Renormalization

Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. Even if no infinities arose in loop diagrams in quantum field theory, it can be shown that it is necessary to renormalize the mass and fields appearing in the original Lagrangian. This is the dominant method used in theoretical physics to treat these divergent quantities due its broad applicability, though more limited but rigorous approaches like causal perturbation theory are also used.

For example, an electron theory may begin by postulating an electron with an initial mass and charge. In quantum field theory a cloud of virtual particles, such as photons, positrons, and others surrounds and interacts with the initial electron. Accounting for the interactions of the surrounding particles (e.g. collisions at different energies) shows that the electron-system behaves as if it had a different mass and charge than initially postulated. Renormalization, in this example, mathematically replaces the initially postulated mass and charge of an electron (the bare particle) with the experimentally observed mass and charge (the dressed particle). Mathematics and experiments prove that positrons and more massive particles such as protons exhibit precisely the same observed charge as the electron – even in the presence of much stronger interactions and more intense clouds of virtual particles. Renormalization procedures are based on the requirement that certain physical quantities (such as the mass and charge of an electron) equal observed (experimental) values.

Renormalization specifies relationships between parameters in the theory when parameters describing large distance scales differ from parameters describing small distance scales. Physically, the pileup of contributions from an infinite number of scales involved in a problem may then result in further infinite quantities. When describing spacetime as a continuum, certain statistical and quantum mechanical constructions are not well-defined. To define them unambiguously, a continuum limit must carefully remove "construction scaffolding" of lattices at various scales.

Renormalization was first developed in quantum electrodynamics (QED) to make sense of infinite integrals in perturbation theory. Initially viewed as a suspect provisional procedure even by some of its originators, renormalization eventually was embraced as an important and self-consistent actual mechanism of scale physics in several fields of physics and mathematics. Despite his later skepticism, it was Paul Dirac who pioneered renormalization.

Today, on the basis of the breakthrough renormalization group insights of Nikolay Bogolyubov and Kenneth Wilson, the focus of studies of renormalization is on variation of physical quantities across contiguous scales; distant scales are instead related to each other through "effective" descriptions. All scales are linked in a broadly systematic way, and the actual physics pertinent to each is extracted with suitable computational techniques appropriate for each. Wilson clarified which variables of a system are crucial and which are redundant.

Renormalization is performed by setting a "renormalization scheme", equations which determine how the physical parameters are rescaled within the theory. Different approaches such as on-shell or minimal subtraction are required depending on the type of interaction being considered. Renormalization is distinct from regularization, another technique to control infinities by assuming the existence of new unknown physics at new scales, though the two can be used in tandem.