Billiard Field Theory

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It is well known, from daily experience, that any physical system (e.g. a gas, a liquid) will reach equilibrium when left on its own for a long period of time. The process of reaching equilibrium is often called thermalisation. This can stand for the melting of an ice cube on a plate at room temperature, or the dissipation of small waves at the surface of a pond. Thermalisation is usually explained by assuming that there is a very complex dynamics at smaller scales. This is sometimes dubbed as 'microscopic chaos'. Another important ingredient for thermalisation is that the environment may act upon the considered system (the warmer air surrounding the ice cube, the still surface around the perturbation in a pond). Recently the role of both those fundamental ingredients to describe thermalisation has been challenged. The first ingredient, (microscopic) chaos, can be proved to be absent in the specfic case of an integrable system. Integrability is a very specific property, which claims e.g. that, apart from the total energy, there are infinitely many conserved quantities during the time evolution. The second ingredient can nowadays be made effectively absent in cold atom experiments. At very low temperature it is possible to observe a system where the interaction with its environment is negligible. Indeed it was observed that standard thermalisation fails!This project aims to tackle the question of thermalisation for a new class of models. Those models are especially relevant as they can be tuned to be integrable or fully chaotic at the microscopic level. Hence they sit in a unique position to enable one to fully understand the relevant and required assumptions for thermalisation to occur. For the sake of simplicity our models deal with isolated systems so our predictions will be of direct relevance for the experiments described above. A very powerful tool to describe the possible equilibria of a system is called statistical field theory. This has been successful to analyse the effects of the symmetry on the possible equilibria of a given system. Our models sits in a group of models called (non)linear sigma models. The main idea is to enforce the symmetry effects in a geometrical manner. It is remarkable that the standard sigma models have consisted only of geometries without edges (e.g. the surface of a torus or a sphere). One central aspect of this project is to study the effects of having a boundary (hard wall). Those effects connect sigma models to mathematical billiards. Those consist of tracing a ray of light trapped inside a table with an arbitrarily chosen shape. For a rectangular billiard table, the ray will have an integrable time evolution. If two half-disks are glued to the smallest sides, one gets a stadium-like shape for which the time evolution meets the strongest criterion for chaos. In particular two rays starting from neighbouring positions will depart quickly from each other.The second important aspect of the project is to focus on the subtle regime where quantum particles (or fields) start to show similarity with their non-quantum (classical) counterpart, typically at moderate or high energy. This regime is called semiclassical, for which a specific toolbox has been used to study the quantum version of mathematical billiards. Our aim is to transfer this accumulated expertise to fields in sigma models. We shall start with simpler billiard shapes, also to compare with numerous alternative approaches. Then we will implement the semiclassical tools for fields trapped in a billiard table of arbitrary shape. We believe that this can lead to field theories of new symmetry class and enable one to use non-perturbative techniques for non-integrable field theories.