13 December 2015

A well-known thereom at the heart of mathematical logic, Gödel’s Incompleteness Theorem, has deep implications for physics because it makes a fundamental question about matter literally unanswerable.

In 1931, Austrian-born mathematician Kurt Gödel proved that some statements are ‘undecidable’, meaning that it is impossible to prove them either true or false within the confines of a set of axioms adopted by mathematics.

Three researchers have now found that the same principle makes it impossible to calculate the gaps between the lowest energy levels of its electrons from an idealized model of its atoms.

Toby Cubitt, a quantum-information theorist at University College London, and his collaborators have focused on calculating the ‘spectral gap’: the gap between the lowest energy level that electrons can occupy in a material, and the next one up.

The quantum states of the atoms in an atomic lattice of a material contain the information needed to find the material’s spectral gap. Cubitt and his colleagues showed that for an infinite lattice, the question of whether the gap exists is undecidable.

Cubitt says that the team ultimately wants to study a related problem in particle physics called the Yang–Mills mass-gap problem, which the Clay Mathematics Institute in Peterborough, New Hampshire, has named one of its Millennium Prize Problems. The institute is offering $1 million to anyone who is able to solve it.

The mass-gap problem relates to the observation that the particles that carry the weak and strong nuclear force have mass. This is also why the weak and strong nuclear forces have limited range, unlike gravity and electromagnetism, and why quarks are only found as part of composite particles such as protons or neutrons, never in isolation. The problem is that there is no rigorous mathematical theory which explains why the force-carriers have mass, when photons, the carriers of the electromagnetic force, are massless.

Cubitt hopes that eventually, his team’s methods and ideas will show that the Yang–Mills mass-gap problem is undecidable.

Read more here.