Abstract
Scale invariance is a concept encountered in many branches of physics, from the behavior of a system in the vicinity of a phase transition to high-energy physics. In this latest lecture, we approach it from the angle of the classical field theory used so far to describe a two-dimensional Bose gas. It takes advantage of the fact that the various contributions to the fluid's energy behave in a very simple way when several thermodynamic variables are multiplied by the same scaling factor. This invariance of scale also has consequences for the dynamic properties of the system. One example is the "breathing mode" of a gas confined in an isotropic harmonic trap, which always oscillates at twice the frequency of the trap, whatever the strength of the interactions. The results we have just mentioned relate to a fluid described by a classical field, in which the interactions are taken into account by a constant, dimensionless parameter. Describing interactions in this way is tantamount to assuming that the potential between atoms is proportional to a two-dimensional Dirac distribution. However, this approach is quantum singular, and it is necessary to regularize the Dirac distribution. This leads to a phenomenon known as the "quantum anomaly": we start with a problem that has exact symmetry at the classical level, in this case scale invariance. But quantization of the problem requires regularization to eliminate certain discrepancies. The anomaly then lies in the fact that the regularized version necessarily leads to a breaking of the initial symmetry. In the case of 2D gas, regularization of the contact potential necessarily introduces an energy that breaks the scale invariance found with the classical field. After describing this regularization procedure in detail, we ended this lecture by examining the consequences of scale invariance breaking on the breathing mode.
