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.”


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.


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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Juno mission: Jupiter’s magnetic field is even weirder than expected

It has long been known that Jupiter has the most intense magnetic field in the solar system, but the first round of results from NASA’s Juno mission has revealed that it is far stronger and more misshapen than scientists predicted.

Announcing the findings of the spacecraft’s first data-collection pass, which saw Juno fly within 2,600 miles (4,200km) of Jupiter on 27th August 2016, NASA mission scientists revealed that the planet far surpassed the expectations of models.

Measuring Jupiter’s magnetosphere using Juno’s magnetometer investigation (MAG) tool, they found that the planet’s magnetic field is even stronger than models predicted, at 7.766 Gaus: 10 times stronger than the strongest fields on Earth.

Furthermore, it is far more irregular in shape, prompting a re-think about how it could be generated.

“Juno is giving us a view of the magnetic field close to Jupiter that we’ve never had before,” said Jack Connerney, Juno deputy principal investigator and magnetic field investigation lead at NASA’s Goddard Space Flight Center in Greenbelt, Maryland.

“Already we see that the magnetic field looks lumpy: it is stronger in some places and weaker in others.

An enhanced colour view of Jupiter’s south pole. Image courtesy of NASA/JPL-Caltech/SwRI/MSSS/Gabriel Fiset. Featured image courtesy of NASA/SWRI/MSSS/Gerald Eichstädt/Seán Doran

At present, scientists cannot say for certain why or how Jupiter’s magnetic field is so peculiar, but they do already have a theory: that the field is not generated from the planet’s core, but in a layer closer to its surface.

“This uneven distribution suggests that the field might be generated by dynamo action closer to the surface, above the layer of metallic hydrogen,” said Connerney.

However, with many more flybys planned, the scientists will considerable opportunities to learn more about this phenomenon, and more accurately pinpoint the bizarre magnetic field’s cause.

“Every flyby we execute gets us closer to determining where and how Jupiter’s dynamo works,” added Connerney.

With each flyby, which occurs every 53 days, the scientists are treated to a 6MB haul of newly collected information, which takes around 1.5 days to transfer back to Earth.

“Every 53 days, we go screaming by Jupiter, get doused by a fire hose of Jovian science, and there is always something new,” said Scott Bolton, Juno principal investigator from the Southwest Research Institute in San Antonio.

A newly released image of Jupiter’s stormy south pole. Image courtesy of NASA/JPL-Caltech/SwRI/MSSS/Betsy Asher Hall/Gervasio Robles

An unexpected magnetic field was not the only surprise from the first data haul. The mission also provided a first-look at Jupiter’s poles, which are unexpectedly covered in swirling, densely clustered storms the size of Earth.

“We’re puzzled as to how they could be formed, how stable the configuration is, and why Jupiter’s north pole doesn’t look like the south pole,” said Bolton. “We’re questioning whether this is a dynamic system, and are we seeing just one stage, and over the next year, we’re going to watch it disappear, or is this a stable configuration and these storms are circulating around one another?”

Juno’s Microwave Radiometer (MWR) also threw up some surprises, with some of the planet’s belts appearing to penetrate down to its surface, while others seem to evolve into other structures. It’s a curious phenomenon, and one which the scientists hope to better explore on future flybys.

“On our next flyby on July 11, we will fly directly over one of the most iconic features in the entire solar system – one that every school kid knows – Jupiter’s Great Red Spot,” said Bolton.

“If anybody is going to get to the bottom of what is going on below those mammoth swirling crimson cloud tops, it’s Juno and her cloud-piercing science instruments.”