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

We introduce the basic ideas and the framework of the non-perturbative reno rmalization group particularly for pedestrians using elementary examples. F irst we briefly review the history of the renormalization theory and the re normalization group. We will make it clear that the modern renormalization theory is constructed on the idea of the renormalization group and it is qu ite a new type of theory in physics. Then we derive the exact non-pecturbative renormalization group equation an d set up its systematic approximation method. The lowest order approximatio n called the local potential approximation is applied to scalar theories wi th the ferromagnetic transition and quantum mechanics with tunneling. We co mpare our results with other methods, and will show that the non-perturbati ve renormalization group method is promising since it gives fairly good res ults already in the lowest order approximation and it does not suffer any d ivergent series expansion. As a typical application in high energy physics, we analyze the dynamical c hiral symmetry breaking in gauge theories and investigate the chiral phase structures. Our new method improves results by the ladder Schwinger-Dyson e quation so that the physical results might be less gauge dependent.