Renormalization is a crucial technique in quantum field theory (QFT) that allows us to handle infinities that arise in calculations involving quantum fields. In QFT, interactions are described by Feynman diagrams, which represent different possible ways particles can interact and exchange energy and momentum. When calculating the probability amplitudes for these interactions, certain terms can lead to infinite results.
The purpose of renormalization is to remove these infinities and obtain meaningful, finite predictions from quantum field theories. It involves two main steps: regularization and renormalization proper.
Regularization: Infinities arise because QFT calculations involve integrals that can diverge at certain points. Regularization is a technique used to give these divergent integrals a well-defined mathematical meaning. It introduces a regulator, typically a parameter or cutoff, which effectively sets a finite limit on the momentum or energy that can be exchanged in the interaction.
Renormalization: After the regularization step, the divergent integrals can be expressed as a sum of finite parts and divergent terms. Renormalization is the process of absorbing these divergent terms into the parameters of the theory, such as masses and coupling constants. By redefining these parameters appropriately, the theory can still yield finite, meaningful predictions.
Renormalization differs from traditional perturbation methods in the sense that it goes beyond perturbation theory's usual approach of expanding calculations in a series of powers of a small parameter. While perturbation theory can be used to calculate approximate results to a certain order, renormalization is required to handle the divergences that arise at each order of the perturbative expansion.
Renormalization is a powerful and successful technique that has been extensively used in QFT, particularly in the framework of the Standard Model of particle physics. It allows us to obtain meaningful and accurate predictions that can be tested against experimental observations.