Changing Conceptualization of Renormalization Theory

Abstract

Quantum field theory (QFT) was developed during the late 1920s to describe the interaction of charged particles with the electromagnetic field. During the 1930s the formalism was extended by Fermi to model β-decay phenomena and by Yukawa to explain nuclear forces. All these field theories have one feature in common: the interaction between the fields takes place at a single point in space-time. Such local quantum field theories seem to be the most efficient way to obtain a synthesis of quantum mechanics and special relativity consistent with the principle of causality. However, local quantum field theories are flawed: their perturbative solutions are divergent, the infinities that are encountered being a consequence of the locality of the interaction. Stimulated by important experimental advances after World War II (the measurement of the Lamb shift and of the hyperfine structure of hydrogen), renormalization theory was formulated to circumvent the divergence difficulties encountered in higher order calculations in quantum electrodynamics (QED). In this approach QED is defined by a limiting procedure. A cutoff is introduced into the theory so that the physics at momenta higher than some momentum—or equivalently the physics at distances shorter than some cut-off length—is altered and all calculated quantities thereby rendered finite but cut-off dependent. The parameters of the cut-off theory are then expressed in terms of physically measurable quantities (such as the mass and charge of the particles which are described by the theory).