In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization refers to a collection of techniques used to take a continuum limit.

When describing space and time as a continuum, certain statistical and quantum mechanical constructions are ill defined. In order to define them, the continuum limit has to be taken carefully.

Renormalization determines the relationship between parameters in the theory, when the parameters describing large distance scales differ from the parameters describing small distances. Renormalization was first developed in quantum electrodynamics (QED) to make sense of infinite integrals in perturbation theory. Initially viewed as a suspect, provisional procedure by some of its originators, renormalization eventually was embraced as an important and self-consistent tool in several fields of physics and mathematics.

Self-interactions in classical physics

Figure 1. Renormalization in quantum electrodynamics: The simple electron-photon interaction that determines the electron's charge at one renormalization point is revealed to consist of more complicated interactions at another.

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. Assume that the particle is a charged spherical shell of radius r_e. The energy in the field is

and it is infinite when r_e is zero, which is obviously absurd (because it implies that the point particle would have infinite inertia, and be unable to be accelerated). Incidentally, the value of r_e that makes m_em equal to the electron mass is called the classical electron radius, which (restoring factors of c and ε₀) turns out to be α times smaller than the 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 aforementioned mass 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.[1] 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, it is tempting to ignore the back-reaction of a particle's own field on itself. But this back reaction 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 had a noncausal "pre-acceleration". Sometimes an electron would start moving before the force is applied. This is a sign that the point limit is inconsistent. An extended body will start moving when a force is applied within one radius of the center of mass.

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 log of the radius of the particle.

Many physicists^[who?] believe^{[citation needed]} that when the fine structure constant is much greater than one, so that the classical electron radius is bigger than the quantum wavelength, the same problems that plague classical electrodynamics are still present in quantum electrodynamics.

Divergences in quantum electrodynamics

Figure 2. A diagram contributing to electron-electron scattering in QED. The loop has an ultraviolet divergence.

Vacuum polarization, a.k.a. charge screening. This loop has a logarithmic ultraviolet divergence.

Self energy diagram in QED.

When developing quantum electrodynamics in the 1930s, Max Born, Werner Heisenberg, Pascual Jordan, and Paul Dirac discovered that in perturbative calculations many integrals were divergent.

One way of describing the divergences was discovered in the 1930s by Ernst Stueckelberg, in the 1940s by Julian Schwinger, Richard Feynman, and Shin'ichiro Tomonaga, and systematized by Freeman Dyson. The divergences appear in calculations 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 one that is not allowed by the relativistic energy-momentum relation for the observed mass of that particle. (That is, E² − p² is not necessarily the mass of the particle in that process (e.g. for a photon it could be nonzero).) Such a particle is called off-shell. 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 variation in the energy of another particle in the loop. So to find the amplitude for the loop process one must integrate over all possible combinations of energy and momentum that could travel around the loop.

These integrals are often divergent, that is, they give infinite answers. The divergences which are significant are the "ultraviolet" (UV) ones. An ultraviolet divergence can be described as one 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.

So these divergences are short-distance, short-time phenomena.

There are exactly three one-loop divergent loop diagrams in quantum electrodynamics.

1. a photon creates a virtual electron-positron pair which then annihilate, this is a vacuum polarization diagram.
2. an electron which quickly emits and reabsorbs a virtual photon, called a self-energy.
3. An electron emits a photon, emits a second photon, and reabsorbs the first. This process is shown in figure 2, and it is called a vertex renormalization.

The three divergences correspond to the three parameters in the theory:

1. the field normalization Z.
2. the mass of the electron.
3. the charge of the electron.

A second class of divergence, called an infrared divergence, is due to massless particles, like the photon. 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 photons, these divergences are well understood. For example, at the 1-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. 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.

A loop divergence

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 four-momentum p^μ and ends up with four-momentum r^μ. It emits a virtual photon carrying r^μ − p^μ to transfer energy and momentum to the other electron. But in this diagram, before that happens, it emits another virtual photon carrying four-momentum q^μ, and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the four-momentum q^μ uniquely, so all possibilities contribute equally and we must integrate.