# Physicist untangles problems from the border of string theory and the theory of knots

Dr. Piotr Sułkowski is looking for new links between physics and mathematics in the prestigious ERC grant. He investigates issues associated with both the theory of knots fascinating for mathematicians, and with string theory that intrigues physicists.

Forgot tie a tie again? Wondering how to untangle the headphone cable from which became entangled in your pocket? Many of us will sometimes note shortages of information about tying and untying various knots. White spots in this area of knowledge intrigued mathematicians so much that they created a separate branch of mathematics - the theory of knots. Within it, they attempt, for example, to classify knots, describe them, determine how many there are and what properties they have.

Examples of knots

(K)NOT THAT SIMPLE

"It would seem that theory of knots is just a game, a puzzle. But it turns out that it has non-trivial relationships with other branches of science" - said in an interview with PAP Piotr Sułkowski from the Faculty of Physics, University of Warsaw. For his research at the border of knot theory, quantum fields and string theory in 2013 he received prestigious Starting Grant from the European Research Council (ERC).

The researcher wants to use the theory of knots in studies of elementary particles and so-called quantum field theory. "Regardless of more interesting to me aspects of mathematical physics, the theory that I study is used to describe the phenomena that occur in the physical world. These are systems with which can be found in the condensed matter physics, at very low temperatures, e.g. fractional quantum Hall effect" - explained Sułkowski.

KNOT MOLECULE

"In this physical theory can calculate the probability of a particle having a certain trajectory. As a result you get a function. And it turns out that it looks exactly like the functions that mathematicians use to describe knots..." - revealed the researcher. He added that the probabilities do not depend on the shape of the path of the particle, but only on how this path is entangled.

The research of Sułkowski and his team will give mathematicians a reason to be happy, because they will get new functions, the properties of which they can study, and physicists will gain new information on the quantum field theory.

HIT A SENSITIVE STRING

In his research and unusual approach to quantum field theory Piotr Sułkowski also touches the issues related to string theory. This topic is interesting due to the fact that string theory aspires to become a "theory of everything". If that were the case, this theory could consistently describe all the forces and interactions that we observe in the world. For now, however, there is no convincing evidence to support that. It does not prevent scientists from studying this theory from different theoretical angles. And there is no denying that this theory really works on the imagination.

String theory in a nutshell.

"The particles we observe in nature have properties that may seem quite random. Just look at the masses of particles, their charges or spins. It is not quite clear how to explain this randomness" - noted Sułkowski. String theory suggests, however, that there is method to this madness. This theory presupposes that the basic objects that make up our world, are not particles but rather vibrating strings. The idea is that if you look at a vibrating string from a great distance, it may look like a point particle. So researchers are hoping that the apparently random properties of elementary particles are simply manifestations of various types of vibration of the same basic object, which is a string, and that the reality around us can be built of such strings.

EXTRA-DIMENSIONALLY COMPLEX MATTER

String theory predicts that there are more dimensions than the four we already know. Different versions of string theory predict that there should be 10 or even 11 dimensions. "These other dimensions may be very >>small<<, or rather so compacted that they can not be seen" - said Sułkowski and gave the example of a two-dimensional plane - such as a sheet of paper - that you can roll so tight that from a distance it will look like a one-dimensional line. "Our measuring instruments may be unable to detect these unknown dimensions" - he said.

"The properties of dimensions that we can not see should somehow manifest in these four dimensions that we do see. It may turn out that the properties of particles we see in four dimensions, are a consequence of existence of the remaining several dimensions" - said the scientist. He explained that in his work he studies simpler models, in which the 6 hidden dimensions have a relatively simple structure, from which properties of quantum field theory in the 4 known dimensions can be derived. These simplified (from the physics point of view) models are also associated with the aforementioned theory of knots, and, as it turns out, lead to very non-trivial mathematical results.

"In this sense, the physical apparatus of quantum field theory and string theory tells us something new about math. This is a very interesting situation, worth a deeper understanding. Also, when we better understand this type of simple (from a physical perspective) examples, the next step will be to try to generalize them into more realistic, but also more complex models" - said the scientist.

Members of Dr. Sułkowski’s team are scientists from the Faculty of Physics, University of Warsaw, as well as researchers from the USA (Caltech, the Institute for Advanced Study in Princeton, University of California, Davis), France (Institut de Physique Theorique in Saclay) and Portugal (Instituto Superior Tecnico in Lisbon).

In addition to conducting advanced research, Piotr Sułkowski is also involved in the promotion and popularisation of science - including the website "Ask a physicist".

PAP - Science and Scholarship in Poland, Ludwika Tomala

lt/ ula/ agt/ mrt/

tr. RL