AI and String Theory: Bridging the Cosmic Divide

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Charles Wood

Can String Theory Truly Depict Reality? Artificial Intelligence Might Provide Answers

This story first made its appearance in Quanta Magazine.

String theory won over many scientists years ago with its elegant simplicity. If you look closely enough at a piece of space, according to the theory, you won't find a variety of particles or unstable quantum fields. Instead, you'll find uniform strings of energy that vibrate, combine, and split apart. By the late 1980s, researchers discovered that these strings could only behave in a limited number of ways, opening up the exciting possibility that the journey from these oscillating strings to the fundamental particles of our universe could be mapped. The fundamental vibrations of these strings were thought to create gravitons, theoretical particles that are supposed to weave the gravitational field of spacetime. Different vibrations were believed to lead to the creation of electrons, quarks, and neutrinos. This led to string theory being hailed as a "theory of everything."

"Many believed it was inevitable that we'd eventually be able to calculate all there is to understand," remarked Anthony Ashmore, a string theory expert at Sorbonne University in Paris.

However, as physicists delved into string theory, they encountered a daunting complexity.

As the perspective broadens beyond the simple realm of strings, each transition closer to our complex universe of particles and forces unleashes a vast array of potential outcomes. For the sake of mathematical harmony, strings must navigate a realm of 10-dimensional spacetime. However, since our universe is composed of four dimensions (three of spatial dimensions and one of temporal dimension), string theorists have deduced that the additional six dimensions must be minuscule—wrapped into tiny, loofah-like configurations. These invisible six-dimensional structures exist in an almost infinite diversity. Within these loofah structures, strings converge into what we recognize as the oscillations of quantum fields, and there could be countless methods for these fields to form. Consequently, our universe could be viewed as composed of the elements of these fields that extend out from the miniature loofahs into our expansive four-dimensional existence.

String theory researchers aimed to explore if the elements and dynamics within string theory could form the basis for the array of fundamental particles observed in the actual universe. However, the challenge was not just the incredibly vast number of potential configurations to examine—estimated at 10^500 particularly credible microscopic arrangements—but also the difficulty in transitioning from a detailed setup of dimensions and strings to understanding the resultant universe of particles that would appear.

"Is string theory capable of producing distinct forecasts? Can it genuinely be considered a part of physics? The verdict is yet to be determined," stated Lara Anderson, a physicist from Virginia Tech who has dedicated a significant portion of her career to connecting string theory with particle physics.

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Recently, a new wave of scientists has employed an innovative approach to tackle a longstanding issue: leveraging neural networks, the driving force behind recent breakthroughs in artificial intelligence. In the past few months, collaborations involving physicists and computer scientists have successfully utilized neural networks to accurately predict, for the first time, the type of large-scale universe that would result from a particular microscopic realm of strings. This significant achievement revitalizes a pursuit that had largely lost momentum years ago: the endeavor to ascertain if string theory can indeed encapsulate the reality of our universe.

"According to Anderson, we haven't yet established definitive rules for our universe, but we've made significant progress towards that goal."

The Peculiar Realm of Strings

The key characteristic dictating the formation of the observable universe from string theory hinges on the configuration of the six tiny spatial dimensions.

The most basic configurations of such structures are complex six-dimensional forms known as Calabi-Yau manifolds, which bear a resemblance to loofah sponges. These manifolds are named after Eugenio Calabi, a mathematician who proposed their existence in the 1950s, and Shing-Tung Yau, who initially aimed to disprove Calabi's theory in the 1970s but instead ended up confirming it. Calabi-Yau manifolds are six-dimensional spaces distinguished by two specific features that appeal to physicists.

Initially, these entities are capable of supporting quantum fields characterized by a symmetry referred to as supersymmetry, making the study of these supersymmetric fields less complex compared to their more chaotic counterparts. Findings from experiments conducted at the Large Hadron Collider have indicated that the overarching laws of physics do not adhere to supersymmetry. However, the characteristics of the microscopic realm that extends beyond the conventional model are still a mystery. The majority of string theorists operate on the belief that the universe operates on a supersymmetric basis at this smaller scale, with some leaning on scientific reasons for this belief, while others are driven by the demands of mathematical consistency.

Secondly, Calabi-Yau spaces are characterized by their "Ricci-flat" nature. This concept hails from Albert Einstein's general theory of relativity, which tells us that matter or energy can warp spacetime, creating what is known as Ricci curvature. However, Calabi-Yau spaces do not exhibit this specific type of curvature, even though they may still exhibit other forms of curvature that are not directly related to their matter and energy composition. To grasp the idea of Ricci flatness, imagine a doughnut, which represents a simpler version of a Calabi-Yau space. This doughnut can be flattened out and displayed on a flat surface, where moving beyond the right edge brings you to the left edge, and moving beyond the top edge brings you to the bottom, illustrating the unique spatial properties of these manifolds.

Thus, the overall strategy for string theory essentially involves identifying the particular manifold that would explain the microscopic makeup of spacetime in our universe. One approach to this search involves selecting a likely 6-dimensional torus and determining if it corresponds with the particles observed.

The initial phase involves determining the correct category of six-dimensional toroids. The quantifiable characteristics of Calabi-Yau spaces, like their hole counts, dictate the quantifiable aspects of our universe, including the number of unique particles of matter it contains. (There are 12 in our universe.) Therefore, scientists begin by looking for Calabi-Yau spaces that possess the appropriate mix of quantifiable characteristics to account for the observed particles.

Over recent years, a team based in the United Kingdom has made significant advancements in the precise art of choosing doughnuts, elevating it to a scientific level. Through the use of various computational methods in 2019 and 2020, this collaboration has pinpointed several key equations that reveal categories of Calabi-Yau manifolds. These categories mirror a simplified version of the standard model, accurately incorporating the correct quantity of matter particles. Although these models sometimes predict the existence of long-range forces that are not observed in reality, the British scientists have succeeded in largely automating previously complex and time-consuming calculations.

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"The effectiveness of these techniques is truly astonishing," remarked Andrei Constantin, a physicist from the University of Oxford responsible for uncovering these equations. He noted that these equations "cut down the analysis time required for string theory models from months of computing work to merely an instant."

The next phase presents greater challenges. Proponents of string theory strive to refine their quest, focusing on a specific type among the Calabi-Yau spaces. Their goal is to pinpoint its exact size and the exact positions of all its curves and indentations. These specific geometric characteristics are believed to dictate all other aspects of the larger universe, such as the exact strength of particle interactions and their precise masses.

To advance to the next phase, one must understand the metric of the manifold, which is essentially a function capable of determining the distance between any two points on the given shape. An example of a metric we commonly encounter is the Pythagorean theorem, which outlines the geometry of a two-dimensional plane. However, as we delve into spacetimes that are higher-dimensional and curved, the metrics evolve into more intricate and complex representations of the geometry. While physicists have managed to derive the metric for a singular rotating black hole within our four-dimensional reality, the challenge of deciphering metrics in six-dimensional spaces has proven too daunting. Toby Wiseman, a physicist at Imperial College London, expressed a poignant sentiment regarding this limitation, noting, “It’s one of the most disheartening experiences as a physicist to encounter situations where, despite the sophistication of mathematics, we find ourselves constrained in our ability to formulate solutions to certain equations.”

In the early 2000s, while working as a postdoctoral researcher at Harvard University, Wiseman became intrigued by the elusive metrics of Calabi-Yau manifolds, a topic often considered legendary within the field. The groundbreaking work of Yau, which confirmed the existence of these metrics and contributed to his winning the prestigious Fields Medal in mathematics, had yet to be quantified by anyone. Wiseman, who was already employing computational methods to approximate the metrics of spacetimes around unique black holes, wondered if these same techniques could be applied to uncover the metrics of Calabi-Yau spacetimes.

As there is no

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Wiseman recounted, "People were skeptical, telling us, 'That's impossible.' But Matthew Headrick, an expert in string theory, and I proved them wrong by demonstrating it was feasible."

Digitized Geometries

In their work, Wiseman and Headrick (the latter being affiliated with Brandeis University) were aware that for a Calabi-Yau metric to be valid, it had to satisfy the equations Einstein set forth for a vacuum. A metric that meets this criterion ensures that the spacetime it describes is devoid of any curvature in the Ricci sense. Choosing a four-dimensional space as their experimental field, they utilized a numerical method often introduced in senior high school math courses. Their findings in 2005 demonstrated that it was possible to closely approximate a 4D Calabi-Yau metric. Although it wouldn't be completely uniform across all points, it would be nearly so, akin to a doughnut displaying only minutely noticeable flaws.

"The idea occurred to me that if a neural network has the capability to surpass the world champion in Go, it might also have the potential to excel beyond mathematicians, or at the very least, physicists such as myself."

During the same period, Simon Donaldson, a renowned mathematician at Imperial College, was examining Calabi-Yau metrics for mathematical purposes and quickly developed an alternative algorithm for metric approximation. String theorists, including Anderson, began attempting to compute specific metrics using these methods. However, the processes were time-consuming and resulted in excessively irregular doughnuts, thwarting efforts to accurately predict particle behavior.

Efforts to finalize the second phase came to a halt for almost ten years. Meanwhile, as scientists concentrated on the initial phase and addressed various challenges within string theory, a significant breakthrough in computational science emerged—neural networks. This innovation involves tweaking vast arrays of numerical values until they can effectively represent an otherwise indeterminate function.

Neural networks discovered algorithms capable of recognizing objects within pictures, converting spoken language across different tongues, and conquering the most challenging board games known to humans. The development of the AlphaGo algorithm by the team at DeepMind, an artificial intelligence firm, captured the attention of physicist Fabian Ruehle, especially after it defeated a leading Go champion in 2016.

Ruehle, currently at Northeastern University, reflected, "Considering its ability to surpass the world champion in Go, it led me to wonder if it could also excel beyond mathematicians, or at the very least, physicists such as myself."

Ruehle and their team revisited the longstanding challenge of finding approximate solutions for Calabi-Yau metrics. Similarly, Anderson and his team reengaged with their previous efforts to address a significant hurdle in their research. They discovered that leveraging neural networks could provide the efficiency and adaptability that past methods were missing. These algorithms had the capability to propose a metric, evaluate the curvature at numerous points across a six-dimensional space, and refine the prediction until the curvature was effectively zero across the entire manifold. The only requirement for the researchers was to modify existing machine learning tools to fit their needs; by the year 2020, several research groups had developed and shared their own specialized software for calculating Calabi-Yau metrics.

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Having access to metrics enabled physicists to explore the detailed characteristics of vast universes associated with each manifold. "Once I had it, the initial step I took was to compute the particle masses," Ruehle mentioned.

In 2021, Ruehle and Ashmore made significant progress by determining the masses of theoretical heavy particles, which are solely influenced by the shapes of Calabi-Yau spaces. However, these particles are predicted to be so massive that they are beyond the reach of current detection methods. To tackle the longstanding challenge of calculating the masses of known particles, such as electrons, which string theorists have been pursuing for many years, the application of machine learning techniques would need to be enhanced.

Particles of minimal mass gain their mass by interacting with the Higgs field, an omnipresent energy field. The interaction's intensity with the Higgs field determines the particle's mass; the more attention it pays to the Higgs field, the more massive it becomes. This interaction strength is measured by something known as the Yukawa coupling for each particle. In the realm of string theory, the Yukawa coupling is influenced by two main factors. The first is the geometry of the Calabi-Yau manifold, which can be likened to the contours of a doughnut. The second factor involves how quantum fields, which emerge as clusters of strings, are distributed across this manifold. These quantum fields can be thought of as similar to sprinkles on a doughnut, their placement not only depends on the doughnut's shape but also has a degree of independence.

Ruehle and his team of physicists introduced software solutions that were able to model the toroidal shape. The final challenge was to model the toppings, and artificial neural networks were up to the task. Earlier in the year, two groups successfully integrated all components.

A team spearheaded by Challenger Mishra from the University of Cambridge pioneered the use of a custom-built neural network to determine the shape and structure of a torus. Following this, they developed and used unique algorithms to analyze how quantum fields interact and intertwine over the shape, similar to the way toppings cover a doughnut. Crucially, their research took place under conditions where the physical properties of the fields are closely intertwined with the shape's geometry, enabling them to explore new methods for calculating Yukawa couplings, something that hadn't been attempted previously. The team's calculations using both methods aligned perfectly. Additionally, the results they uncovered suggested a distinction in the masses of particles, shedding light on a perplexing aspect of the standard model.

"Mishra mentioned that this has been a desire of people long before his birth in the 1980s."

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A team, including string theory experts Burt Ovrut from the University of Pennsylvania and Andre Lukas from Oxford, took significant steps forward. They began their work utilizing Ruehle’s software for metric calculation, a project Lukas had contributed to. From this initial step, they integrated a sophisticated set of 11 neural networks designed to manage various "sprinkles" configurations. This innovation enabled them to compute multiple fields that could adopt a wider range of shapes, thus providing a more accurate simulation that is unachievable by other methods. This collective of neural networks mastered the metric and field layouts, determined the Yukawa couplings, and determined the masses of three different quark types. This comprehensive analysis was performed across six uniquely shaped Calabi-Yau manifolds. Anderson highlighted the unprecedented accuracy of these calculations, marking a first in the field.

None of the Calabi-Yau spaces form the foundation of our universe due to the fact that two quarks share the same mass, in contrast to the six types existing in our universe which are distinguished by three levels of mass. Instead, these findings demonstrate a conceptual validation that machine-learning techniques can guide physicists from a Calabi-Yau space down to the precise masses of particles.

"Up until this point, performing these calculations was considered impossible," remarked Constantin, who is part of the team stationed at Oxford.

Game of Digits

The artificial neural systems struggle to process doughnuts that contain more than a few holes, and the scientists aim to eventually examine manifolds that consist of hundreds. Up to this point, the scientists have only explored relatively straightforward quantum fields. To fully reach the standard model, according to Ashmore, "a more advanced neural network might be necessary."

Greater challenges are on the verge of emerging. The quest to locate aspects of particle physics within string theory's solutions — assuming they exist there — is akin to playing a lottery. The more varieties of decorated doughnuts you examine, the higher your chances of discovering a corresponding match. After years of persistent work, string theory researchers are now capable of analyzing these doughnuts and comparing their findings with the actual properties of fundamental particles that we detect. However, even the most hopeful among these theorists admit that stumbling upon a match purely by chance is incredibly unlikely. The sheer quantity of Calabi-Yau manifold varieties, which might be endless, further complicates matters. “You need to figure out how to manipulate the system,” Ruehle advised.

A strategy involves examining myriad Calabi-Yau manifolds to identify any recurring themes that might guide the investigation. By manipulating these manifolds, altering their dimensions in various manners, physicists could gain a fundamental understanding of how specific configurations correspond to certain particles. "Ideally, after scrutinizing specific models, you're armed with solid logic," Ashmore explained, "leading you to discover the accurate model that represents our universe."

Lukas and his team at Oxford are set to embark on a journey of discovery, focusing their efforts on their top doughnut models and tweaai-allcreator.com">king the details to identify a mathematical structure capable of generating a believable array of quarks. Constantin is optimistic that within a few years, they will discover a structure that accurately reflects the masses of other known particles.

"To spice things up, it would be great to include a few fresh predictions about physical phenomena."

Some other researchers specializing in string theory believe it's too early to begin examining specific manifolds closely. Thomas Van Riet, a researcher at KU Leuven focused on string theory, is exploring the "swampland" research initiative. This project aims to pinpoint common characteristics across all mathematically coherent solutions in string theory, including the notably feeble nature of gravity compared to other forces. Van Riet and his team are working to dismiss large segments of potential string theory solutions—essentially, possible universes—prior to delving into detailed analyses of individual shapes and features.

Van Riet expressed approval for the engagement in machine learning, acknowledging its future necessity. However, he emphasized the importance of first understanding the foundational concepts and patterns, suggesting that the focus on specifics comes secondary.

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Numerous physicists have shifted their focus from string theory to explore alternative quantum gravity theories. The advancements in machine learning are not expected to draw them back into the fold. Renate Loll, a physicist working at Radboud University in the Netherlands, mentioned that for string theory to genuinely capture attention, it must not only predict but also verify new phenomena that go beyond the established standard model. "The quest is akin to finding a needle in a haystack, and it's uncertain what insights we would gain from it, even if there were compelling, quantitative proof that achieving the standard model is feasible," she stated. "For it to be truly compelling, it must offer some novel physical predictions."

The primary aim for numerous machine learning enthusiasts is to achieve groundbreaking forecasts. Their aspiration is that string theory will demonstrate a level of inflexibility, implying that any versions resembling our cosmos will share certain traits. Such versions could potentially all possess a unique type of particle, presenting an opportunity for experimental investigation. However, at this stage, this remains a hopeful ambition, with no guarantee of success.

"String theory is incredible. A lot of string theorists do amazing work. However, when it comes to making qualitatively accurate predictions about the universe, their success rate is quite poor," stated Nima Arkani-Hamed, a theoretical physicist based at the Institute for Advanced Study in Princeton, New Jersey.

In the end, the mystery of string theory's predictions continues to be unresolved. Currently, as string theorists utilize the capabilities of neural networks to bridge the six-dimensional micro-universes of strings and the four-dimensional macro-universes of particles, they are more likely to eventually find an answer.

Anderson remarked, “Certainly, a multitude of string theories exist that bear no relevance to the natural world. The real inquiry is whether any of them actually relate to it. While the response could be negative, I find it quite fascinating to challenge the theory to make a determination.”

This article is republished with authorization from Quanta Magazine, a publication autonomously operated by the Simons Foundation. Its goal is to broaden the public's grasp of science through reports on emerging research and advancements in the fields of mathematics, physics, and the biological sciences.

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