Quantum physics problem proved unsolvable

A mathematical problem underlying fundamental questions in particle and quantum physics is provably unsolvable, according to scientists at UCL, Universidad Complutense de Madrid – ICMAT and Technical University of Munich. It is the first major problem in physics for which such a fundamental limitation could be proven. The findings are important because they show that even a perfect and complete description of the microscopic properties of a material is not enough to predict its macroscopic behavior.

A small spectral gap – the energy needed to transfer an electron from a low-energy state to an excited state – is the central property of semiconductors. In a similar way, the spectral gap plays an important role for many other materials. 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. An example of this is when a material becomes superconducting.

Mathematically extrapolating from a microscopic description of a material to the bulk solid is considered one of the key tools in the search for materials exhibiting superconductivity at ambient temperatures or other desirable properties. A study, published today in Nature, however, shows crucial limits to this approach. 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, in fact, an undecidable question.

“Alan Turing is famous for his role in cracking the Enigma code,” says Co-author, Dr. Toby Cubitt from UCL Computer Science. “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 behavior of quantum materials, and potentially even fundamental particle physics.”

One million dollars to win!

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.

Dr. Cubitt added, “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 $1m prize. But our results do raise the prospect that some of these big open problems in theoretical physics could be provably unsolvable.”

"We knew about the possibility of problems that are undecidable in principle since the works of Turing and Gödel in the 1930s,” added Co-author Professor Michael Wolf from Technical University of Munich. “So far, however, this only concerned the very abstract corners of theoretical computer science and mathematical logic. No one had seriously contemplated this as a possibility right in the heart of theoretical physics before. But our results change this picture. From a more philosophical perspective, they also challenge the reductionists’ point of view, as the insurmountable difficulty lies precisely in the derivation of macroscopic properties from a microscopic description."

Not all bad news

Co-author, Professor David Pérez-García from Universidad Complutense de Madrid and ICMAT, said: “It's not all bad news, though. The reason this problem is impossible to solve in general is because models at this level exhibit extremely bizarre behavior that essentially defeats any attempt to analyze them. But this bizarre behavior 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 researchers are now seeing whether their findings extend beyond the artificial mathematical models produced by their calculations to more realistic quantum materials that could be realized in the laboratory.

The research has been funded by the John Templeton Foundation, the Royal Society (UK), the Spanish Ministry of Economics and Competitiveness (MINECO), the Madrid Regional Government and the European Research Council (ERC).

Technical University of Munich

Supercomputing the Strange Difference Between Matter and Antimatter

Supercomputers such as Brookhaven Lab's Blue Gene/Q were essential for completing the complex calculation of direct CP symmetry violation. The same calculation would have required two thousand years using a laptop.

An international team of physicists including theorists from the U.S. Department of Energy's (DOE) Brookhaven National Laboratory has published the first calculation of direct "CP" symmetry violation—how the behavior of subatomic particles (in this case, the decay of kaons) differs when matter is swapped out for antimatter. Should the prediction represented by this calculation not match experimental results, it would be conclusive evidence of new, unknown phenomena that lie outside of the Standard Model—physicists' present understanding of the fundamental particles and the forces between them.

The current result—reported in the November 20 issue of Physical Review Letters—does not yet indicate such a difference between experiment and theory, but scientists expect the precision of the calculation to improve dramatically now that they've proven they can tackle the task. With increasing precision, such a difference—and new physics—might still emerge.

"This so called 'direct' symmetry violation is a tiny effect, showing up in just a few particle decays in a million," said Brookhaven physicist Taku Izubuchi, a member of the team performing the calculation. Results from the first, less difficult part of this calculation were reported by the same group in 2012.  However, it is only now, with completion of the second part of this calculation—which was hundreds of times more difficult than the first—that a comparison with the measured size of direct CP violation can be made.  This final part of the calculation required more than 200 million core processing hours on supercomputers, "and would have required two thousand years using a laptop," Izubuchi said.

The calculation determines the size of the symmetry violating effect as predicted by the Standard Model, and was compared with experimental results that were firmly established in 2000 at the European Center for Nuclear Research (CERN) and Fermi National Accelerator Laboratory.

"This is an especially important place to compare with the Standard Model because the small size of this effect increases the chance that other, new phenomena may become visible," said Robert Mawhinney of Columbia University.

"Although the result from this direct CP violation calculation is consistent with the experimental measurement, revealing no inconsistency with the Standard Model, the calculation is on-going with an accuracy that is expected to increase two-fold within two years," said Peter Boyle of the University of Edinburgh. "This leaves open the possibility that evidence for new phenomena, not described by the Standard Model, may yet be uncovered."

Matter-antimatter asymmetry

Physicists' present understanding of the universe requires that particles and their antiparticles (which have the same mass but opposite charge) behave differently. Only with matter-antimatter asymmetry can they hope to explain why the universe, which was created with equal parts of matter and antimatter, is filled mostly with matter today. Without this asymmetry, matter and antimatter would have annihilated one another leaving a cold, dim glow of light with no material particles at all.

The first experimental evidence for the matter-antimatter asymmetry known as CP violation was discovered in 1964 at Brookhaven Lab. This Nobel-Prize-winning experiment also involved the decays of kaons, but demonstrated what is now referred to as "indirect" CP violation. This violation arises from a subtle imperfection in the two distinct types of neutral kaons.

The target of the present calculation is a phenomenon that is even more elusive: a one-part-in-a-million difference between the matter and antimatter decay probabilities. The small size of this "direct" CP violation made its experimental discovery very difficult, requiring 36 years of intense experimental effort following the 1964 discovery of "indirect" CP violation.

This calculation required more than 200 million core processing hours on supercomputers and would have required two thousand years using a laptop.

While these two examples of matter-antimatter asymmetry are of very different size, they are related by a remarkable theory for which physicists Makoto Kobayashi and Toshihide Maskawa were awarded the 2008 Nobel Prize in physics. The theory provides an elegant and simple explanation of CP violation that manages to explain both the 1964 experiment and later CP-violation measurements in experiments at the KEK laboratory in Japan and the SLAC National Accelerator Laboratory in California.

"This new calculation provides another test of this theory—a test that the Standard Model passes, at least at the present level of accuracy," said Christoph Lehner, a Brookhaven Lab member of the team.

Although the Standard Model does successfully relate the matter-antimatter asymmetries seen in the 1964 and later experiments, this Standard-Model asymmetry is insufficient to explain the preponderance of matter over antimatter in the universe today.

"This suggests that a new mechanism must be responsible for the preponderance of matter of which we are made," said Christopher Kelly, a member of the team from the RIKEN BNL Research Center (RBRC). "This one-part-per-million, direct CP violation may be a good place to first see it. The approximate agreement between this new calculation and the 2000 experimental results suggests that we need to look harder, which is exactly what the team performing this calculation plans to do."

This calculation was carried out on the Blue Gene/Q supercomputers at the RIKEN BNL Research Center (RBRC), at Brookhaven National Laboratory, at the Argonne Leadership Class Computing Facility (ALCF) at Argonne National Laboratory, and at the DiRAC facility at the University of Edinburgh. The research was carried out by Ziyuan Bai, Norman Christ, Robert Mawhinney, and Daiqian Zhang of Columbia University; Thomas Blum of the University of Connecticut; Peter Boyle and Julien Frison of the University of Edinburgh; Nicolas Garron of Plymouth University; Chulwoo Jung, Christoph Lehner, and Amarjit Soni of Brookhaven Lab; Christopher Kelly, and Taku Izubuchi of the RBRC and Brookhaven Lab; and Christopher Sachrajda of the University of Southampton. The work was funded by the U.S. Department of Energy's Office of Science, by the RIKEN Laboratory of Japan, and the U.K. Science and Technology Facilities Council.  The ALCF is a DOE Office of Science User Facility.

Nanotechnology World Association