Editing Quantum field theory (section)

==History==
{{Main|History of quantum field theory}}
Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between [[light]] and [[electrons]], culminating in the first quantum field theory—[[quantum electrodynamics]]. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the [[renormalization]] procedure. A second major barrier came with QFT's apparent inability to describe the [[weak interaction|weak]] and [[strong interaction]]s, to the point where some theorists called for the abandonment of the field theoretic approach. The development of [[gauge theory]] and the completion of the [[Standard Model]] in the 1970s led to a renaissance of quantum field theory.

===Theoretical background===
[[File:Magnet0873.png|thumb|200px|[[Magnetic field lines]] visualized using [[iron filings]]. When a piece of paper is sprinkled with iron filings and placed above a bar magnet, the filings align according to the direction of the magnetic field, forming arcs allowing viewers to clearly see the poles of the magnet and to see the magnetic field generated.]]

Quantum field theory results from the combination of [[classical field theory]], [[quantum mechanics]], and [[special relativity]].<ref name="peskin"/>{{rp|xi}} A brief overview of these theoretical precursors follows.

The earliest successful classical field theory is one that emerged from [[Newton's law of universal gravitation]], despite the complete absence of the concept of fields from his 1687 treatise ''[[Philosophiæ Naturalis Principia Mathematica]]''. The force of gravity as described by Isaac Newton is an "[[action at a distance]]"—its effects on faraway objects are instantaneous, no matter the distance. In an exchange of letters with [[Richard Bentley]], however, Newton stated that "it is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact".<ref name=Hobson/>{{rp|4}} It was not until the 18th century that mathematical physicists discovered a convenient description of gravity based on fields—a numerical quantity (a [[vector (mathematics and physics)|vector]] in the case of [[gravitational field]]) assigned to every point in space indicating the action of gravity on any particle at that point. However, this was considered merely a mathematical trick.<ref name="weinberg">{{cite journal |last=Weinberg |first=Steven |author-link=Steven Weinberg |date=1977 |title=The Search for Unity: Notes for a History of Quantum Field Theory |journal=Daedalus |volume=106 |issue=4 |pages=17–35 |jstor=20024506 }}</ref>{{rp|18}}

Fields began to take on an existence of their own with the development of [[electromagnetism]] in the 19th century. [[Michael Faraday]] coined the English term "field" in 1845. He introduced fields as properties of space (even when it is devoid of matter) having physical effects. He argued against "action at a distance", and proposed that interactions between objects occur via space-filling "lines of force". This description of fields remains to this day.<ref name=Hobson>{{cite journal
 | last =Hobson
 | first =Art
 | title =There are no particles, there are only fields 
 | journal =[[American Journal of Physics]]
 | volume =81
 | issue =211
 | pages =211–223
 | year =2013
 | doi =10.1119/1.4789885
 | arxiv =1204.4616
 | bibcode =2013AmJPh..81..211H
 | s2cid =18254182
 }}</ref><ref name="Heilbron2003">{{Cite book |url=https://archive.org/details/oxfordcompaniont0000unse_s7n3 |title=The Oxford companion to the history of modern science |date=2003 |publisher=[[Oxford University Press]] |isbn=978-0-19-511229-0 |editor-last=Heilbron |editor-first=J. L. |editor-link=John L. Heilbron |location=Oxford; New York}}</ref>{{rp|301}}<ref name="Thomson1893">{{Cite book |last1=Thomson |first1=Joseph John |author-link1=Joseph John Thomson |url=https://archive.org/details/notesonrecentres00thom |title=Notes on recent researches in electricity and magnetism, intended as a sequel to Professor Clerk-Maxwell's 'Treatise on Electricity and Magnetism' |last2=Maxwell |first2=James Clerk |publisher=[[Clarendon Press]] |year=1893}}</ref>{{rp|2}}

The theory of [[classical electromagnetism]] was completed in 1864 with [[Maxwell's equation]]s, which described the relationship between the [[electric field]], the [[magnetic field]], [[electric current]], and [[electric charge]]. Maxwell's equations implied the existence of [[electromagnetic waves]], a phenomenon whereby electric and magnetic fields propagate from one spatial point to another at a finite speed, which turns out to be the [[speed of light]]. Action-at-a-distance was thus conclusively refuted.<ref name=Hobson/>{{rp|19}}

Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in [[emission spectrum|atomic spectra]], nor for the distribution of [[blackbody radiation]] in different wavelengths.<ref name="weisskopf">{{cite journal |last=Weisskopf |first=Victor |author-link=Victor Weisskopf |date=November 1981 |title=The development of field theory in the last 50 years |journal=[[Physics Today]] |volume=34 |issue=11 |pages=69–85 |doi=10.1063/1.2914365 |bibcode=1981PhT....34k..69W }}</ref> [[Max Planck]]'s study of blackbody radiation marked the beginning of quantum mechanics. He treated atoms, which absorb and emit [[electromagnetic radiation]], as tiny [[oscillator]]s with the crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as [[quantum harmonic oscillator]]s. This process of restricting energies to discrete values is called quantization.<ref name="Heisenberg1999">{{Cite book |last=Heisenberg |first=Werner |author-link=Werner Heisenberg |url=https://archive.org/details/physics-and-philosophy-the-revolution-in-modern-scirnce-werner-heisenberg-f.-s.-c.-northrop |title=Physics and philosophy: the revolution in modern science |publisher=[[Prometheus Books]] |year=1999 |isbn=978-1-57392-694-2 |series=Great minds series |location=Amherst, N.Y}}</ref>{{rp|Ch.2}} Building on this idea, [[Albert Einstein]] proposed in 1905 an explanation for the [[photoelectric effect]], that light is composed of individual packets of energy called [[photon]]s (the quanta of light). This implied that the electromagnetic radiation, while being waves in the classical electromagnetic field, also exists in the form of particles.<ref name="weisskopf" />

In 1913, [[Niels Bohr]] introduced the [[Bohr model]] of atomic structure, wherein [[electrons]] within atoms can only take on a series of discrete, rather than continuous, energies. This is another example of quantization. The Bohr model successfully explained the discrete nature of atomic spectral lines. In 1924, [[Louis de Broglie]] proposed the hypothesis of [[wave–particle duality]], that microscopic particles exhibit both wave-like and particle-like properties under different circumstances.<ref name="weisskopf" /> Uniting these scattered ideas, a coherent discipline, [[quantum mechanics]], was formulated between 1925 and 1926, with important contributions from [[Max Planck]], [[Louis de Broglie]], [[Werner Heisenberg]], [[Max Born]], [[Erwin Schrödinger]], [[Paul Dirac]], and [[Wolfgang Pauli]].{{r|weinberg|page1=22–23}}

In the same year as his paper on the photoelectric effect, Einstein published his theory of [[special relativity]], built on Maxwell's electromagnetism. New rules, called [[Lorentz transformations]], were given for the way time and space coordinates of an event change under changes in the observer's velocity, and the distinction between time and space was blurred.{{r|weinberg|page1=19}} It was proposed that all physical laws must be the same for observers at different velocities, i.e. that physical laws be invariant under Lorentz transformations.

Two difficulties remained. Observationally, the [[Schrödinger equation]] underlying quantum mechanics could explain the [[stimulated emission]] of radiation from atoms, where an electron emits a new photon under the action of an external electromagnetic field, but it was unable to explain [[spontaneous emission]], where an electron spontaneously decreases in energy and emits a photon even without the action of an external electromagnetic field. Theoretically, the Schrödinger equation could not describe photons and was inconsistent with the principles of special relativity—it treats time as an ordinary number while promoting spatial coordinates to [[linear operator]]s.<ref name="weisskopf" />

===Quantum electrodynamics===
Quantum field theory naturally began with the study of electromagnetic interactions, as the electromagnetic field was the only known classical field as of the 1920s.{{r|shifman|page1=1}}

Through the works of Born, Heisenberg, and [[Pascual Jordan]] in 1925–1926, a quantum theory of the free electromagnetic field (one with no interactions with matter) was developed via [[canonical quantization]] by treating the electromagnetic field as a set of [[quantum harmonic oscillator]]s.{{r|shifman|page1=1}} With the exclusion of interactions, however, such a theory was yet incapable of making quantitative predictions about the real world.{{r|weinberg|page1=22}}

In his seminal 1927 paper ''The quantum theory of the emission and absorption of radiation'', Dirac coined the term [[quantum electrodynamics]] (QED), a theory that adds upon the terms describing the free electromagnetic field an additional interaction term between electric [[current density]] and the [[electromagnetic four-potential|electromagnetic vector potential]]. Using first-order [[perturbation theory (quantum mechanics)|perturbation theory]], he successfully explained the phenomenon of spontaneous emission. According to the [[uncertainty principle]] in quantum mechanics, quantum harmonic oscillators cannot remain stationary, but they have a non-zero minimum energy and must always be oscillating, even in the lowest energy state (the [[ground state]]). Therefore, even in a perfect [[vacuum]], there remains an oscillating electromagnetic field having [[zero-point energy]]. It is this [[quantum fluctuation]] of electromagnetic fields in the vacuum that "stimulates" the spontaneous emission of radiation by electrons in atoms. Dirac's theory was hugely successful in explaining both the emission and absorption of radiation by atoms; by applying second-order perturbation theory, it was able to account for the [[scattering]] of photons, [[resonance fluorescence]] and non-relativistic [[Compton scattering]]. Nonetheless, the application of higher-order perturbation theory was plagued with problematic infinities in calculations.<ref name="weisskopf" />{{rp|71}}

In 1928, Dirac wrote down a [[wave equation]] that described relativistic electrons: the [[Dirac equation]]. It had the following important consequences: the [[Spin (physics)|spin]] of an electron is 1/2; the electron [[g-factor (physics)|''g''-factor]] is 2; it led to the correct Sommerfeld formula for the [[fine structure]] of the [[hydrogen atom]]; and it could be used to derive the [[Klein–Nishina formula]] for relativistic Compton scattering. Although the results were fruitful, the theory also apparently implied the existence of negative energy states, which would cause atoms to be unstable, since they could always decay to lower energy states by the emission of radiation.<ref name="weisskopf" />{{rp|71–72}}

The prevailing view at the time was that the world was composed of two very different ingredients: material particles (such as electrons) and [[Field (physics)#Quantum fields|quantum fields]] (such as photons). Material particles were considered to be eternal, with their physical state described by the probabilities of finding each particle in any given region of space or range of velocities. On the other hand, photons were considered merely the [[excited state]]s of the underlying quantized electromagnetic field, and could be freely created or destroyed. It was between 1928 and 1930 that Jordan, [[Eugene Wigner]], Heisenberg, Pauli, and [[Enrico Fermi]] discovered that material particles could also be seen as excited states of quantum fields. Just as photons are excited states of the quantized electromagnetic field, so each type of particle had its corresponding quantum field: an electron field, a proton field, etc. Given enough energy, it would now be possible to create material particles. Building on this idea, Fermi proposed in 1932 an explanation for [[beta decay]] known as [[Fermi's interaction]]. [[Atomic nucleus|Atomic nuclei]] do not contain electrons ''per se'', but in the process of decay, an electron is created out of the surrounding electron field, analogous to the photon created from the surrounding electromagnetic field in the radiative decay of an excited atom.{{r|weinberg|page1=22–23}}

It was realized in 1929 by Dirac and others that negative energy states implied by the Dirac equation could be removed by assuming the existence of particles with the same mass as electrons but opposite electric charge. This not only ensured the stability of atoms, but it was also the first proposal of the existence of [[antimatter]]. Indeed, the evidence for [[positron]]s was discovered in 1932 by [[Carl David Anderson]] in [[cosmic ray]]s. With enough energy, such as by absorbing a photon, an electron-positron pair could be created, a process called [[pair production]]; the reverse process, annihilation, could also occur with the emission of a photon. This showed that particle numbers need not be fixed during an interaction. Historically, however, positrons were at first thought of as "holes" in an infinite electron sea, rather than a new kind of particle, and this theory was referred to as the [[Dirac hole theory]].<ref name="weisskopf" />{{rp|72}}{{r|weinberg|page1=23}} QFT naturally incorporated antiparticles in its formalism.{{r|weinberg|page1=24}}

===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}}

===Non-renormalizability===
Given the tremendous success of QED, many theorists believed, in the few years after 1949, that QFT could soon provide an understanding of all microscopic phenomena, not only the interactions between photons, electrons, and positrons. Contrary to this optimism, QFT entered yet another period of depression that lasted for almost two decades.{{r|weinberg|page1=30}}

The first obstacle was the limited applicability of the renormalization procedure. In perturbative calculations in QED, all infinite quantities could be eliminated by redefining a small (finite) number of physical quantities (namely the mass and charge of the electron). Dyson proved in 1949 that this is only possible for a small class of theories called "renormalizable theories", of which QED is an example. However, most theories, including the [[Fermi's interaction|Fermi theory]] of the [[weak interaction]], are "non-renormalizable". Any perturbative calculation in these theories beyond the first order would result in infinities that could not be removed by redefining a finite number of physical quantities.{{r|weinberg|page1=30}}

The second major problem stemmed from the limited validity of the Feynman diagram method, which is based on a series expansion in perturbation theory. In order for the series to converge and low-order calculations to be a good approximation, the [[coupling constant]], in which the series is expanded, must be a sufficiently small number. The coupling constant in QED is the [[fine-structure constant]] {{math|''α'' ≈ 1/137}}, which is small enough that only the simplest, lowest order, Feynman diagrams need to be considered in realistic calculations. In contrast, the coupling constant in the [[strong interaction]] is roughly of the order of one, making complicated, higher order, Feynman diagrams just as important as simple ones. There was thus no way of deriving reliable quantitative predictions for the strong interaction using perturbative QFT methods.{{r|weinberg|page1=31}}

With these difficulties looming, many theorists began to turn away from QFT. Some focused on [[symmetry (physics)|symmetry]] principles and [[conservation law]]s, while others picked up the old S-matrix theory of Wheeler and Heisenberg. QFT was used heuristically as guiding principles, but not as a basis for quantitative calculations.{{r|weinberg|page1=31}}

=== Source theory ===
Schwinger, however, took a different route. For more than a decade he and his students had been nearly the only exponents of field theory,<ref name=MiltonMehra/>{{rp|p=454}} but in 1951<ref>{{Cite journal |last=Schwinger |first=Julian |date=July 1951 |title=On the Green's functions of quantized fields. I |journal=Proceedings of the National Academy of Sciences |language=en |volume=37 |issue=7 |pages=452–455 |doi=10.1073/pnas.37.7.452 |issn=0027-8424 |pmc=1063400 |pmid=16578383 |doi-access=free |bibcode=1951PNAS...37..452S }}</ref><ref>{{Cite journal |last=Schwinger |first=Julian |date=July 1951 |title=On the Green's functions of quantized fields. II |journal=Proceedings of the National Academy of Sciences |language=en |volume=37 |issue=7 |pages=455–459 |doi=10.1073/pnas.37.7.455 |issn=0027-8424 |pmc=1063401 |pmid=16578384 |doi-access=free |bibcode=1951PNAS...37..455S }}</ref> he found a way around the problem of the infinities with a new method using ''external sources'' as currents coupled to [[gauge field]]s.<ref>{{Cite journal |last=Schweber |first=Silvan S. |date=2005-05-31 |title=The sources of Schwinger's Green's functions |journal=Proceedings of the National Academy of Sciences |language=en |volume=102 |issue=22 |pages=7783–7788 |doi=10.1073/pnas.0405167101 |issn=0027-8424 |pmc=1142349 |pmid=15930139 |doi-access=free }}</ref> Motivated by the former findings, Schwinger kept pursuing this approach in order to "quantumly" generalize the [[Lagrangian mechanics#Lagrange multipliers and constraints|classical process]] of coupling external forces to the configuration space parameters known as Lagrange multipliers. He summarized his [[Source field|source theory]] in 1966<ref>{{cite journal |last1=Schwinger |first1=Julian |title=Particles and Sources |journal=Phys Rev |date=1966 |volume=152 |issue=4 |page=1219|doi=10.1103/PhysRev.152.1219 |bibcode=1966PhRv..152.1219S }}</ref> then expanded the theory's applications to quantum electrodynamics in his three volume-set titled: ''Particles, Sources, and Fields.''<ref name="Perseus Books" /><ref>{{Cite book |last=Schwinger |first=Julian |title=Particles, sources, and fields. 2 |date=1998 |publisher=Advanced Book Program, Perseus Books |isbn=978-0-7382-0054-5 |edition=1. print |location=Reading, Mass}}</ref><ref>{{Cite book |last=Schwinger |first=Julian |title=Particles, sources, and fields. 3 |date=1998 |publisher=Advanced Book Program, Perseus Books |isbn=978-0-7382-0055-2 |edition=1. print |location=Reading, Mass}}</ref> Developments in pion physics, in which the new viewpoint was most successfully applied, convinced him of the great advantages of mathematical simplicity and conceptual clarity that its use bestowed.<ref name="Perseus Books">{{cite book |last1=Schwinger |first1=Julian |title=Particles, Sources and Fields vol. 1 |date=1998 |publisher=Perseus Books |location=Reading, MA |isbn=0-7382-0053-0 |page=xi}}</ref>

In source theory there are no divergences, and no renormalization. It may be regarded as the calculational tool of field theory, but it is more general.<ref>{{cite book |editor=C.R. Hagen |display-editors=etal |title=Proc of the 1967 Int. Conference on Particles and Fields |date=1967 |publisher=Interscience |location=NY |page=128}}</ref>  Using source theory, Schwinger was able to calculate the anomalous magnetic moment of the electron, which he had done in 1947, but this time with no ‘distracting remarks’ about infinite quantities.<ref name=MiltonMehra/>{{rp|p=467}}

Schwinger also applied source theory to his QFT theory of gravity, and was able to reproduce all four of Einstein's classic results: gravitational red shift, deflection and slowing of light by gravity, and the perihelion precession of Mercury.<ref>{{cite book |last1=Schwinger |first1=Julian |title=Particles, Sources and Fields vol. 1 |date=1998 |publisher=Perseus Bookks |location=Reading, MA |pages=82–85}}</ref>  The neglect of source theory by the physics community was a major disappointment for Schwinger:<blockquote>The lack of appreciation of these facts by others was depressing, but understandable.  -J. Schwinger<ref name="Perseus Books"/></blockquote>See "[[Julian Schwinger#Career|the shoes incident]]" between J. Schwinger and [[Steven Weinberg|S. Weinberg]].<ref name=MiltonMehra>{{Cite book |last1=Milton |first1=K. A. |url=https://books.google.com/books?id=9SmZSN8F164C |title=Climbing the Mountain: The Scientific Biography of Julian Schwinger |last2=Mehra |first2=Jagdish |date=2000 |publisher=Oxford University Press |isbn=978-0-19-850658-4 |edition=Repr |location=Oxford |language=en}}</ref>

===Standard model===
[[File:Standard Model of Elementary Particles.svg|thumb|300px|[[Elementary particles]] of the [[Standard Model]]: six types of [[quark]]s, six types of [[lepton]]s, four types of [[gauge boson]]s that carry [[fundamental interaction]]s, as well as the [[Higgs boson]], which endow elementary particles with mass.]]
In 1954, [[Yang Chen-Ning]] and [[Robert Mills (physicist)|Robert Mills]] generalized the [[gauge theory|local symmetry]] of QED, leading to [[Yang–Mills theory|non-Abelian gauge theories]] (also known as Yang–Mills theories), which are based on more complicated local [[symmetry group]]s.<ref name="thooft">{{Cite book |last='t Hooft |first=Gerard |author-link=Gerard 't Hooft |arxiv=1503.05007 |chapter=The Evolution of Quantum Field Theory |title=The Standard Theory of Particle Physics |volume=26 |pages=1–27 |date=2015-03-17 |bibcode=2016stpp.conf....1T |doi=10.1142/9789814733519_0001 |series=Advanced Series on Directions in High Energy Physics |isbn=978-981-4733-50-2 |s2cid=119198452 }}</ref>{{rp|5}} In QED, (electrically) charged particles interact via the exchange of photons, while in non-Abelian gauge theory, particles carrying a new type of "[[charge (physics)|charge]]" interact via the exchange of massless [[gauge boson]]s. Unlike photons, these gauge bosons themselves carry charge.{{r|weinberg|page1=32}}<ref>{{cite journal |last1=Yang |first1=C. N. |last2=Mills |first2=R. L. |author-link1=Chen-Ning Yang |author-link2=Robert Mills (physicist) |date=1954-10-01 |title=Conservation of Isotopic Spin and Isotopic Gauge Invariance |journal=[[Physical Review]] |volume=96 |issue=1 |pages=191–195 |doi=10.1103/PhysRev.96.191 |bibcode=1954PhRv...96..191Y |doi-access=free }}</ref>

[[Sheldon Glashow]] developed a non-Abelian gauge theory that unified the electromagnetic and weak interactions in 1960. In 1964, [[Abdus Salam]] and [[John Clive Ward]] arrived at the same theory through a different path. This theory, nevertheless, was non-renormalizable.<ref name="coleman">{{cite journal |last=Coleman |first=Sidney |author-link=Sidney Coleman |date=1979-12-14 |title=The 1979 Nobel Prize in Physics |journal=[[Science (journal)|Science]] |volume=206 |issue=4424 |pages=1290–1292 |jstor=1749117 |bibcode=1979Sci...206.1290C |doi=10.1126/science.206.4424.1290 |pmid=17799637 }}</ref>

[[Peter Higgs]], [[Robert Brout]], [[François Englert]], [[Gerald Guralnik]], [[C. R. Hagen|Carl Hagen]], and [[T. W. B. Kibble|Tom Kibble]] proposed in their famous [[1964 PRL symmetry breaking papers|''Physical Review Letters'' papers]] that the gauge symmetry in Yang–Mills theories could be broken by a mechanism called [[spontaneous symmetry breaking]], through which originally massless gauge bosons could acquire mass.{{r|thooft|page1=5–6}}

By combining the earlier theory of Glashow, Salam, and Ward with the idea of spontaneous symmetry breaking, [[Steven Weinberg]] wrote down in 1967 a theory describing [[electroweak interaction]]s between all [[lepton]]s and the effects of the [[Higgs boson]]. His theory was at first mostly ignored,<ref name="coleman" />{{r|thooft|page1=6}} until it was brought back to light in 1971 by [[Gerard 't Hooft]]'s proof that non-Abelian gauge theories are renormalizable. The electroweak theory of Weinberg and Salam was extended from leptons to [[quark]]s in 1970 by Glashow, [[John Iliopoulos]], and [[Luciano Maiani]], marking its completion.<ref name="coleman" />

[[Harald Fritzsch]], [[Murray Gell-Mann]], and [[Heinrich Leutwyler]] discovered in 1971 that certain phenomena involving the [[strong interaction]] could also be explained by non-Abelian gauge theory. [[Quantum chromodynamics]] (QCD) was born. In 1973, [[David Gross]], [[Frank Wilczek]], and [[Hugh David Politzer]] showed that non-Abelian gauge theories are "[[asymptotic freedom|asymptotically free]]", meaning that under renormalization, the coupling constant of the strong interaction decreases as the interaction energy increases. (Similar discoveries had been made numerous times previously, but they had been largely ignored.) {{r|thooft|page1=11}} Therefore, at least in high-energy interactions, the coupling constant in QCD becomes sufficiently small to warrant a perturbative series expansion, making quantitative predictions for the strong interaction possible.{{r|weinberg|page1=32}}

These theoretical breakthroughs brought about a renaissance in QFT. The full theory, which includes the electroweak theory and chromodynamics, is referred to today as the [[Standard Model]] of elementary particles.<ref>{{cite web |url=https://www.britannica.com/science/Standard-Model |title=Standard model |last=Sutton |first=Christine |author-link=Christine Sutton |website=britannica.com |publisher=[[Encyclopædia Britannica]] |access-date=2018-08-14}}</ref> The Standard Model successfully describes all [[fundamental interaction]]s except [[gravity]], and its many predictions have been met with remarkable experimental confirmation in subsequent decades.{{r|shifman|page1=3}} The [[Higgs boson]], central to the mechanism of spontaneous symmetry breaking, was finally detected in 2012 at [[CERN]], marking the complete verification of the existence of all constituents of the Standard Model.<ref>{{cite arXiv |last=Kibble |first=Tom W. B. |author-link=Tom Kibble |eprint=1412.4094 |title=The Standard Model of Particle Physics |class=physics.hist-ph |date=2014-12-12 }}</ref>

===Other developments===
The 1970s saw the development of non-perturbative methods in non-Abelian gauge theories. The [['t Hooft–Polyakov monopole]] was discovered theoretically by 't Hooft and [[Alexander Markovich Polyakov|Alexander Polyakov]], [[flux tube]]s by [[Holger Bech Nielsen]] and [[Poul Olesen]], and [[instanton]]s by Polyakov and coauthors. These objects are inaccessible through perturbation theory.{{r|shifman|page1=4}}

[[Supersymmetry]] also appeared in the same period. The first supersymmetric QFT in four dimensions was built by [[Yuri Golfand]] and [[Evgeny Likhtman]] in 1970, but their result failed to garner widespread interest due to the [[Iron Curtain]]. Supersymmetry theories only took off in the theoretical community after the work of [[Julius Wess]] and [[Bruno Zumino]] in 1973,{{r|shifman|page1=7}} but to date have not been widely accepted as part of the Standard Model due to lack of experimental evidence.<ref name="Wolchover">{{cite journal |last=Wolchover |first=Natalie |title=Supersymmetry Fails Test, Forcing Physics to Seek New Ideas |journal=Quanta Magazine |date=November 20, 2012 |url=https://www.quantamagazine.org/20121120-as-supersymmetry-fails-tests-physicists-seek-new-ideas/}}</ref>

Among the four fundamental interactions, gravity remains the only one that lacks a consistent QFT description. Various attempts at a theory of [[quantum gravity]] led to the development of [[string theory]],{{r|shifman|page1=6}} itself a type of two-dimensional QFT with [[conformal symmetry]].<ref name="polchinski1" /> [[Joël Scherk]] and [[John Henry Schwarz|John Schwarz]] first proposed in 1974 that string theory could be ''the'' quantum theory of gravity.<ref>{{cite arXiv |last=Schwarz |first=John H. |author-link=John Henry Schwarz |eprint=1201.0981 |title=The Early History of String Theory and Supersymmetry |class=physics.hist-ph |date=2012-01-04 }}</ref>

===Condensed-matter-physics===
Although quantum field theory arose from the study of interactions between elementary particles, it has been successfully applied to other physical systems, particularly to [[many-body system]]s in [[condensed matter physics]].

Historically, the Higgs mechanism of spontaneous symmetry breaking was a result of [[Yoichiro Nambu]]'s application of [[superconductor]] theory to elementary particles, while the concept of renormalization came out of the study of second-order [[phase transition]]s in matter.<ref>{{cite web |url=https://science.energy.gov/~/media/hep/pdf/Reports/HEP-BES_Roundtable_Report.pdf |title=Common Problems in Condensed Matter and High Energy Physics |date=2015-02-02 |website=science.energy.gov |publisher=Office of Science, [[U.S. Department of Energy]] |access-date=2018-07-18}}</ref>

Soon after the introduction of photons, Einstein performed the quantization procedure on vibrations in a crystal, leading to the first [[quasiparticle]]—[[phonon]]s. Lev Landau claimed that low-energy excitations in many condensed matter systems could be described in terms of interactions between a set of quasiparticles. The Feynman diagram method of QFT was naturally well suited to the analysis of various phenomena in condensed matter systems.<ref name="wilczek">{{Cite journal |last=Wilczek |first=Frank |author-link=Frank Wilczek |arxiv=1604.05669 |title=Particle Physics and Condensed Matter: The Saga Continues |journal=Physica Scripta |volume=2016 |issue=T168 |pages=014003 |date=2016-04-19 |bibcode=2016PhST..168a4003W |doi=10.1088/0031-8949/T168/1/014003 |s2cid=118439678 }}</ref>

Gauge theory is used to describe the quantization of [[magnetic flux]] in superconductors, the [[resistivity]] in the [[quantum Hall effect]], as well as the relation between frequency and voltage in the AC [[Josephson effect]].<ref name="wilczek" />