Renormalization is a powerful technique in quantum field theory (QFT) used to handle infinities that arise in calculations involving quantum interactions. In QFT, particles and fields are described by mathematical objects called Feynman diagrams, which represent possible interactions and their associated probabilities.
However, when performing calculations using Feynman diagrams, certain integrals can produce divergent or infinite results. These infinities typically arise from quantities such as the self-energy of particles, the interaction vertices, or vacuum fluctuations. These divergences pose a significant challenge and require a systematic approach to address them.
Renormalization provides a framework to deal with these infinities and obtain meaningful, finite results. The idea behind renormalization is to absorb the infinities into redefined parameters of the theory while still preserving the physically observable quantities.
There are several steps involved in the renormalization procedure:
UV Divergences: The infinities that arise in QFT calculations are primarily due to ultraviolet (UV) divergences. These divergences occur when integrating over high-energy virtual particles that are inaccessible to direct measurement. UV divergences appear as infinite contributions to quantities like particle masses or coupling constants.
Regularization: To handle these infinities, a regularization scheme is employed to introduce a cutoff parameter. This parameter restricts the integration over momenta to a limited range, avoiding the problematic high-energy contributions. Common regularization techniques include dimensional regularization or momentum cutoff regularization.
Counterterms: The cutoff-dependent infinities are absorbed by introducing additional terms called counterterms into the Lagrangian of the theory. Counterterms include parameters like mass counterterms or coupling counterterms. These counterterms mimic the divergent behavior of the original quantities but with opposite signs to cancel out the infinities.
Renormalization Conditions: To fix the values of the counterterms, renormalization conditions are imposed. These conditions are typically chosen by comparing theory predictions with experimental data or by requiring specific physical properties to hold. For example, in electrodynamics, the electric charge can be renormalized by setting it equal to the measured value at a specific energy scale.
Renormalization Group: Once the counterterms are determined, the renormalized theory is obtained. However, the counterterms now depend on the cutoff scale, which introduces a dependence on the energy scale at which the theory is probed. The renormalization group equations describe how the parameters of the theory, including the counterterms, evolve with the energy scale.
Renormalization ensures that the infinities encountered in QFT calculations are absorbed into the redefined parameters while maintaining the agreement between theory and experimental observations. It allows for meaningful calculations of physical observables, such as cross-sections or decay rates, and provides a consistent framework for understanding and predicting particle interactions.