Renormalization from Classical to Quantum Physics

The concept of renormalization was first introduced by Dirac to investigate the innite self energy of an electron classically. This radical theory was probably the first time when an infinity occurring in a physical system was systematically investigated. This thesis presents a new perspective of renormalization by introducing methods from metric geometry to control divergences. We start by extending Dirac's work and analyzing how the radiation reaction due to the precision of the electron's magnetic moment affects its motion. This is followed by modeling scalar field theory on lattices of various kinds. Scale invariance, which plays a major role in the very few renormalizable theories in nature, is inbuilt in our formalism. We also use Wilson's ideas of effective theory and finite element methods to study continuum systems. Renormalization group transformations form the central theme in this picture. By incorporating finite element methods, an idea borrowed from mechanical engineering, we study scalar fields on triangular lattices in a hierarchal manner. In our case, the cotangent formula turns out to be a fixed point of the renormalization group transformations. We end our thesis by introducing a new metric for space-time which emerges from the scalar field itself. The standard techniques used in the theory of renormalization so far attempt to redefine coupling constants of the theory to remove divergences at short distance scales. In our formalism, we deduce the distance scale itself. In our notion of distance, built from correlation functions of the fields, the divergences disappear.