Editing Renormalization (section)

== Self-interactions in classical physics ==
[[Image:Renormalized-vertex.png|thumbnail|upright=1.3|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 [[Elementary particle|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 {{math|''r''<sub>e</sub>}}. The mass–energy in the field is
<math display="block">m_\text{em} = \int \frac{1}{2} E^2 \, dV = \int_{r_\text{e}}^\infty \frac{1}{2} \left( \frac{q}{4\pi r^2} \right)^2 4\pi r^2 \, dr = \frac{q^2}{8\pi r_\text{e}},</math>
which becomes infinite as {{math|''r''<sub>e</sub> → 0}}. This implies that the point particle would have infinite [[inertia]] and thus cannot be accelerated. Incidentally, the value of {{math|''r''<sub>e</sub>}} that makes <math>m_\text{em}</math> equal to the electron mass is called the [[classical electron radius]], which (setting <math>q = e</math> and restoring factors of {{mvar|c}} and <math>\varepsilon_0</math>) turns out to be
<math display="block">r_\text{e} = \frac{e^2}{4\pi\varepsilon_0 m_\text{e} c^2} = \alpha \frac{\hbar}{m_\text{e} c} \approx 2.8 \times 10^{-15}~\text{m},</math>
where <math>\alpha \approx 1/137</math> is the [[fine-structure constant]], and <math>\hbar/(m_\text{e} c)</math> is the reduced [[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.{{Citation needed|date=March 2015}} This was called ''renormalization'', and [[Hendrik Lorentz|Lorentz]] and [[Max Abraham|Abraham]] attempted to develop a classical theory of the electron this way. This early work was the inspiration for later attempts at [[regularization (physics)|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 [[electromagnetism|electromagnetic]] interactions of [[electric charge|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 law|inverse-square]].

The [[Abraham–Lorentz force|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]]{{citation needed|date=May 2025}} 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.