Scientists have proven for the very first time that one of the most fundamental problems of particle and quantum physics is mathematically unsolvable.
In short, they show that regardless of how no matter how perfectly we can mathematically describe a material on the microscopic level, we are never going to be able to predict its macroscopic behavior. Never.
But let’s break this down a bit more. To begin, the problem goes back to the spectral gap, which is the energy required for an electron to transition from a low-energy state to an excited state. Spectral gaps are important because they’re a central property of semiconductors, which are crucial components of most electrical circuits. Ultimately, physicists were hoping that they’d be able to discover whether a material is superconductive at room temperature simply by extrapolating from a complete-enough microscopic description.
But an international team of scientists has now shown that proving that a material has a spectral gap is what’s known as “an undecidable question”.
Summing up the research, Toby Cubitt, from University College London in the UK and one of the researchers, said in a press release, “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,”.
He goes on to note how this relates to determining what is happening on a quantum level, stating, “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.”
How can you deduce that something is unsolvable? More to the point, how do you prove it? Well, using a lot of very complex mathematics. In short, in order to be “provably unsolvable,” no algorithm will be able to determine the spectral gap – so you can’t determine the transition – so regardless of how the math is broken down, information about energy of the system does not confirm its presence.
While physicists have known since the 1930s about the possible existence of these undecidable questions, this is the first time that it has been put in place for such a fundamental problem. Significantly, however, the discovery also suggests that there are even stranger problems out there for physicists and mathematicians to solve.
Later in the release, co-author, Professor David Pérez-García from Universidad Complutense de Madrid and ICMAT, clarifies what this may mean for new physics: “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.”
So time will tell where this discovery leads us as far as revealing even stranger forms of physics.