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.”

Driverless cars are learning how to go faster in dangerous conditions

A Georgia Institute of Technology research team has discovered a way to make self-driving cars safe when they’re driven in hazardous road conditions or at high speeds.

Up until now we’ve seen driverless cars performing comfortably on roads in good condition, but by using advanced algorithms and onboard computing, together with installed sensor devices, the Georgia Tech team was able to maintain control of a driverless vehicle when roadway adhesion was limited.

So a driverless car would be able to perform in icy or, as the researchers tested, rally-style conditions.

“An autonomous vehicle should be able to handle any condition, not just drive on the highway under normal conditions,” said School of Aerospace Engineering professor and expert on the mathematics behind rally-car racing control, Panagiotis Tsiotras.

“One of our principal goals is to infuse some of the expert techniques of human drivers into the brains of these autonomous vehicles.”

The Georgia Tech researchers used a method called model predictive path integral control (MPPI) to keep their cars at the edge of their limits.

To create their MPPI control algorithm the team combined large amounts of car handling information with data on the dynamics of the vehicle, to calculate the most stable trajectories from the numerous possibilities.

“Aggressive driving in a robotic vehicle – manoeuvring at the edge – is a unique control problem involving a highly complex system,” said School of Aerospace Engineering assistant professor and project leader, Evangelos Theodorou.

“However, by merging statistical physics with control theory, and utilising leading-edge computation, we can create a new perspective, a new framework, for control of autonomous systems.”

Images courtesy of Rob Felt, Georgia Tech

Images courtesy of Rob Felt, Georgia Tech

The MPPI control algorithm was tested by racing, sliding and jumping one-fifth scale, fully autonomous auto-rally cars at the equivalent of 90 mph.

The cars carried a motherboard with a quad-core processor, a potent GPU and a battery. Each vehicle was also fitted with two forward-facing cameras, an inertial measurement unit and sophisticated wheel-speed sensors.

In order to maintain balance in the hazardous testing conditions the cars had to balance a desire to stay on the track with achieving the desired velocity.

The researchers refer to these two separate desires, which they managed to coordinate, as costs.

“What we’re talking about here is using the MPPI algorithm to achieve relative entropy minimisation, and adjusting costs in the most effective way is a big part of that,” said James Rehg, a professor in the Georgia Tech School of Interactive Computing.

“To achieve the optimal combination of control and performance in an autonomous vehicle is definitely a non-trivial problem.”

Augmented reality helmet to revolutionise firefighting

A team from Swiss research institute EPFL is developing a visor that will help firefighters see everything around them in real time.

While firefighters battle with flames on a daily basis, they also face dangers in the form of toxic smoke and darkness, which can slow their progress, adding critical time to rescue operations. The protective gear they wear can weigh over 20kg, and between dragging a firehose and carrying a thermal camera to help them analyse their surroundings, their job is not an easy one.

So, in an attempt to lighten the load, two engineers at EPFL – Adrien Birbaumer and Martijn Bosch – have come up with a solution that frees up firefighters’ hands.

Their solution, which is part of the VIZIR project at the Images and Visual Representation Laboratory, involves placing a mini infrared camera on the firefighter’s helmet and incorporating a transparent screen in the oxygen mask.

“[Firefighters] really count on the thermal imaging camera, but it gets in the way and forces them to interrupt their search if they want to analyse a room,” Birbaumer and Bosch explained.

But with this smart visor, firefighters are able to move around more easily and avoid obstacles without having to interrupt their search process.

Wearing the mask, firefighters see two images in their field of vision: what their eyes see and what the thermal imaging camera records and displays in real time. The infrared camera uses the standard red and blue colours to represent hot and cold zones but, Birbaumer noted, “we had to find the right tones that would be visible on a transparent surface.”

Images courtesy of Alain Herzog / EPFL

Images courtesy of Alain Herzog / EPFL

The engineers are testing their prototype in collaboration with ECA, the public fire and natural disaster insurance company in Vaud Canton. Firefighters wear the prototype visor during training sessions to test its effectiveness and give feedback.

And according to Jean-Marc Pittet, the man in charge of training firefighters in Vaud Canton: “At first it’s hard to know what you’re seeing, if it’s the real thing or not, but you get used to it surprisingly fast and can easily handle the two overlapping views.”

So while the double vision effect might take some getting used to, the next step is to incorporate the screen into the oxygen mask itself and make this augmented reality, well, a reality for firefighters.

Beyond design limits: How to harness new materials and fabrication methods

New technologies such as additive manufacturing are improving the ability to turn advanced materials that combine extreme strength with super lightness into previously unimaginable shapes.

However, generating new designs that are able to fully exploit the properties of these advanced materials has proven challenging, and today’s design technologies are unable to recreate the level of physical detail and complexity made possible with innovative manufacturing capabilities and materials.

This is where DARPA comes in. In April of this year, the US Defense Advanced Research Projects Agency (DARPA) announced its TRAnsformative DESign (TRADES) programme – a research effort to develop new mathematics and algorithms that can fully take advantage of this new design space.

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“The structural and functional complexities introduced by today’s advanced materials and manufacturing methods have exceeded our capacity to simultaneously optimize all the variables involved,” explained Jan Vandenbrande, DARPA programme manager.

“We have reached the fundamental limits of what our computer-aided design tools and processes can handle, and need revolutionary new tools that can take requirements from a human designer and propose radically new concepts, shapes and structures that would likely never be conceived by even our best design programs today, much less by a human alone.”

DARPA uses a phased array radar and an aircraft skin as an example of a difficult structure to design with currently available tools, as its components vary considerably in their physical or functional properties.

While the components in this structure are usually designed separately before being joined, the TRADES programme envisions a more unified design – for example embedding the radar directly into the aircraft skin itself – which could reduce cost, size and weight of future military systems.

DARPA’s TRAnsformative DESign (TRADES) programme is a fundamental research effort to develop new mathematics and algorithms that can more fully take advantage of the almost boundless design space that has been enabled by new materials and fabrication methods. Image courtesy of DARPA

DARPA’s TRAnsformative DESign (TRADES) programme is a fundamental research effort to develop new mathematics and algorithms that can more fully take advantage of the almost boundless design space that has been enabled by new materials and fabrication methods. Image courtesy of DARPA

And what’s more, existing design tools are unable to take full advantage of the unique properties and processing requirements of the advanced materials, such as carbon fibre composites, which have their own shaping requirements. If these requirements are not taken into account during the design process, it could result in production difficulties or defects.

These problems could be mitigated or even eliminated if designers had the tools to account for these specific characteristics and requirements.

“Much of today’s design is really re-design based on useful but very old ideas. The design for building aircraft fuselages today, for example, is based on a spar-and-rib concept that dates back to design ideas from four thousand years ago when ancient ships such as the Royal Barge of Khufu used this basic design concept for its hull,” added Vandenbrande.

“TRADES could revolutionize such well-worn designs.”