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.”

Factor’s Gift Guide: 10 Gifts for Geeks and Scifi Supremos

Stuck trying to find a present for the geek or science fiction fan in your life? We’ve got the answer, with 10 fun but nerdy ideas for gifts this Christmas.

Pluto Plush

£14.95 from getDigital

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The change of Pluto’s status from planet to dwarf planet was hard to bear and has to have been an emotional blow to the planet itself. Luckily you now have the chance to provide poor Pluto some comfort with this 1:17,000,000 plush. With the heart and arms based on the famous images sent from NASA’s New Horizons space probe, this is probably the friendliest the ice planet has ever been.

Cross Marvel Tech2 Iron Man Gift Set

£65 from Cross

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To commemorate three of the most iconic characters from the Marvel franchise – Captain America, Spider-Man, and Iron Man – Cross have paired up their Tech2 Marvel pens with high quality journals to produce the perfect gift set for the Marvel fan in your life. And just in case the looks alone weren’t enough to sell you on this, the pen itself switches between stylus and ballpoint to work with touchscreens.

Spiral Planetary Nebula Mural

£25+ from Murals Wallpaper

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Showing the planetary nebula NGC 5189, this mural is a beautiful addition to any astronomically inclined home. The image, captured by NASA’s Hubble Wide Field Camera, shows the star as it consumes the last remaining fuel from its core and expels glowing clouds of gas. Lighting up the darkness of space with the bright gases of sulphur, hydrogen and oxygen, this mural adds some stellar colour to any home.

Robocup

€20.95 from MonsterZeug

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A great gift for fans of the ‘80s classic, this Robocop mug combines high quality ceramic with the grim visage of a cyborg to satisfy all caffeine needs. Once designed to fight crime, the part-man part-machine now serves to deliver your morning coffee. So whether you’re a fan yourself, or just trying to help a friend get over the reboot, be sure to add this to your Christmas list.

Zeitia Polygon Chrome Chandelier

£1,000 from The Chandelier & Mirror Company

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If you want a lighting fixture for a geometrically inclined friend, this is the chandelier for you. Finished in chrome and composed of LED lights, the polygonal piece is sure to be as much of a talking piece as a source of illumination. Suitable for any room in the home, it may be pricey, but you’re unlikely to see much else like it.

Zero Gravity Fridge Rover

£4.99 from Mulberry Bush

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A brilliant little wind up car for any space geeks out there, the Fridge Rover makes use of magnets and a clockwork mechanism to defy gravity. A nice stocking filler, just wind it up and put it on your fridge to see the rover at work as it grips and goes. The science can be kept a secret, but the fun is certain to be obvious.

Scientific Lab Flasks

£14 from Not Another Bill

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Whether your recipients are just after décor that’s a little quirkier than usual or needing some new glassware for those secret experiments, these flasks have you covered. Each with 100ml or 250ml measurements, the glass homeware set features one beaker and two flasks. Bring a little science to your sauces by using them for measuring, or make your centrepiece a little more experimental and place them as a vase.

Townsend Star Wars R2-D2 Fountain Pen

£575 from Cross

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You don’t need to go to a galaxy far, far away to get a great gift for the Star Wars fan in your life this Christmas. Instead, grab this limited edition pen to commemorate the original movie. With deep-etched engravings on brushed platinum plate and presented in a luxury gift box, each of these pens is individually serialised. Well worthy of a triumphant whistle.

LEGO Doctor Who Set

£49.99 from Forbidden Planet

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LEGO continues its trend of producing the best licensed products out there with this Doctor Who set. Created by fan-designer Andrew Clark, this set brings together the Doctor and Clara inside the Tardis to face some of their iconic enemies. Choose from the Eleventh or Twelfth Doctor and then set him against a pair of Daleks or a Weeping Angel. Don’t blink, though, or you might miss this amazing gift.

Human Ingredients T-Shirt

£20 from the Science Museum Shop

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Ever wondered just what it is that goes into making you, well, you? Wonder no more, as the Science Museum has you covered with this handy t-shirt. Presented in the manner of a food label, the t-shirt’s design lists the most commonly found elements in the human body. Your nearest and dearest will never be caught out by the pub quiz again as right in front of their nose is the answer: they’re 65% oxygen.

If you had to retake control of a driverless car would you be ready?

Researchers from Stanford University have concluded that drivers who retake control of an autonomous car are more likely to be involved in an accident.

The study, published in the first issue of Science Robotics, found that roads could become especially dangerous when drivers made the transition back to being in control of autonomous vehicles due to changes in speed and other changes in driving conditions.

“Many people have been doing research on paying attention and situation awareness. That’s very important,” said lead author of the research and former graduate student in the Dynamic Design Lab at Stanford University, Holly Russell.

“But, in addition, there is this physical change and we need to acknowledge that people’s performance might not be at its peak if they haven’t actively been participating in the driving.”

Featured image courtesy of Steve Jurvetson

Featured image courtesy of Steve Jurvetson

During testing all drivers were given advance warning that they would be put back in control of a driverless car and were given the opportunity to drive around the testing track a number of times, so they could feel for themselves changes in speed or steering that may occur while the car drives itself.

However, during the trial itself, the drivers’ steering manoeuvres differed significantly from their ability when in control of the car from start to finish.

“Even knowing about the change, being able to make a plan and do some explicit motor planning for how to compensate, you still saw a very different steering behaviour and compromised performance,” said co-author of the research and research associate in the Revs Program at Stanford, Lene Harbott.

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Although no driver was so thrown off by the changes in steering that they drove off-course, the fact that there was a period of altered steering behaviour increased the likelihood of an accident occurring.

However, the Stanford study only addressed one specific example of the autonomous car to human driver handover; there is still a lot to learn about how people respond in other circumstances, depending on the type of car, the driver and how the driving conditions have changed.

“If someone is designing a method for automated vehicle handover, there will need to be detailed research on that specific method,” said Harbott. “This study is tip of an iceberg.”