Editing Quantum field theory (section)

===Infinities and renormalization===
[[Robert Oppenheimer]] showed in 1930 that higher-order perturbative calculations in QED always resulted in infinite quantities, such as the electron [[self-energy]] and the vacuum zero-point energy of the electron and photon fields,<ref name="weisskopf" /> suggesting that the computational methods at the time could not properly deal with interactions involving photons with extremely high momenta.{{r|weinberg|page1=25}} It was not until 20 years later that a systematic approach to remove such infinities was developed.

A series of papers was published between 1934 and 1938 by [[Ernst Stueckelberg]] that established a relativistically invariant formulation of QFT. In 1947, Stueckelberg also independently developed a complete renormalization procedure. Such achievements were not understood and recognized by the theoretical community.<ref name="weisskopf" />

Faced with these infinities, [[John Archibald Wheeler]] and Heisenberg proposed, in 1937 and 1943 respectively, to supplant the problematic QFT with the so-called [[S-matrix theory]]. Since the specific details of microscopic interactions are inaccessible to observations, the theory should only attempt to describe the relationships between a small number of [[observable]]s (''e.g.'' the energy of an atom) in an interaction, rather than be concerned with the microscopic minutiae of the interaction. In 1945, [[Richard Feynman]] and Wheeler daringly suggested abandoning QFT altogether and proposed [[action-at-a-distance]] as the mechanism of particle interactions.{{r|weinberg|page1=26}}

In 1947, [[Willis Lamb]] and [[Robert Retherford]] measured the minute difference in the <sup>2</sup>''S''<sub>1/2</sub> and <sup>2</sup>''P''<sub>1/2</sub> energy levels of the hydrogen atom, also called the [[Lamb shift]]. By ignoring the contribution of photons whose energy exceeds the electron mass, [[Hans Bethe]] successfully estimated the numerical value of the Lamb shift.<ref name="weisskopf" />{{r|weinberg|page1=28}} Subsequently, [[Norman Myles Kroll]], Lamb, [[James Bruce French]], and [[Victor Weisskopf]] again confirmed this value using an approach in which infinities cancelled other infinities to result in finite quantities. However, this method was clumsy and unreliable and could not be generalized to other calculations.<ref name="weisskopf" />

The breakthrough eventually came around 1950 when a more robust method for eliminating infinities was developed by [[Julian Schwinger]], [[Richard Feynman]], [[Freeman Dyson]], and [[Shinichiro Tomonaga]]. The main idea is to replace the calculated values of mass and charge, infinite though they may be, by their finite measured values. This systematic computational procedure is known as [[renormalization]] and can be applied to arbitrary order in perturbation theory.<ref name="weisskopf" />  As Tomonaga said in his Nobel lecture:<blockquote>Since those parts of the modified mass and charge due to field reactions [become infinite], it is impossible to calculate them by the theory. However, the mass and charge observed in experiments are not the original mass and charge but the mass and charge as modified by field reactions, and they are finite. On the other hand, the mass and charge appearing in the theory are… the values modified by field reactions. Since this is so, and particularly since the theory is unable to calculate the modified mass and charge, we may adopt the procedure of substituting experimental values for them phenomenologically... This procedure is called the renormalization of mass and charge… After long, laborious calculations, less skillful than Schwinger's, we obtained a result... which was in agreement with [the] Americans'.<ref>{{cite journal |last1=Tomonaga |first1=Shinichiro |title=Development of Quantum Electrodynamics |url=https://www.nobelprize.org/prizes/physics/1965/tomonaga/lecture/ |journal=Science|year=1966 |volume=154 |issue=3751 |pages=864–868 |doi=10.1126/science.154.3751.864 |pmid=17744604 |bibcode=1966Sci...154..864T }}</ref></blockquote>

By applying the renormalization procedure, calculations were finally made to explain the electron's [[anomalous magnetic moment]] (the deviation of the electron [[g-factor (physics)|''g''-factor]] from 2) and [[vacuum polarization]]. These results agreed with experimental measurements to a remarkable degree, thus marking the end of a "war against infinities".<ref name="weisskopf" />

At the same time, Feynman introduced the [[path integral formulation]] of quantum mechanics and [[Feynman diagrams]].{{r|shifman|page1=2}} The latter can be used to visually and intuitively organize and to help compute terms in the perturbative expansion. Each diagram can be interpreted as paths of particles in an interaction, with each vertex and line having a corresponding mathematical expression, and the product of these expressions gives the [[scattering amplitude]] of the interaction represented by the diagram.{{r|peskin|page1=5}}

It was with the invention of the renormalization procedure and Feynman diagrams that QFT finally arose as a complete theoretical framework.{{r|shifman|page1=2}}