page_title: Renormalization
section_title: Self-interactions in classical physics
hierarchical_section_title: Renormalization / Self-interactions in classical physics
caption_attribution_description: Renormalization in QED
context_page_description: Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, 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. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian.
  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 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 with the experimentally observed mass and charge.
context_section_description: 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 rₑ. The mass–energy in the field is
  which becomes infinite as rₑ → 0. This implies that the point particle would have infinite inertia, making it unable to be accelerated. Incidentally, the value of rₑ that makes  equal to the electron mass is called the classical electron radius, which (setting  and restoring factors of c and ) turns out to be
  where  is the fine-structure constant, and  is the Compton wavelength of the electron.
  Renormalization: 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.
  (See also regularization (physics) for an alternative way to remove infinities from this classical problem, assuming new physics exists at small scales.)
  When calculating the electromagnetic interactions of charged particles, it is tempting to ignore the back-reaction of a particle's own field on itself. (Analogous to the back-EMF of circuit analysis.)  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.
  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.
caption: 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.
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