AI breakthrough in theoretical physics: model bridges theory and measurement

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One of the greatest challenges — and perhaps the most important challenge of all — for any theoretical physicist is to connect something you can predictively calculate to something you can go out and measure experimentally or observationally. This is the key not just to testing your current best understanding of reality, but to discovering any anomalies that could pave the way forward to the next revolution in our conception of the Universe. Most of the great revolutions in physics history have arisen where theory and observation/experiment have failed to match, including:

the mismatch in Uranus’s predicted orbit from Kepler’s laws leading to the discovery of Netpune,

the extra precession observed over what Newton predicted for Mercury’s orbit leading to general relativity ,

and the failure of classical physics to predict the short-wavelength behavior of a radiating blackbody leading to the start of the quantum revolution.

Over the past few generations, quantum field theory is how we arrive at the most precise physical predictions ever made. The act of testing those predictions against particle physics experiments has led to incredible discoveries, the spectacular confirmation of the Standard Model, and drives our great hopes for finally going beyond it. But calculating observables like cross-sections and scattering amplitudes to greater and greater precisions is computationally hard; physicists have been stuck for many years, unable to surpass the current limits.

That’s why it’s so remarkable that an LLM-based AI has just done what humans hadn’t been able to do on their own: calculate an important theoretical result in quantum field theory to a greater loop-order than any human, even with the aid of computers, had ever done previously. Here’s the story of what just happened, and why it matters.

This diagram illustrates the inherent uncertainty relation between position and momentum. When one is known more accurately, the other is inherently less able to be known accurately. Both position and momentum are better described by a probabilistic wavefunction than by a single value. Other pairs of conjugate variables, including energy and time, spin in two perpendicular directions, or angular position and angular momentum, also exhibit this same uncertainty relation.

Credit : Maschen/Wikimedia Commons

One of the most important ways of conceiving of our Universe is through the lens of quantum field theory. In order to understand what this breakthrough is all about, we have to understand what’s so important (and different) about quantum field theory from all prior conceptions of reality.

Our original view of physics was Newtonian: where reality is made up of particles with well-defined properties like mass, position, and velocity, and where those particles move through space with the passage of time. As a consequence of Einstein, we learned that space and time are linked, and that space itself is not flat, static, and unchanging, but rather can be curved and can evolve. (That notion of spacetime now serves as the backdrop in both classical and quantum physics.)

The main insight of the quantum revolution was that what we think of as particles don’t necessarily have fixed properties, but rather things like “where they are,” “how fast they’re moving,” and even (if they’re unstable) “how much they weigh” follow a probability distribution instead. Only when you make a measurement — via the interaction of that quantum with another one — to actually find out what a particular particle’s properties are is a particular value determined for one particular property. But if you repeat the experiment over and over again, even with identical initial conditions, you won’t get that same result over and over again; you’ll get a probability distribution for your results.

The inherent width, or half the width of the peak in the above image when you’re halfway to the crest of the peak, is measured to be 2.5 GeV: an inherent uncertainty of about +/- 3% of the total mass. The mass of the particle in question, the Z boson, is peaked at 91.187 GeV, but that mass is inherently uncertain by a significant amount owing to its excessively short lifetime. This result is remarkably consistent with Standard Model predictions, and illustrates the energy-time uncertainty inherent to quantum physics.

Credit : J. Schieck for the ATLAS Collaboration, JINST7, 2012

You can make quantum physics relativistically invariant, and arrive at things like the fine-structure of atoms, intrinsic angular momentum, magnetic moments, and even negative-energy states: identified as the existence of antimatter. But there’s a further step we can take beyond making the matter, antimatter, and energy of the Universe quantum in nature: we can make the underlying fields behind the fundamental forces quantum in nature, too.

That’s how quantum field theory goes a step further than quantum physics . If you imagine you have a particle that’s electrically charged, if its position and motion are perfectly well-determined, you can write down values at every point in space and at every moment in time for the electric and magnetic fields that it generates. But if there’s an uncertainty in those values (position and motion) for the particle, due to its quantum nature, then it isn’t so simple to just write down values for those fields through space and time. If the fields are generated by quantum particles, then the fields must be quantum in nature, too.

In fact, even if you take the particles away, and just leave yourself with empty space, those fields — still quantum in nature — must persist. That is the essence of quantum field theory: that there’s an inherently quantum reality that doesn’t just apply to particles, but to the fields associated with each of the fundamental forces that make up our reality.

If you have two conductors with equal and opposite charges on them, it’s an exercise in classical physics alone to calculate the electric field and its strength at every point in space. In regular (Schrodinger-like) quantum mechanics, we discuss how particles respond to that electric field, but the field itself is not quantized as well. This seems to be the biggest flaw in the original formulation of quantum mechanics.

Credit : Geek3/Wikimedia Commons

This advance — of treating the fields that underpin reality as inherently quantum in nature — is profound. Just like a particle don’t have “properties” like position and momentum, but rather has its position and momentum described by a quantum operator, so too do properties that we think of classically like an electric or magnetic field. Instead of having values at all points in spacetime, they behave as quantum operators everywhere in space and all across time. When you treat the fields that govern reality as quantum mechanical operators, you immediately recover a whole slew of predictions that accurately describe our reality:

the creation and annihilation of particles and antiparticles,

the radioactive decay of unstable particles,

the spontaneous production of electron-positron pairs,

and the emergence of quantum corrections to the intrinsic magnetic moment of even fundamental particles, like the electron and muon.

These quantum fields are present everywhere, and allow for the spontaneous creation and/or annihilation of quantum particles in probabilistic fashion. This implies that even a quantum particle just doing something simple — traveling through space freely, interacting with electric and/or magnetic fields, or encountering another quantum — can be visualized as having the potential to interact with spontaneously generated particles, antiparticles, and other quanta, albeit merely “virtually” or in a calculational sense.

Visualization of a quantum field theory calculation showing virtual particles as field fluctuations in the quantum vacuum. (Specifically, for the strong interactions.) Even in empty space, this vacuum energy is non-zero. If there are additional particles or fields beyond what the Standard Model predicts, they will affect the quantum vacuum and will change the properties of many quantities away from their Standard Model predictions. However, the QCD contribution cannot be calculated perturbatively, the way electromagnetism can.

Credit : Derek Leinweber

One set of mathematical tools that we use to represent this process are known as Feynman diagrams. In Feynman diagrams, we know what the incoming and outgoing quantum states are: we know what’s “coming in” to the interaction (which could be as simple as one particle entering, or as complex as many particles), and we know, and can measure, what’s “exiting” the interaction (which could be the same number of particles that entered, or more or fewer), while still obeying all of the quantum laws of nature.

The simplest Feynman diagram that we can draw is known as a tree-level diagram: where you connect the incoming and outgoing particles together, where they simply interact at an intersection point that we know mathematically as a vertex. But you can also have more complicated diagrams that arise from the nature of these quantum fields: what are known as loop diagrams, with more than one vertex and paths for particles to “loop” around the connections between vertices. In a quantum field theory, the tree-level diagram represents the standard quantum-mechanical prediction, but adding loops into it — which we visualize as the tree-level diagram’s branches exchanging additional quanta between them — allows us to begin to probe the quantum field theory predictions that take us beyond the capabilities of quantized particles alone.

Today, Feynman diagrams are used in calculating every fundamental interaction spanning the strong, weak, and electromagnetic forces, including in high-energy and low-temperature/condensed conditions. The electromagnetic interactions, shown here, are all governed by a single force-carrying particle: the photon, but weak, strong, and Higgs couplings can also occur. Diagrams here are shown for tree-level, one-loop, and up to two-loop order.

Credit : V. S. de Carvalho and H. Freire, Nucl. Phys. B, 2013

This is important for physical reasons! When the number of incoming and outgoing particles remains unchanged from the tree-level diagram, we get contributions from the loop diagrams that alter the expected, theoretical values for observable properties like:

cross-sections for interactions between the incoming quanta

and scattering amplitudes for the outgoing quanta.

When the number of outgoing particles changes, usually by adding more outgoing particles, we arrive at predictions for observable properties like:

decay pathways for unstable particles,

and branching ratios, which are the probability percentages for each decay pathway.

Whenever you can connect your theoretical predictions, which you arrive at by performing these quantum field theory calculations, and your observables, which you arrive at by conducting experiments and measuring the outcomes of particle interactions, that is the battlefield on which the science of physics is conducted. That’s what the enterprise of particle physics is all about: making theoretical predictions on one hand, and comparing them with experimental or observable results from the measurements you can take involving actual particles.

The total proton-antiproton cross section as a function of energy, versus the comparable proton-proton cross-section. At high energies, at ~200 GeV and up, these cross-sections are identical. But at lower energies, the opposite electric charges of the proton and antiproton, as well as their potential to form a bound state, dominate and lead to larger cross-sections for protons and antiprotons. Quantum field theory is needed to calculate these quantities like cross-sections and scattering amplitudes as a function of energy, so that theory and observations can be compared.

Credit : R.M. Godbole, A. Grau, & G. Pancheri, AIP Conf. Proc., 2001

The beauty and power of the Feynman diagram approach is that, particularly for forces like electromagnetism and the weak interaction, you get more and more precise and accurate results with each successive “loop order” that you go to. This means that:

if you can write down (i.e., without omitting/forgetting any) and correctly calculate all of the one-loop diagrams in addition to the tree calculation, you’ll get a more accurate answer than if you only do the tree calculation,

if you go to two-loop order, writing down and correctly calculating all of the two-loop diagrams and the one-loop diagrams and the tree-level diagram, you’ll get a more accurate answer than from the one-loop and tree diagrams alone,

and so on. For electromagnetism, you’d have to go to more than the thousandth-loop order before this method broke down!

For most real-world-relevant calculations, going to two-loop order is very impressive. The scattering amplitudes at the LHC typically go no farther than that. The most precise calculation ever done in particle physics, for the magnetic moment of the electron , has gotten up to five-loop order, which took an enormous effort over many years by a whole community of people. Even for electromagnetism, which is the simplest of the fundamental forces in terms of calculational ease, calculating high-loop order diagrams is an incredibly complex task.

In quantum electrodynamics, higher-order loop diagrams contribute progressively smaller and smaller effects. However, as the energy increases, these higher-order processes become more efficient, and thus the value of the fine-structure constant increases with energy. The five-loop contributions in QED, shown here, represent the highest loop order ever achieved for the physical theories that describe and are applicable to our Universe.

Credit : T. Aoyama et al., Phys. Rev. Lett., 2012

That’s why physicists who want to study high-loop order aspects of quantum field theories develop special “toy models” that don’t actually reflect the real world where calculations are easier. By calculating what’s impractical to calculate within our Standard Model, they hope to reveal a key aspect of nature that’s beyond our current means to calculate conventionally, but that leads to some insight that’s actually relevant to our own Universe. Examples include scalar field theory, scalar quantum electrodynamics, and more recently, supergravity and super Yang-Mills theories with different numbers of “supercharges” (or number of supersymmetric particles for every normal particle, defined by the letter 𝒩) inherent to them.

Of particular interest to theorists working in this field these days is 𝒩 = 8 supergravity and 𝒩 = 4 super Yang-Mills, which led theoretical physicist and science writer Matt von Hippel to issue a challenge to LLM-based AIs: can you go past the current limits of 𝒩 = 8 supergravity and calculate it to seven loops, or 𝒩 = 4 super Yang-Mills and calculate it to nine loops?

Only a few weeks after issuing that challenge, the results have come back. Whereas almost everyone in the field had expected the problem to be a computational bottleneck, regardless of technique, several theorists, working in collaboration with a frontier model of Claude AI at Anthropic labs, successfully calculate 𝒩 = 4 super Yang-Mills to nine loops . And, in addition to achieving what was widely thought to be impractical, they did it on a budget: using only thousands of dollars worth of compute.

When the LLMs o3, Claude Sonnet 4, and Gemini 2.5 Pro were asked to reconstruct force laws for a variety of mock solar systems back in 2025, they were all unable to recover something equivalent to Newton’s law of universal gravitation, despite the LLMs themselves having been trained on Newton’s laws. There have been fantastic improvements since then, with Claude Fable, running 5.1 and in science mode, achieving theoretical results that even professionals in even more advanced fields were unable to make.

Credit : K. Vafa et al., ICML 2025/arXiv:2507.06952, 2025

Context, of course, is important here. This is only a toy model, but one that can be very insightful for learning about the Universe we actually inhabit in many ways. This particular theoretical model is finite and scale invariant, and is done in the planar (two-dimensional) limit of the full four-dimensional theory. But 𝒩 = 4 super Yang-Mills is interesting among theorists for many reasons: it is the most supersymmetric theory possible, and it is a mandatory part of the story of string theory.

It’s also worth noting that, from 2011 to 2023 , the ability to calculate this six-vertex diagram went from three-loops up to eight-loops, and that Lance Dixon , a leading figure in this specialty, already had built the theoretical framework necessary to check the nine-loop solution if one were presented to him, and had developed two techniques — a bootstrap technique and a form factor technique — to increase the loop-order that was possible in the past.

Two physicists at Anthropic, Liam Fitzpatrick and Siddharth Mishra-Sharma, reported to Matt von Hippel that his challenge was successfully accomplished, and with minimal human input. They prompted Fable 5.1 within Claude Science, which is a specialized platform that scientists can pay to use, and that has built-in structural rules and additional built-in prompting that is designed to yield more scientifically robust results. They gave the LLM a single, simple initial prompt:

“The problem is to compute the Six-particle (hexagon) amplitude in planar 𝒩=4 SYM at nine loops.”

They would tell it to keep going, asking for updates, and not much else. After enough compute time and enough persistence, Claude got the answer two different ways: through the bootstrap and the form-factor approach. Afterwards, Lance Dixon indeed verified it. It was correct.

This set of diagrams shows the tree-level 6-vertex diagram in 𝒩 = 4 super Yang-Mills, alongside the various loop-order diagrams up to nine-loop order. After achieving eight-loop order in 2023, humans had yet to reach nine-loop order until Claude did it in August and September of 2026.

Credit : Anthropic

Lance reported receiving the solution on September 1, 2026, which means it was just 25 days from when the challenge was issued to when the solution was presented to the validator: an incredible achievement. Sure, there are limitations here: the LLM relied heavily on existing work, delivered no fundamentally new insights, developed no new techniques, and was working on a problem that other, independent groups had also mostly already solved . However, according to Dixon, who also developed the technique of leveraging what he and Andy Liu called antipodal duality to get 2023’s eight-loop results:

“I thought it would be too hard to do the [nine-loop] amplitude directly. So I was really quite impressed that Claude could do it directly. Not so much because it was a big computational task, but because the whole setup is very fragile: if you make any mistake at all in the computational recipe, it all crashes down like a failed soufflé, and you are left to wonder why (and debug). Also, there are so many details of the construction that are too boring to document fully in a publication. So Claude had to develop all that code from scratch.”

Fitzpatrick and Mishra-Sharma have published the results they obtained with Claude here , and one can only marvel at this accomplishment. In many ways, this is the most significant breakthrough in theoretical physics ever made by an LLM, and it makes one wonder just what sorts of problems, thought to be too difficult to solved, can be tackled by existing LLMs. Perhaps, someday soon, greater theoretical precisions will arrive for quantum electrodynamics, frontier physics at the Large Hadron Collider, and potentially even non-perturbative problems like lattice QCD or current approaches to quantum gravity. With this first meaningful breakthrough now behind us, one can only wonder at what might come next.

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