Introduction to the non-perturbative renormalization group and its recent applications

We introduce the basic ideas and the framework of the non-perturbative renormalization group particularly for pedestrians using elementary examples. First we briefly review the history of the renormalization theory and the renormalization group. We will make it clear that the modern renormalization theory is constructed on the idea of the renormalization group and it is quite a new type of theory in physics. Then we derive the exact non-pecturbative renormalization group equation and set up its systematic approximation method. The lowest order approximation called the local potential approximation is applied to scalar theories with the ferromagnetic transition and quantum mechanics with tunneling. We compare our results with other methods, and will show that the non-perturbative renormalization group method is promising since it gives fairly good results already in the lowest order approximation and it does not suffer any divergent series expansion. As a typical application in high energy physics, we analyze the dynamical chiral symmetry breaking in gauge theories and investigate the chiral phase structures. Our new method improves results by the ladder Schwinger-Dyson equation so that the physical results might be less gauge dependent.