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A mathematical problem underlying fundamental questions in particle physics and quantum physics is provably unsolvable, according to scientists at UCL, Universidad Complutense de Madrid - ICMAT and Technische Universität München. The findings show that even a perfect and complete description of the microscopic properties of a material is not enough to predict its macroscopic behaviour.
The study, published today in Nature, investigated the problem of the ‘spectral gap’ – the energy needed for a material to transfer from its lowest-energy state to an excited state. When this energy becomes very small (i.e. the ‘spectral gap closes’), it becomes possible for the material to transition to a completely different state and the properties of the material can undergo dramatic changes. For example, when a material at a very low temperature transitions from insulating to superconducting, this dramatic change is the result of its spectral gap closing.
Using sophisticated mathematics, the authors proved that even with a complete microscopic description of a quantum material, determining whether it has a spectral gap is an 'undecidable' question.
Co-author, Dr Toby Cubitt from UCL Computer Science, said: “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.
“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.”
The most famous problem concerning spectral gaps is whether the theory governing the fundamental particles of matter itself – the standard model of particle physics – has a spectral gap (the `Yang-Mills mass gap' conjecture). Particle physics experiments such as CERN and numerical calculations on supercomputers suggest that there is a spectral gap. Although there is a $1m prize at stake from the Clay Mathematics Institute for whoever can, no one has yet succeeded in proving this mathematically from the equations of the standard model.
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