Paradox at the heart of mathematics makes physics problem
unanswerable
Gödel’s incompleteness theorems are connected to unsolvable calculations in quantum physics.
09 December 2015
A logical paradox at the heart of mathematics and computer science turns out to have implications for the real world, making a basic
question about matter fundamentally unanswerable.
In 1931, Austrian-born mathematician Kurt Gödel shook the academic world when he announced that some statements are
‘undecidable’, meaning that it is impossible to prove them either true or false. Three researchers have now found that the same
principle makes it impossible to calculate an important property of a material — the gaps between the lowest energy levels of its
electrons — from an idealized model of its atoms.
The result also raises the possibility that a related problem in particle physics — which has a US$1-million
prize attached to it — could be similarly unsolvable, says Toby Cubitt, a quantum-information theorist at
University College London and one of the authors of the study.
The finding, published on 9 December in Nature1, and in a longer, 140-page version on the arXiv preprint
server2, is “genuinely shocking, and probably a big surprise for almost everybody working on condensedmatter theory”, says Christian Gogolin, a quantum information theorist at the Institute of Photonic Sciences in
Barcelona, Spain.
From logic to physics
Gödel’s finding was first connected to the physical world in 1936, by British mathematician Alan Turing. “Turing thought more clearly
about the relationship between physics and logic than Gödel did,” says Rebecca Goldstein, a US author who has written a biography of
Gödel3.
Turing reformulated Gödel’s result in terms of algorithms executed by an idealized computer that can read or write one bit at a time. He
showed that there are some algorithms that are undecidable by such a ‘Turing machine’: that is, it’s impossible to tell whether the
machine could complete the calculations in a finite amount of time. And there is no general test to see whether any particular algorithm
is undecidable. The same restrictions apply to real computers, since any such devices are mathematically equivalent to a Turing
machine.
Since the 1990s4, theoretical physicists have tried to embody Turing’s work in idealized models of physical
phenomena. But "the undecidable questions that they spawned did not directly correspond to concrete
problems that physicists are interested in”, says Markus Müller, a theoretical physicist at Western University
in London, Canada, who published one such model with Gogolin and another collaborator in 20125.
“I think it’s fair to say that ours is the first undecidability result for a major physics problem that people would
really try to solve,” says Cubitt.
Spectral gap
Cubitt and his collaborators 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. This determines some of a material’s basic
properties. In some materials, for example, lowering the temperature causes the gap to close, which leads
Answers & Comments
Paradox at the heart of mathematics makes physics problem
unanswerable
Gödel’s incompleteness theorems are connected to unsolvable calculations in quantum physics.
09 December 2015
A logical paradox at the heart of mathematics and computer science turns out to have implications for the real world, making a basic
question about matter fundamentally unanswerable.
In 1931, Austrian-born mathematician Kurt Gödel shook the academic world when he announced that some statements are
‘undecidable’, meaning that it is impossible to prove them either true or false. Three researchers have now found that the same
principle makes it impossible to calculate an important property of a material — the gaps between the lowest energy levels of its
electrons — from an idealized model of its atoms.
The result also raises the possibility that a related problem in particle physics — which has a US$1-million
prize attached to it — could be similarly unsolvable, says Toby Cubitt, a quantum-information theorist at
University College London and one of the authors of the study.
The finding, published on 9 December in Nature1, and in a longer, 140-page version on the arXiv preprint
server2, is “genuinely shocking, and probably a big surprise for almost everybody working on condensedmatter theory”, says Christian Gogolin, a quantum information theorist at the Institute of Photonic Sciences in
Barcelona, Spain.
From logic to physics
Gödel’s finding was first connected to the physical world in 1936, by British mathematician Alan Turing. “Turing thought more clearly
about the relationship between physics and logic than Gödel did,” says Rebecca Goldstein, a US author who has written a biography of
Gödel3.
Turing reformulated Gödel’s result in terms of algorithms executed by an idealized computer that can read or write one bit at a time. He
showed that there are some algorithms that are undecidable by such a ‘Turing machine’: that is, it’s impossible to tell whether the
machine could complete the calculations in a finite amount of time. And there is no general test to see whether any particular algorithm
is undecidable. The same restrictions apply to real computers, since any such devices are mathematically equivalent to a Turing
machine.
Since the 1990s4, theoretical physicists have tried to embody Turing’s work in idealized models of physical
phenomena. But "the undecidable questions that they spawned did not directly correspond to concrete
problems that physicists are interested in”, says Markus Müller, a theoretical physicist at Western University
in London, Canada, who published one such model with Gogolin and another collaborator in 20125.
“I think it’s fair to say that ours is the first undecidability result for a major physics problem that people would
really try to solve,” says Cubitt.
Spectral gap
Cubitt and his collaborators 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. This determines some of a material’s basic
properties. In some materials, for example, lowering the temperature causes the gap to close, which leads
the material to become a superconductor.