Renormalization

In quantum field theory (QFT) and the statistical mechanics of fields, renormalization refers to a collection of techniques used to express physical calculations in terms of observable quantities that already include some field effects. Renormalization arose in quantum electrodynamics as a means of making sense of the infinite results of various calculations and extracting finite answers to properly posed physical questions. Initially viewed as a suspect, provisional procedure by most of its originators, renormalization eventually was embraced as an important tool in several fields of physics, as a result of work in effective field theory and the renormalization group.

Divergences in quantum electrodynamics

When developing quantum electrodynamics in the 1940s, Shin'ichiro Tomonaga, Julian Schwinger, Richard Feynman, and Freeman Dyson discovered that, in perturabative calculations, problems with divergent integrals abounded. One way of describing the divergences in QED is that they appear as the consequence of calculations involving Feynman diagrams with closed loops of virtual particles in them. These diagrams appear in the perturbative approximation of quantum field theory. Each looped diagram represents a perturbation, or small correction, to a diagram without loops. Intuitively, diagrams with more and more loops should give smaller and smaller corrections to the values of diagrams which do not contain any loops. However, when the contributions of these loop diagrams are naively calculated, they become infinitely large. One type of loop would be a situation in which a virtual electron-positron pair appear out of the vacuum, interact with various photons, and then annihilate. Another would be an electron-photon interaction as in Figure 2.

Related Topics:
1940s - Shin'ichiro Tomonaga - Julian Schwinger - Richard Feynman - Freeman Dyson - Feynman diagram - Virtual particle - Positron - Vacuum - Photon

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While virtual particles obey conservation of energy and momentum, they can possess combinations of energies and momenta not allowed by the classical laws of motion; physicists say that they are not on shell. Furthermore, whenever a loop appears, the particles involved in the loop are not individually constrained by the energies and momenta of incoming and outgoing particles, since a variation in, say, the energy of one particle in the loop can be balanced by an equal and opposite variation in the energy of another. Therefore, in order to calculate the contribution to a probability amplitude, one must integrate over all possible combinations of energy and momentum in the loop—and these integrals are often divergent, that is, they give infinite answers. The most theoretically troublesome divergences are the "ultraviolet" (UV) ones associated with large energies and momenta of the virtual particles in the loop, or, equivalently, very short wavelengths and high frequencies of the fields for which these particles are the quanta. These divergences are, therefore, fundamentally short-distance, short-time phenomena. (There are infrared divergences as well, but they are not as hard to interpret and are beyond the scope of this article.)

Related Topics:
Conservation of energy - Momentum - Classical - On shell - Probability amplitude - Integrate - Ultraviolet - Wavelength - Frequencies - Infrared divergence

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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^mu and ends up with four-momentum r^mu. It emits a virtual photon carrying r^mu - p^mu 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^mu, and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the four-momentum q^mu uniquely, so all possibilities contribute equally and we must integrate.

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This diagram's amplitude ends up with, among other things, a factor from the loop of

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:

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