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.”

Police want to 3D print a dead man's fingers to access his phone

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Tesla's autopilot prevented a driver from hitting a pedestrian

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NASA analyses of ground-based observations and satellite data has confirmed that two key climate change indicators – global surface temperatures and Arctic sea ice extent – have broken numerous records through the first half of 2016.

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Lampposts and churches will recharge Amazon's fleet of drones

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Are we 10 years away from finding life outside our Solar System?

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DesignSpace has been developed for the HTC Vive to enable architects and designers to draft concepts in virtual reality, as well as allowing colleagues to work together in a shared virtual space.

While DesignSpace is just a prototype at the minute it could be an amazing tool for architects, and shows that both AR and VR have an exciting future.

Elon Musk has already disrupted the automotive industry. Now he plans to do it again

Elon Musk has announced that he is at the end of the first stage of his Tesla Motors masterplan, and is anxious to get on with stage two.

Despite the technology mogul finding himself under pressure in recent weeks, following the death of a driver in a self-driving Tesla car, Musk has found time to publicly announce his ‘Master Plan, Part Deux’.

In response to the death of Joshua Brown in a Tesla Model S, Musk’s revamped masterplan includes an ambition to “develop a self-driving capability that is 10 times safer than manual via massive fleet learning”.

“As the technology matures, all Tesla vehicles will have the hardware necessary to be fully self-driving with fail-operational capability, meaning that any given system in the car could break and your car will still drive itself safely,” wrote Musk in a blog post.

Image courtesy of Maurizio Pesce. Featured image by Earth Day Network

Image courtesy of Maurizio Pesce. Featured image by Earth Day Network

Musk has also pointed out why Tesla is deploying partial autonomy now even though it has come in for criticism in recent weeks.

“The most important reason is that, when used correctly, it is already significantly safer than a person driving by themselves and it would therefore be morally reprehensible to delay release simply for fear of bad press or some mercantile calculation of legal liability,” said Musk.

The other three achievements Musk is looking to make in stage two of his plan to accelerate the advent of sustainable energy are: to create solar roofs with seamlessly integrated battery storage, to expand the electric vehicle product line to address all major segments and to enable your car to make money for you when you aren’t using it.

Image courtesy of Jusdafaxderivative

Image courtesy of Jusdafaxderivative

To achieve the first point Musk plans to merge Tesla with his other sustainable energy company, SolarCity, as they are in “pursuit of the same overarching goal of sustainable energy”.

Musk imagines the second point, of expanding the electric vehicle product line, will be achieved by introducing a fleet of autonomous buses that can accommodate wheelchairs, strollers and bikes.

It’s in  the third point though where Musk has told of his plans to disrupt the automotive industry as we know it.

Musk wants to develop a Tesla shared fleet where cars can generate income for their owners while they work or are on vacation.

“Since most cars are only in use by their owner for 5% to 10% of the day, the fundamental economic utility of a true self-driving car is likely to be several times that of a car which is not,” said Musk.