Arithmetical surprises from the quantum universe.
The classical notion of trajectory in physics has its foundation in common sense: the position and the velocity of an object can be predicted by computations and measured at a given time. As opposed to the classical universe, the behaviour of elementary particles is elusive. Only the probabilities of physical events can be predicted. These probabilities are sums of all possible trajectories of the quantum system from the initial to the final state.
The Quantum Field Theory (QFT) aims to predict the probabilities of the interactions of elementary particles. Its predictions have been put through the test many times, and have come through unscathed. Almost 50 years ago, QFT predicted the existence of the Higgs particle to explain why fundamental particles have mass. The CERN particle accelerator in Switzerland/France made headlines last year when it finally confirmed the existence of this particle.
Despite its very precise theoretical predictions that have been confirmed time and again in experiments, the mathematical aspects of QFT have been surprising researchers. The FP7 project "From QFT to motives and 3-manifolds", in which the University of Luxembourg is involved, targets one of these intriguing mathematical problems arising from QFT.
Some numbers are more equal than others, according to the quantum universe
The numbers arising from the computations of probabilities in general are not particularly notable. They can be fractions of natural numbers or transcendental numbers, and their decimal representations can have any number of digits including infinity.
While considering the quantum events, however, the known computations in quantum field theory result in a very particular class of numbers, which are known as multiple ?-values. These numbers are of great interest in number theory. Even though they are transcendental in general, they are described by finite data, in other words, they are not as generic as transcendental numbers are expected to be.
A priori, there is no reason for obtaining these numbers. Their presence hints at new arithmetic structures of the quantum system that we were not aware of. Up until recent progress had been made with the project, it was not known whether these particular numbers, multiple ?-values, must always be present or whether it is just a coincidence for the computations that we are capable of realising today.
Geometry behind the numbers
The “From QFT to motives and 3-manifolds” project investigates the geometry behind the elementary particle physics to uncover the mystery behind the persistence of multiple ?-values. The idea behind the solution to the puzzle is not more sophisticated than the elementary computations of the area of geometric shapes. Together with Professor Matilde Marcolli from Caltech, the project consortium has started an ambitious programme aimed at reformulating the problem in a suitable geometric setting. This approach proved successful in explaining the puzzle of numbers in the quantum universe.
The arithmetic properties investigated in the project suggest that the quantum universe has some additional symmetries that we cannot completely formulate yet. The advances in this research area, and this project in particular, represent interesting progress towards a better understanding of the quantum universe.