A fundamental quantum physics problem has been proved unsolvable

For the first time a major physics problem has been proved unsolvable, meaning that no matter how accurately a material is mathematically described on a microscopic level, there will not be enough information to predict its macroscopic behaviour.

The research, by an international team of scientists from UCL, the Technical University of Music and the Universidad Complutense de Madrid – ICMAT, concerns the spectral gap, a term for the energy required for an electron to transition from a low-energy state to an excited state.

Spectral gaps are a key property in semiconductors, among a multitude of other materials, in particular those with superconducting properties. It was thought that it was possible to determine if a material is superconductive by extrapolating from a complete enough microscopic description of it, however this study has shown that determining whether a material has a spectral gap is what is known as “an undecidable question”.

“Alan Turing is famous for his role in cracking the Enigma, but amongst mathematicians and computer scientists, he is even more famous for proving that certain mathematical questions are `undecidable’ – they are neither true nor false, but are beyond the reach of mathematics code,” said co-author Dr Toby Cubitt, from UCL Computer Science.

“What we’ve shown is that the spectral gap is one of these undecidable problems. This means a general method to determine whether matter described by quantum mechanics has a spectral gap, or not, cannot exist. Which limits the extent to which we can predict the behaviour of quantum materials, and potentially even fundamental particle physics.”

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The research, which was published today in the journal Nature, used complex mathematics to determine the undecidable nature of the spectral gap, which they say they have demonstrated in two ways:

“The spectral gap problem is algorithmically undecidable: there cannot exist any algorithm which, given a description of the local interactions, determines whether the resulting model is gapped or gapless,” wrote the researchers in the journal paper.

“The spectral gap problem is axiomatically independent: given any consistent recursive axiomatisation of mathematics, there exist particular quantum many-body Hamiltonians for which the presence or absence of the spectral gap is not determined by the axioms of mathematics.”

In other words, no algorithm can determine the spectral gap, and no matter how the maths is broken down, information about energy of the system does not confirm its presence.

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The research has profound implications for the field, not least for the Clay Mathematics Institute’s infamous $1m prize to prove whether the standard model of particular physics, which underpins the behaviour of the most basic particulars of matter, has a spectral gap using standard model equations.

“It’s possible for particular cases of a problem to be solvable even when the general problem is undecidable, so someone may yet win the coveted $1m prize. But our results do raise the prospect that some of these big open problems in theoretical physics could be provably unsolvable,” said Cubitt.

“We knew about the possibility of problems that are undecidable in principle since the works of Turing and Gödel in the 1930s,” agreed co-author Professor Michael Wolf, from the Technical University of Munich.

“So far, however, this only concerned the very abstract corners of theoretical computer science and mathematical logic. No one had seriously contemplated this as a possibility right in the heart of theoretical physics before. But our results change this picture. From a more philosophical perspective, they also challenge the reductionists’ point of view, as the insurmountable difficulty lies precisely in the derivation of macroscopic properties from a microscopic description.”

“It’s not all bad news, though,” added Professor David Pérez-García, from the Universidad Complutense de Madrid and ICMAT. “The reason this problem is impossible to solve in general is because models at this level exhibit extremely bizarre behaviour that essentially defeats any attempt to analyse them.

“But this bizarre behaviour also predicts some new and very weird physics that hasn’t been seen before. For example, our results show that adding even a single particle to a lump of matter, however large, could in principle dramatically change its properties. New physics like this is often later exploited in technology.”

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Researchers have found a running style for six-legged robots that significantly improves on the traditional nature-inspired method of movement.

The research, conducted by scientists at the École Polytechnique Fédérale de Lausanne (EPFL) and the University of Lausanne (UNIL) in Switzerland, found that as long as the robots are not equipped with insect-like adhesive pads, it is faster for them to move with only two legs on the ground at any given time.

Robotics has in the past few years made heavy use of biomimicry – the practice of mimicking natural systems – resulting in six-legged robots being designed to move like insects. In nature, insects use what is known as a tripod gait, where they have three legs on the ground at a time, so it had been assumed that this was the most efficient way for similarly legged robots to move.

However, by undertaking a series of computer simulations, tests on robots and experiments on Drosophila melanogaster – better known as the common fruit fly – the scientists found that the two-legged approach, which they have dubbed the bipod gait, results in faster and more efficient movement.

The core goal of the research, which is published today in the journal Nature Communications, was to confirm whether the long-held assumption that a tripod gait was best was indeed correct.

“We wanted to determine why insects use a tripod gait and identify whether it is, indeed, the fastest way for six-legged animals and robots to walk,” said Pavan Ramdya, study co-lead and corresponding author.

Initially, this involved the use of a simulated insect model based on the common fruit fly and an algorithm designed to mimic different evolutionary stages. This algorithm simulated different potential gaits to create a shortlist of those that it deemed to be the fastest.

This, however, shed light on why insects have a tripod gait – and why it may not be the best option for robots. The simulations showed that the traditional tripod gait works in combination with the adhesive pad found on the ends of insects’ legs to make climbing over vertical surfaces such as rocks easier and quicker.

Robots, however, are typically designed to walk along flat surfaces, and so the benefits of such a gait are lost.

“Our findings support the idea that insects use a tripod gait to most effectively walk on surfaces in three dimensions, and because their legs have adhesive properties. This confirms a long-standing biological hypothesis,” said Ramdya. “Ground robots should therefore break free from only using the tripod gait”.

Study co-lead authors Robin Thandiackal (left) and Pavan Ramdya with the six-legged robot used in the research. Images courtesy of EPFL/Alain Herzog

To for always corroborate the simulation’s findings, the researchers built a six-legged robot that could move either with a bipod or tripod gait, and which quickly confirmed the research by being faster when moving with just two legs on the ground at once.

However, they went further by confirming that the adhesive pads were in fact playing a role in the insect’s tripod movement.

They did this by equipping the fruit flies with tiny polymer boots that would cover the adhesive pads, and so remove their role in the way the insects moved. The flies’ responses confirms their theory: they began moving with a bipod-like gate rather than their conventional tripod-style movement.

“This result shows that, unlike most robots, animals can adapt to find new ways of walking under new circumstances,” said study co-lead author Robin Thandiackal.

As bizarre as the research sounds, it provides valuable new insights both for roboticists and biologists, and could lead to a new standard in the way that six legged robots are designed to move.

“There is a natural dialogue between robotics and biology: Many robot designers are inspired by nature and biologists can use robots to better understand the behavior of animal species,” added Thandiackal. “We believe that our work represents an important contribution to the study of animal and robotic locomotion.”