed.
The reason is related to vacuum, which constantly produces transient pairs of positive and negative charge fluctuations. These fluctuations usually attract each other and disappear. But if the α becomes negative, these fluctuations push each other away and turn into real particles. Particles emerging from nothing trigger the collapse of the universe, as Dyson called "the explosive disintegration of a vacuum."
From a physical point of view, any negative α poses a problem. However, mathematically, the sign of x is not important: if a series diverges at a small negative x value, it should also diverge at a smaller positive x value. Therefore, for smaller positive α (i.e. 1/137), this series should also diverge. Dyson's catastrophic physics scenario meant that Feynman's famous approach to quantum electromagnetism eventually predicted infinity.
- An unruly infinity. Physicists usually study quantum fields, such as electromagnetic fields, through infinite summation. The first few of these summations are getting smaller and smaller, giving us an approximate answer. However, the latter items proliferate, causing the sum to seem meaningless.
Now, physicists expect that quantum electrodynamics (the quantum field theory of electromagnetism) may begin to diverge around item 137. Other words
Factoring it into the summation makes the prediction less accurate, not better.
The problem is that higher terms lead to an explosive growth in the number of Feynman plots. This means that calculating a_9 takes approximately 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 (about 362,880) Feynman plots, while calculating a_10 requires about 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 (3,628,800) Feynman diagrams. This factorial growth rate of the number of Feynman graphs contributing to the A term eventually outpaces the shrinkage of the power of α, allowing the sum to grow uncontrollably toward infinity.
For most physicists, the inevitable divergence of even the simplest quantum field theory is still an abstract problem. When calculating (let alone testing) that the 10th item of this progression all seems like science fiction, why worry about the dangers lurking far away from 100?
But for a handful of physicists, the most understood theories in modern physics technically give endless answers to any question remain deeply troubling. Even with unlimited computing resources in principle, we do not know how to simulate the world.
Demon-like divergence
Meanwhile, mathematicians had been pondering divergent series for more than a century before Dyson began worrying about quantum theory. "Divergent progression is a devilish invention, and it would be shameful to use them to prove anything," Abel said in 1828.
Abel died the following year at the age of 26. But at the end of the century, Henry Poincaré took an important step towards understanding why divergent progressions were so tricky: they were not diabolical evil, just incomplete.
Poincaré is exploring an age-old question: How do the three celestial bodies orbit each other? He tried to solve this problem with perturbation theory, as Feynman and Dyson did a century later when they encountered quantum fields. Poincaré attempted to construct complex functions describing the trajectories of three celestial bodies using the sum of simple units of infinite length. One hopes that this progression converges to a finite answer.
At first, he thought he had succeeded. In 1890, King Oscar II of Sweden and Norway presented an award for Poincaré's progress on this famous issue. But just before his book was about to be published, he demanded that printing be stopped. This series is divergent. Further analysis (which laid the groundwork for chaos theory) revealed that it matched not one but two different functions. Today, physicists know that three celestial bodies can interact in countless very different ways, and no simple equation can encompass all possibilities.
Carl Bend, a mathematical physicist at Washington University in St. Louis, likened the divergent series Poincaré encountered to a fuzzy view of a function. Fuzzy encompasses many possible functions. When you expand a complex function into such an "asymptotic" series, "you've lost information," Bender said.
Since Poincaré's time, mathematicians and physicists have realized that there are other types of terms that are "beyond all orders" and that are smaller than the smallest power. These "exponentially small" items can take the form e^(−1/x), for example, because they provide missing information. If you include them in the series and use a suitable "re-sum" method to make the series finite, you can reduce some (and possibly all) of the ambiguity.
Physicists call these additional terms "non-perturbation terms" because they are beyond the scope of perturbation theory. You can spend trillions of years drawing Feynman diagrams and calculating term a, but you can never understand some of the physical events contained in these non-perturbation terms. Although the effects described by these tiny items may be rare or subtle, they can make a huge difference in the real world.
Take the Schrödinger equation of quantum mechanics, for example, which describes the wave behavior of particles. This is a complex equation that physicists usually approximate using perturbation theory. Although the resulting infinite series predicts many experiments very precisely, it completely ignores a highly unlikely (but not impossible) event, the tunneling effect. In such an event, particles are teleported through obstacles.
The tunneling effect is one of many non-perturbation phenomena in quantum physics, but non-perturbation effects are everywhere: the branching growth of snowflakes, the flow of liquid through perforated pipes, the orbits of planets in the solar system, the ripples of waves around circular islands, and countless other physical phenomena are non-perturbated. They do exist, and they are so vital that perturbation theory alone is not enough.
Due to the universal nature of non-perturbation phenomena, a large number of mathematicians and physicists have devoted themselves to the problem of how to calculate non-perturbation terms. At the end of the 20th century, a group of researchers began to discover some fascinating clues that the perturbation series seemed to contain more information than expected.
Among these researchers, in the 1980s, a group at the Thackeray Center for Nuclear Research in France developed a method of combining perturbation power terms with non-perturbation exponential terms to obtain precise results of quantum mechanical tunneling effects. Their technique was successful in relying on a key mathematical technique called Borel resummation. Borrell resummation was the most powerful tool for deriving finite numbers from divergent series at the time, but it also had its limitations. It occasionally gives errors, which frustrates physicists who want a series to correctly predict the outcome of an experiment.
When physicists found a series that could not be summed with Borrell, they basically gave up. What they didn't know was that an eccentric mathematician just a few miles away from Thackeray's group had begun an unprecedented exploration of asymptotic series.
Feynman Tu's counterattack
In the early 70s of the 20th century, Ecalé's curiosity drove him into Poincaré's footsteps. He began to analyze more abstract mathematical objects that appeared in the study of celestial bodies. Asymptotic series gradually appeared, as did the more general derivatives that he had speculated in high school. Ecaler eventually developed what he described as "an exact, well-defined structure—alien calculus, that arises spontaneously from divergence." ”
Ecaler's heterogeneous calculus is abstract and multifaceted. But for physicists, its message is very clear. Perturbation series, despite divergence, hides a whole set of non-perturbation information. The progression contains everything needed to upgrade to remove blur and restore a clear image with the unique corresponding function.
Despite its far-reaching impact, at first, Ecale's work was not taken seriously. It is too obscure and abstract for physicists, and not rigorous enough for mathematicians.
Ecalle first outlined the core concepts of pick-up theory in three papers in 1976, and wrote three of his textbooks between 1981 and 1985 detailing the heterogeneous calculus of regression theory. These textbooks were never published in mathematical journals.
If physicists had begun studying his books at the time, their experience might have been like contact with a highly intelligent alien civilization. They will encounter mathematical tools far beyond what they are used to.
When Ecaler was contacted by Quanta Magazine and asked questions about the history of pick-up theory, he spent six days writing a 24-page monograph on the topic, a rare gem for researchers eager to learn more about uptick theory and its development.
Here's a very rough simplified version of this approach:
First, write out the typical perturbation series. Terms shrink at first, but eventually they grow rapidly as the A value becomes very large. Plot the growth of the values of A, and you will see that they grow upwards almost as fast as the factorial growth. Study the difference between the line depicted by the a-value and the factorial growth curve to understand the first non-perturbation term.
This is just the beginning, though. The first step in applying Borrell resum. This eliminates factorial growth and allows you to see the behavior of perturbations in more detail. The result plot of the modified A should grow exponentially. But take a closer look at it and you'll notice that the perturbation data is a bit off. This deviation comes from a completely new asymptotic series, which you need to multiply by the first non-perturbation term.
The process continues. Remove exponential growth from the perturbation data, and if you look closely enough, you may find further deviations that reveal a second non-perturbation term. If you look closely, you'll see that this non-perturbation term is accompanied by another asymptotic series.
Eventually, there may be any number of non-perturbation terms, each with an asymptotic progression. Find as many of these items as you can, and you'll have an object in your hand called a span number. Spanning series starts with the familiar perturbation series, followed by a non-perturbation term (with a series), followed by another non-perturbation term, and so on.
Ecaler 's cross-series overcame the difficulty of Borrell's resumation that had previously plagued physicists. If you know the spans that describe a measurement (e.g., the g-factor of an electron), Borrell resummation will give you a unique, correct answer. Furthermore, pick-up theory asserts that subtle deviations in the familiar perturbation series at the beginning of the span series will tell you everything you need to know about the next potentially infinite series.
This mathematical picture has two compelling implications for physicists. First, it suggests that quantum fields and other complex systems may have precise results — not just approximations. If so, it would make quantum theory finite and reasonable. This would be a major step forward.
Second, this implies that a potentially infinite number of non-perturbated parts can be derived entirely from the perturbation series, which divergence bothers Dyson. What seemed like separate fields of physics for decades are actually closely related.
Pick-up theory enters physics
Ecaler's discoveries (knowledge of non-perturbation through perturbation theory) have gradually penetrated into the world of mathematical physics. Here, physicists have used it to find hidden new pieces in two of the most closely studied theories of the 21st century: the theory of forces and string theory. Forcefully polymerizes quarks together to form protons and other particles
Mithat Ünsal, a physicist at North Carolina State University, is dedicated to the study of power. In 2008, he sought to understand Ecaler's work after reading about the theory of pick-up in an article on divergent series. Later, he met Gerald Dunne of the University of Connecticut at a conference, and while chatting, they found that the same article inspired the two of them to start teaching themselves the theory of pick-up. They decided to join forces.
The two physicists were motivated by their attempts to understand something more complex than the problems facing Dyson and Feynman. These physicists are lucky when it comes to electromagnetic fields. The electromagnetic field is very weak, only 1/137 of the α. Another fundamental force, weak interaction, is equally easy to handle. For these two forces, perturbation theory works because they are so weak that it can almost be said that they do not exist at all.
But that luck ended when physicists tried to solve the strong force. The strength is about 100 times stronger than the electromagnetic force, and its α analogue is about 1, which cannot be ignored. Square or cubic 1 does not produce any reduction effect at all, so the perturbation series goes directly to infinity from the earliest terms. Physicists have spent decades using supercomputers to develop an alternative way to deal with powerful forces, with amazing results in the process. However, numerical calculations do not explain well how strong forces work.
Ünsal and Dunne realized that a recovery theory with the ability to tame divergent progressions might take them a step toward their dream of unforcing forces. In particular, they set out to solve a mystery that had plagued the theory of force for 40 years.
In 1979, physicists Gerard Hofter and Parisi deduced that there are tiny and bizarre terms in powerful calculations. They call them renormalons, and no one knows what to do with them. The reorganizer does not seem to have any correspondence with specific field behavior. But they were there, anyway, they ruined the calculation.
Ünsal and Dunne use pick-up to solve the reorganization sub-problem. Although they were working on powerful 2D analogues, it took them about a year. But in 2012, they showed that, at least in their simplified model, Hofter and Parisi's renormalizon matched the behavior that physicists understand.
Last year, however, researchers used the pick-up to add further complexity. Marino, a mathematical physicist at the University of Geneva, and his collaborators made more rigorous calculations (albeit also in simplified theory) and discovered new reorganizers. Marino now suspects that the reorganization is just the tip of the non-perturbation iceberg. If he's right, the quantum world could one day be even harder to imagine than it is now.
Marino also played a key role in the discovery of a new non-perturbation effect in string theory, a speculative idea that the universe is not made up of point-like particles, but of extended objects such as strings. The vibration of this string will determine the nature of the particles we observe.
String theory, like quantum theory, is often seen as a perturbation series that uses Feynman-like diagrams to show that strings merge and split in increasingly complex ways. However, unlike quantum theorists, string theorists lack even the most basic guidance on non-perturbation effects in theory. They argue that just as quantum theory contains tunneling and reregularization, the completely non-perturbation formula of string theory also contains unknown physical phenomena.
A striking example of a non-perturbation phenomenon in string theory is the discovery of a sheet-like object called a D-branes in the 90s of the 20th century. D-membranes would later drive some of the greatest developments in string theory.
In 2010, Marino noticed a series of negative counterparties hidden in the shadow of the D-membrane term. It is unclear what physical phenomena these companions might describe.
Six years later, Karan Wafa of Harvard University and his collaborators found clues while exploring a generalized string theory in which certain quantities can be negative. They found a D-membrane with negative tension — a membrane version with negative mass. These bizarre things distort the structure of the reality around them, create multiple time dimensions and violate the basic principle that the sum of probabilities must be 100%. But the team didn't find that these objects should escape from their weird worlds and enter standard string theory.
Now, Ricardo Schiappa, a theoretical physicist at the University of Lisbon, thinks he's found evidence. In recent months, Schiapa and his collaborators have scrutinized several simple string-theoretic models using the pick-up method. They found that the negative tension D-membrane of Wafa exactly matched the exponential small term that Marino discovered in 2010.
Other theorists aren't yet sure what to think of the new finding. Wafa noted that Skiapá's team performed their calculations in a simplified string model, a result that is not guaranteed to hold true in more complex formulations. But if it does hold, and string theory does describe our universe, then it must contain another way to stop negative D-membranes from forming.
Other anomalies
Although physicists have made progress in discovering renormalizers and negative membranes, they believe that there are two huge obstacles to making the uptick theory the official successor to perturbation theory.
First, not all theories have proven to have a pick-up structure. For quantum field theory, the problem is particularly acute, and physicists have been examining it case by case. It's a tedious process.
That's why Serone has spent the last three years stress-testing the pick-up method in some quantum field theory. In 2021, he and his collaborators worked on a theory that shared key features with Strong, but was still simple enough to allow them to calculate the many A's needed to perform the pick-up method. They calculated the energy of space in such a universe using the pick-up method and two other methods, proving that the three methods were consistent. The qualitative argument that the pick-up method should apply to quantum field theory is one of the first concrete calculations has further fueled optimism.
The more serious problem is that to discover non-perturbated parts, you need to know a large number of perturbation terms. For example, in his recent research, Serone chose quantum field theory, which lets him generate thousands of terms. But for power, it is currently impossible to calculate only 8 or 9 items. Even the pioneers of the method made no secret of the time when they expected to see it produce a real number (like a proton mass) (a mathematical feat worth millions of dollars).
New hope
However, the daunting difficulties did not kill the dream of getting a real prediction from the regression method. First, this technique is already producing other unobtainable results in quantum mechanics. As early as the 80s of the 20th century, mathematical physicists in France used the primitive upward method to make accurate predictions of particle tunneling, which was a problem that physicists could only approximate before. Another group examined these results using standard methods. They can only reach six decimal places, a difficult job that takes months and requires considerable computing power.
These dramatic examples inspired Dunne's efforts to develop efficient pick-up practice methods that he hopes will one day be used in quantum field theory. Over the past five years, he has worked with Ovidiu Costin, a mathematician at Ohio State University, to find techniques that can play a bigger role in perturbation theory. In some cases (though far from real-world theory), they found that only 10 to 15 items were enough.
For Ecale, the uplift method is a chapter in the past. Almost 40 years have passed since his original trilogy. For the past 20 years, he has been working on a branch that is more algebraic in nature. If he decides to publish a sequel trilogy that collects all his discoveries in one place, who knows what treasures physicists will find in it.