In quantum field theory, renormalization is a technique used to address infinities that arise in calculations involving quantum fields. It is a mathematical procedure that allows physicists to remove or absorb these infinities into a few well-defined parameters of the theory, such as masses and coupling constants. By doing so, renormalization enables meaningful and finite predictions to be made from the theory.
The need for renormalization arises due to the nature of quantum field theories. In these theories, elementary particles and their interactions are described by fields that permeate all of space and time. When performing calculations involving these fields, such as computing particle interactions or scattering amplitudes, infinities can appear in intermediate steps or final results.
These infinities typically arise from two main sources:
Ultraviolet Divergences: These are divergences that arise from considering quantum fluctuations of the fields at arbitrarily high energies or short distances. Theories without a cutoff or a way to regularize these divergences would produce infinite results.
Infrared Divergences: These divergences occur in certain situations involving massless particles, such as photons or gluons. They arise when particles can have arbitrarily low energies or long wavelengths, leading to divergent integrals.
Renormalization tackles these infinities by introducing systematic procedures to remove them. The process involves several steps:
Regularization: To deal with ultraviolet divergences, a regularization scheme is employed to impose a cutoff on the theory. This effectively limits the energies or momenta of the particles involved. Common regularization methods include dimensional regularization and momentum cutoff regularization.
Counterterms: After regularization, the theory contains finite but divergent quantities. Counterterms are introduced to cancel out these divergences. These counterterms are added to the original Lagrangian of the theory and contain parameters that need to be determined experimentally.
Renormalization Conditions: Renormalization conditions are imposed to fix the values of the parameters in the counterterms. These conditions are typically chosen based on experimental observables or physical requirements. For example, the values of masses and coupling constants can be fixed by measurements.
Renormalization Group: The renormalization group equations describe how the parameters in the theory evolve as the energy scale changes. These equations allow physicists to study the behavior of the theory at different energy scales and make predictions about its behavior in different regimes.
By following these steps, the infinities in the theory can be absorbed into a small set of renormalized parameters, which are then used to make finite predictions. Renormalization has been highly successful in quantum field theory, enabling precise calculations that agree with experimental measurements to remarkable accuracy.