Precisely computing elementary particles

Dirk Kreimer’s working materials could not be simpler: pen and paper, or chalk and blackboard – from numbers and mathematical symbols equations do grow. The calculations done by this internationally respected scientist are, by contrast, complicated: they describe the behaviour of the tiniest particles of matter, elementary particles. And the basis for all this is what is known as quantum field theory.

The professor of mathematical physics, Kreimer, is regarded as one of the top researchers in the field of quantum field theory and a mastermind of particle physics. He identifies mathematical structures in quantum field theory calculations which might help particle physicists to expedite the preparation and evaluation of their experiments. “In physics,” explains Kreimer, “symmetries play an important role. In particle physics, for example, they help to determine the kinds of coupling between particles.” Dirk Kreimer is trying to express these symmetries in precise mathematical terms. In doing so, he has encountered structures that have been used in mathematics for several decades: Hopf algebra.

The physicists’ recipe

Quantum field theory goes back to a problem physicists were struggling with at the beginning of the 20th century: they could not reconcile quantum mechanics, which describe the physical properties of elementary particles, with the specific theory of relativity developed by Albert Einstein. There was a discrepancy between measurements derived from experiments and theoretical predictions. In order to solve this problem, physicists created a mathematical technique called renormalisation. This procedure allowed them to develop a standard model of particle physics which provided a coherent explanatory model for the interactions of elementary particles without, however, factoring in gravity as a force field. It is in this ‘recipe’ used by physicists that Dirk Kreimer discovered the mathematical structures of Hopf algebra. And by using Hopf algebra in nuclear experiments it is possible to predict the behaviour of elementary particles much more precisely.

Take, for example, the ongoing experiments at the largest research centre for particle physics in Geneva, CERN, in which particles are accelerated to high energies and collided. “The calculations for these experiments are large-scale, strategically planned projects,” Kreimer explains. A whole team of scientists often works on them for years using supercomputers. So far, Kreimer’s mathematical innovations have not been incorporated. But the pure researcher is not disheartened by this. “It may take a while for nuclear physicists to abandon their well-tried algorithms.”

Kreimer is quite convinced that collaboration between mathematics and physics is important for both sides. In physical calculations he has found numbers with certain mathematical properties like pi which combine number theoretic properties with geometry. His insights could help mathematicians to understand number theory better. At the same time, he would like to use modern mathematical methods to spark new ideas amongst his colleagues in physics. “But physicists and mathematicians are only just beginning to talk to each other more seriously,” he observes.

Kreimer is the new bridge between the institutes of mathematics and physics at Humboldt-Universität zu Berlin and he therefore wants to promote cooperation between the two disciplines. Only by knowledge sharing can quantum field theory continue to develop, of this the scientist is sure. The university, for its part, is determined that the informal cooperation that has existed so far should be institutionalised in the coming years to create an Interdisciplinary Centre for Mathematical Physics.

Close contact to young researchers

Coming to Berlin meant that Dirk Kreimer had to take his leave of the prestigious French research institute IHÉS, near Paris. “It’s a paradise for research – no administrative bureaucracy or teaching to be done – but the institute is rather isolated. I hardly had any contact with young people.” According to Kreimer, the time had definitely come to pass on his knowledge to junior researchers.

The Humboldt Professor moved into his office in the mathematics building on the Adlershof Campus at the beginning of 2011. Gradually, the doctoral students in his working group are arriving, too. Altogether, he is creating 16 positions for junior researchers and staff. He will also offer seminars for mathematics and physics students. “Berlin is an attractive location because a lot of good mathematicians and physicists work at the three big universities,” says Kreimer. In order to keep track of experiments and have access to high-capacity computers he will also cooperate with theoretical physicists at the Zeuthen site of the DESY particle accelerator, near Berlin.

In Kreimer’s view, the main aim of his mathematical-physical research is not to decipher what some physicists like to call a ‘theory of everything’. “The most important issue is how we can formulate gravitation in terms of quantum field theory. And if it’s not possible, why isn’t it? That’s where we want to go, but it’ll take a while to get there.” For the time being, Kreimer wants to understand quantum field theory in “clean mathematical terms.”