In September 1976 physicist Masao Doi, then working at Tokyo Metropolitan University, published two groundbreaking papers. The work – published in Journal of Physics A – cast classical stochastic many-body problems in the mathematical language samiliar to that of quantum-field theory (9 1465 and 9 1479).

Although this happened at a time when theoretical field methods in statistical mechanics were getting into full swing, this gem of deep scientific insight remained hidden for many years.

Indeed, it was reinvented several times, most notably in 1980 by Peter Grassberger and Martin Scheunert from the University of Wuppertal, who also highlighted the many applications for the theory.

Following conversations with Grassberger at a meeting organised by the Centre Européen de Calcul Atomique et Moléculaire in September 1985, Luca Peliti from the University of Rome went about putting Doi’s theory into a path integral formalism, thus making it amenable to renormalisation.

That same year he published a letter on the renormalisation group treatment of a reaction-diffusion process, using the framework that was named Doi-Peliti field theory (J. Phys. France 46 1469).

The works by Doi and Peliti quickly became viewed as fundamental building blocks to calculate not only asymptotics but also systematic approximations of observables in reaction-diffusion processes.

Theoretical physicist John Cardy from the University of Oxford, used Doi-Peliti theory as a work-horse for reaction-diffusion processes.

In the 2000s, Cardy and colleagues produced not only a string of renormalisation group treatments of reaction-diffusion processes, but also reviews and tutorials at different levels of technicality, which made Doi-Peliti field theory accessible to many of us.

Key insights

In 2018 Doi became chief scientist at the Wenzhou Institute of the University of Chinese Academy of Sciences and was well-known for his work in polymer and soft-matter physics.

Yet when my former PhD student Ziluo Zhang joined the Wenzhou Institute as a postdoc in 2023, he found that Doi was unaware that the field theory bears his name and the many developments since its inception.

Indeed, Doi and Peliti had never met until a meeting from 22 to 25 September at the University of Cambridge that celebrated Doi and Peliti’s work.

“[It was ]the first time that I attend a conference which has my name on it,” noted Doi.

Funded by the Wildcard programme of the German Carl Zeiss Foundation, the meeting featured talks on historic developments and the many achievements and challenges of the Doi-Peliti framework.

In a relaxed atmosphere, some of the key players explained how their insights came about.

Doi-Peliti field theory is a brilliant concept conceived by brilliant minds. One might be tempted to speculate how many more theories have been spun off and developed that are attributed – or even unattributed – to the pair without them ever knowing.

• An upcoming special issue of Journal of Physics A will be dedicated to the Doi-Peliti field theory.

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Two theoretical physicists – including a Nobel laureate – have solved a mathematical problem that flummoxed them for 10 years with help from Claude, the large language model (LLM) developed by the US-based firm Anthropic. In addition to its importance for fundamental physics, the result is significant because it offers further evidence that artificial intelligence (AI) is reshaping the nature of research in some fields.

The problem that kept Francesco Zamponi and Giorgio Parisi awake at night originates in the field of complex systems and relates to a phenomenon known as jamming. “Jamming describes a sudden transition at which a fluid system becomes completely rigid, yet remains disordered,” explains Zamponi, a physicist at the Sapienza Università di Roma, Italy.

To visualize this, Zamponi suggests imagining a box filled with spheres (such as tennis balls) that are floating in zero gravity, free from friction or weight. When there are only a few spheres in the box, they have plenty of room to move, bouncing off each other and the box like a classical gas. However, if you add more spheres or inflate the existing ones, space becomes increasingly scarce, and a sphere that tries to move soon impinges on its neighbours. Eventually, a critical density is reached, and every sphere is securely held by those around it, causing the entire system to “lock up” into a state where no more movement is possible.

If the spheres freeze into a regular, geometric pattern, they form a crystal-like structure that resembles carefully stacked oranges at the market. If they are compressed or added quickly, though, a crystal cannot form. Instead, they freeze into a completely disordered packing state, like a traffic jam.

Generalizing the problem

Moving beyond physical spheres into more abstract territory, Zamponi notes that jamming is an example of a highly general mathematical problem known as constraint satisfaction. “In this framework, the spatial positions of the spheres act as the variables, or degrees of freedom, while the strict rule that no two spheres can overlap serves as the constraint,” he explains. As more variables are incorporated, Zamponi adds, satisfying all the rules simultaneously becomes increasingly difficult.

Because of this precise mathematical mapping, jamming concepts are widely applied in neuroscience and artificial intelligence. In a machine learning model or a biological brain, for example, the synaptic weights represent the degrees of freedom, while the data classifications or memories that must be learned serve as the constraints.

“Just like physical spheres, an artificial neural network undergoes a jamming transition,” Zamponi observes. “In the ‘liquid’ phase, the network easily configures its weights to satisfy all constraints, allowing it to classify data perfectly. But when tasked with too much data, it hits a wall, corresponding to the ‘solid’ phase. It can then no longer satisfy all the conditions, and it begins to make errors.”

A surprising relation

In 2014, Parisi, who received half of the 2021 Nobel Prize in Physics for “the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales”, and Zamponi spotted a surprising relation in the theory of jamming. Working with colleagues in the US and France, they found that in numerical calculations, two mathematical parameters, denoted a and b and related to the scaling of the distribution of contact forces and inter-sphere gaps as the system reaches the jamming point, mysteriously add up to 1. These parameters are very important for characterizing the physical structure of the packing, but try as the researchers might, they could not obtain a formal mathematical proof that a + b = 1.

Asking generative AI for help was Parisi’s idea. The problem was well-suited for machine assistance because it was well-defined, with a clear conjecture and a known numerical answer but no analytical proof. He and Zamponi chose Claude because it boasts more advanced coding and mathematical reasoning capabilities than other models of its kind.

Even so, they didn’t ask it for the proof immediately. “We first prompted Claude to replicate the numerical calculations our group had developed a decade ago,” Zamponi explains. “Once it had successfully reproduced those exact results, we took the natural next step and asked it: ‛If a + b = 1, can you prove why?’”

The model produced an initial conceptual path that was essentially correct, but that contained minor mathematical errors and required several iterative refinements and verifications. Still, Zamponi says, the “core intuition” belonged to the AI: “We had spent years looking for a complicated solution – like deeply hidden structural symmetry – but Claude showed us that the solution was far simpler; it was right there in front of us, but we had just missed it.

“The new proof connects our infinite-dimensional theory, which is more abstract but mathematically rigorous, with a framework developed at the same time by our colleague Matthieu Wyart and his team at the École Polytechnique Fédérale de Lausanne (EPFL),” Zamponi continues. “Wyart’s theory is built on more concrete physical notions, but it relies nevertheless on some assumptions.” The new result, he tells Physics World, confirms that both starting points lead to the same physical laws.

“More important than the industrial revolution or the birth of the Internet”

Zamponi believes that the advent of generative AI could be on par with, or even more important than, the industrial revolution or the birth of the Internet. As a “telescope for the mind”, he says it could become indispensable for research, allowing scientists to test ideas at unprecedented speeds. It could also lower specialization barriers and facilitate interdisciplinary work by unlocking complex literatures that would otherwise take too long to master.

On the downside, though, Zamponi is concerned that LLMs also encourage the production of poor-quality, pseudo-scientific research, straining the peer-review system. “We desperately need to figure out how to filter and review this influx,” he says. “We also need to design novel pedagogical frameworks to responsibly integrate tools like Claude into undergraduate and graduate research courses.

“To use these tools responsibly, we believe transparency is key. This is why we chose to publish our full conversation with Claude alongside our paper. If an AI sparks a genuinely new idea, we believe that its ‘thought processes’ should be public record.”

Jamming and random sequential absorption

Zamponi is now applying the approach detailed in the new work, which is published in Journal of Statistical Mechanics: Theory and Experiment, to a problem involving the random sequential absorption (RSA) of hard hyperspheres. This classic protocol for generating random packings involves introducing spheres one by one at completely random positions. “It is a vital tool for understanding the geometry of void spaces and packing efficiency, which directly relates to finding optimal error-correcting codes and navigating the mathematical ‘curse of dimensionality’,” Zamponi says.

While both RSA and jamming transitions describe how systems of hard spheres lock up, they represent two fundamentally different physical processes, he explains. In traditional jamming, particles are mobile and fluid and they continuously rearrange and relax until the entire system undergoes a collective, sudden transition into a rigid, solid-like state. RSA, on the other hand, is a strictly out-of-equilibrium, irreversible process where particles never move once placed. RSA therefore reaches a “saturation” limit rather than a true collective jamming transition.

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Science fiction has long embraced the idea of travelling backward in time, but the advent of Einstein’s general theory of relativity transformed these ideas from fantasies into potential – albeit contested – realities. In particular, solutions to the equations of general relativity known as closed time-like curves (CTCs) seem to allow a system’s trajectory to return to a previous point in time. Although the existence of CTCs has never been proven, their admissibility within general relativity has inspired many physicists to study their implications, which include increases in information processing speeds as well as retrograde time travel.

Inspired by the 2014 film Interstellar, in which a father sends messages to his daughter in the past, Kaiyuan Ji and Mark Wilde of Cornell University, together with Seth Lloyd at the Massachusetts Institute of Technology, have now investigated the potential advantages of a retrograde messaging channel based on a CTC. As well as deriving an exact expression for the information capacity of such a channel, they showed that this capacity exceeds that of regular channels that do not involve backward time travel.

Post-selected CTC models

To model their CTC, the researchers used an approach that Lloyd and his then-collaborators developed 15 years ago. This approach exploits the mathematical equivalence between a CTC and the combined operations of quantum teleportation and post selection, so it is often referred to as a post-selected CTC model. Previous analyses established that this model disallows time-travel paradoxes such as going back in time to kill your grandfather, while still preserving correlations between the system that travels backward in time and the environment.

The latter feature is important, Ji notes, because quantum CTC models without it have a distinct drawback: “If you travel through that closed time-like curve, you end up in the past, but basically you lost all the memory of what has happened before you do time travel,” he explains.

A further advantage of the team’s chosen model is that it does not prohibit causal loops in which correspondents in the past and future influence each other. To return to the Interstellar scenario, Ji explains that while the daughter is influenced by her father’s message, she also influences her father, because he witnessed how she decoded the message and therefore used that information to optimize his encoding protocols.

The existence of causal loops in the model allowed Ji and his colleagues to formulate an expression for the channel’s bit capacity. It also pointed towards the best way of maximizing that capacity. The optimal strategy, Ji explains, would involve the father using his memory from the past when encoding his message to the past, which makes sending messages to the past more efficient (and capable of a higher bit capacity) than sending messages to the future. “The causal loop is essential to the design of the optimal communication protocol in this retro causal communication setting,” he tells Physics World. “It is not possible in standard communication from the past to the future.”

Capacity for messages

The CTC can also be used to form a quantum channel capable of transmitting either regular classical data or quantum data back in time. Using their expression for the channel’s bit capacity, the researchers were able to show that its classical bit capacity is twice that of its quantum counterpart.

Nicole Yunger Halpern, a theoretical physicist at the University of Maryland who has studied some implications of CTCs for metrology but was not involved in the current work, describes the research as “creative and technically impressive”. She points out that while calculating a communication channel’s capacity is a common challenge in the field of information theory, obtaining straightforward answers can be difficult. “The authors combined this workaday task with a highly unusual (retrocausal) setting and, moreover, obtained a simple solution,” Halpern says.

Not everyone is sold on the mathematical format of post-selected CTC models, though. Scott Aaronson, a computer scientist at the University of Texas at Austin, US, who has previously studied the speed advantages of information processing using CTCs, believes that studies with this approach “attempt to model time travel but don’t fully succeed at it”. However, with respect to the question of channel capacity, Aaronson concedes it may be “interesting to prove things”.

While the current model considered the effects of distortion (“noise”) on messages in the channel, Ji suggests it might be interesting to broaden this to include noise in the memory of the process – like the father in Interstellar having a hazy recall of his daughter’s decoding protocol. Ji also notes that the post-selection CTC model has some mathematical equivalence with black hole final state projections, and he says it might be interesting to explore these connections, too.

The research is described in Physical Review Letters.

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Introduction: “Convenient Notation” In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred  to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta.  The Kronecker delta is simply the indexed components of the identity operator in matrix algebra: [tex]\delta^j_k =\left\{\begin{array}{lcl}1&\text{ if }...