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.See e.g., Weinberg vol I, chapter 10. This is the dominant method used in theoretical physics to treat these divergent quantities due to 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.
Self-interactions in classical physics
The problem of infinities first arose in the classical electrodynamics of point particles in the 19th and early 20th century. The mass of a charged particle should include the mass–energy in its electrostatic field (electromagnetic mass). Assume that the particle is a charged spherical shell of radius . The mass–energy in the field is
which becomes infinite as . This implies that the point particle would have infinite inertia and thus cannot be accelerated. Incidentally, the value of that makes equal to the electron mass is called the classical electron radius, which (setting and ignoring factor of ) turns out to be
where is the fine-structure constant, and is the reduced Compton wavelength of the electron.
The total effective mass of a spherical charged particle includes the actual bare mass of the spherical shell (in addition to the mass mentioned above associated with its electric field). If the shell's bare mass is allowed to be negative, it might be possible to take a consistent point limit. This was called renormalization, and Lorentz and Abraham attempted to develop a classical theory of the electron this way. This early work was the inspiration for later attempts at regularization and renormalization in quantum field theory.
When calculating the electromagnetic interactions of charged particles, the back-reaction of a particle's own field on itself (analogous to the back-EMF of circuit analysis.) is necessary to explain the friction on charged particles when they emit radiation. If the electron is assumed to be a point, the value of the back-reaction diverges, for the same reason that the mass diverges, because the field is inverse-square.
The Abraham–Lorentz theory includes a retrocausal "pre-acceleration", allowing solution where an electron would start moving before the force is applied. These problems remain in the relativistic version of the Abraham-Lorentz equation. This is a sign that the point limit is inconsistent, and/or that a quantum mechanical treatment is required.
The trouble was worse in classical field theory than in quantum field theory, because in quantum field theory a charged particle experiences Zitterbewegung due to interference with virtual particle–antiparticle pairs, thus effectively smearing out the charge over a region comparable to the Compton wavelength. In quantum electrodynamics at small coupling, the electromagnetic mass only diverges as the logarithm of the radius of the particle.
Divergences in quantum electrodynamics
Divergences within quantum field theories are common, but were first encountered in the development of quantum electrodynamics in the 1930s by Max Born, Werner Heisenberg, Pascual Jordan, and Paul Dirac They discovered that, upon adding in perturbative corrections, many integrals within the theory were divergent.
One way of describing the perturbation theory corrections' divergences was discovered in 1947–49 by Hans Kramers,Kramers presented his work at the 1947 Shelter Island Conference, repeated in 1948 at the Solvay Conference. The latter did not appear in print until the Proceedings of the Solvay Conference, published in 1950 (see Laurie M. Brown (ed.), Renormalization: From Lorentz to Landau (and Beyond), Springer, 2012, p. 53). Kramers' approach was nonrelativistic (see Jagdish Mehra, Helmut Rechenberg, The Conceptual Completion and Extensions of Quantum Mechanics 1932–1941. Epilogue: Aspects of the Further Development of Quantum Theory 1942–1999: Volumes 6, Part 2, Springer, 2001, p. 1050). Hans Bethe, Julian Schwinger, Richard Feynman, and Shin'ichiro Tomonaga, and systematized by Freeman Dyson in 1949. The divergences appear in radiative corrections involving Feynman diagrams with closed loops of virtual particles in them.
While virtual particles obey conservation of energy and momentum, they can have any energy and momentum, even off-shell ones that are not allowed by the relativistic energy–momentum relation for the observed mass of that particle. When there is a loop, the momentum of the particles involved in the loop is not uniquely determined by the energies and momenta of incoming and outgoing particles. A variation in the energy of one particle in the loop can be balanced by an equal and opposite change in the energy of another particle in the loop, without affecting the incoming and outgoing particles. Thus many variations are possible. Finding the amplitude for the loop process requires integration over all possible combinations of energy and momentum that could travel around the loop. These integrals are often divergent.
Ultraviolet divergences These divergences have two types; the first is "ultraviolet" (UV), which comes from the region in the integral where all particles in the loop have large energies and momenta, very short wavelengths and high-frequencies fluctuations of the fields, in the path integral for the field, very short proper time between particle emission and absorption, if the loop is thought of as a sum over particle paths.
Shown in the pictures at the right margin, there are exactly three one-loop divergent loop diagrams in quantum electrodynamics:
Infrared divergences The second class of divergence called an infrared divergence. Every process involving charged particles emits infinitely many coherent photons of infinite wavelength, and the amplitude for emitting any finite number of photons is zero. For example, at the one-loop order, the vertex function has both ultraviolet and infrared divergences. In contrast to the ultraviolet divergence, the infrared divergence does not require the renormalization of a parameter in the theory involved. The infrared divergence of the vertex diagram is removed by including a diagram similar to the vertex diagram with the following important difference: the photon connecting the two legs of the electron is cut and replaced by two on-shell (i.e. real) photons whose wavelengths tend to infinity; this diagram is equivalent to the bremsstrahlung process. This additional diagram must be included because there is no physical way to distinguish a zero-energy photon flowing through a loop as in the vertex diagram and zero-energy photons emitted through bremsstrahlung. From a mathematical point of view, the IR divergences can be regularized by assuming fractional differentiation with respect to a parameter, for example:
is well defined at but is UV divergent; if we take the -th fractional derivative with respect to , we obtain the IR divergence
so IR divergences can be converted into UV divergences.
Example calculation
The diagram in Figure 2 shows one of the several one-loop contributions to electron–electron scattering in QED. The electron on the left side of the diagram, represented by the solid line, starts out with 4-momentum and ends up with 4-momentum . It emits a virtual photon carrying to transfer energy and momentum to the other electron (the black curved line). But in this diagram, before that happens, it emits another virtual photon carrying 4-momentum , which is reabsorbed after emitting the first virtual photon. Energy and momentum conservation do not determine the 4-momentum uniquely, so all possibilities contribute equally and we must integrate.
This diagram's amplitude ends up with, among other things, a factor from the loop of The various factors in this expression are gamma matrices as in the covariant formulation of the Dirac equation; they have to do with the spin of the electron. Here is the electric coupling constant, and is the imaginary unit. Key here is the dependence on of the three fractional factors in the integrand, which are from the propagators of the two electron lines and the photon line in the loop.
Expanding the integrand out, there is a term with two powers of in the numerator that dominates at large values of : This integral is divergent, unless the energy and momentum are cut off at some finite value.
Renormalized and bare quantities