Researchers have a whole lot of unanswered questions when it comes to the worlds of particle and quantum physics, but one of the most fundamental of those is going to stay that way, with scientists proving for the first time that the problem is mathematically unsolvable.

The problem in question concerns the spectral gap, which is a term for the energy required for an electron to transition from a low-energy state to an excited state. What that really means is that no matter how perfectly and completely we can mathematically describe a material on the microscopic level, we're never going to be able to predict its macroscopic behaviour. If you listen closely, you can almost hear the dreams of physicists everywhere being shattered. 

Why are spectral gaps so important? They're a central property of semiconductors, which are crucial components of most electrical circuits, and physicists had hoped that if they'd be able to work out whether a material is superconductive at room temperature (a highly desirable trait) simply by extrapolating from a complete-enough microscopic description. 

But publishing their results in Nature, an international team of scientists has now shown that determining whether a material has a spectral gap is what's known as "an undecidable question".

"Alan Turing is famous for his role in cracking the Enigma code. 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," said one of the researchers, Toby Cubitt from University College London in the UK.

"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."

So how do you figure out that something is "provable unsolvable"? The team used a whole lot of complex mathematics, which they describe in the journal article. Or as Lucy Ingham over at Factor-Tech explains

"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."

There are some big implications of this discovery, especially given that there's a US$1 million prize at stake from the Clay Mathematics Institute for anyone who can prove whether the standard model of particle physics – which explains the behaviour of the most basic particles of matter in the Universe – 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 $1 million prize," said Cubitt. "But our results do raise the prospect that some of these big open problems in theoretical physics could be provably unsolvable."

While physicists have known about the possibility of these undecidable questions since the 1930s, this is the first time that the limitation has been put in place for such a fundamental problem.

But it's not all bad news. The discovery also suggests that there are even stranger problems out there for physicists and mathematicians to solve.

"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," said co-author David Pérez-García from the Universidad Complutense de Madrid in Spain.

"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."

The team is now testing whether their mathematical models will hold up when tested in the lab with real quantum materials. Let's hope that problem is a little more solvable.