Renormalization group

The renormalization group (RG) is a mathematical tool used in theoretical physics to study how physical systems change when viewed at different scales. These scales typically describe the interactions of objects, such as variable couplings that measure the strength of various forces, mass parameters, or the size of the system. The RG is closely related to scale invariance and conformal invariance, which are symmetries where a system exhibits self-similarity.

In particle physics, the renormalization group reflects changes in the underlying physical laws as the energy or mass scale at which physical processes occur varies. For instance, in quantum electrodynamics (QED), an electron appears to be composed of electron and positron pairs and photons at very short distances. This composition results in a slightly different electric charge than the dressed electron observed at larger scales.

The concept of fixed points in the renormalization group is crucial. At a fixed point, the parameters of the model can be assigned special values where the field theory becomes conformally invariant, and any running couplings cease to change. This self-similarity at fixed points helps physicists understand and predict the behavior of physical systems across different scales.