Quantum field theory
In 1930, Robert Oppenheimer published a paper showing that calculations in quantum electrodynamics always resulted in infinite quantities. These infinities appeared as electron self-energy and vacuum zero-point energy for both electrons and photons. The mathematical methods of the time could not handle interactions involving photons with extremely high momenta. For twenty years, theoretical physicists struggled to resolve this problem. John Archibald Wheeler and Werner Heisenberg proposed replacing the problematic theory with S-matrix theory in 1937 and 1943 respectively. They argued that since microscopic details are inaccessible to observation, theories should only describe relationships between observables like atomic energy levels.
A breakthrough arrived around 1950 when Julian Schwinger, Richard Feynman, Freeman Dyson, and Shinichiro Tomonaga developed a systematic procedure called renormalization. This method replaced calculated values of mass and charge, which were infinite, with their finite measured values. Hans Bethe had previously estimated the numerical value of the Lamb shift by ignoring contributions from photons exceeding electron mass. Norman Myles Kroll, Willis Lamb, James Bruce French, and Victor Weisskopf confirmed this value using an approach where infinities cancelled other infinities to produce finite results. Although clumsy, this early work paved the way for the robust renormalization procedure. By applying it, calculations finally explained the electron's anomalous magnetic moment and vacuum polarization with remarkable agreement to experimental measurements. This marked the end of what Tomonaga described as a war against infinities.
In 1928, Paul Dirac wrote down a wave equation describing relativistic electrons known as the Dirac equation. The theory produced fruitful results including the correct spin of one-half for electrons and the Sommerfeld formula for hydrogen atom fine structure. However, the equation also implied the existence of negative energy states that would cause atoms to be unstable. Atoms could decay to lower energy states by emitting radiation if such states existed.
Dirac realized in 1929 that these negative energy states could be removed by assuming particles with the same mass as electrons but opposite electric charge existed. This assumption ensured atomic stability and became the first proposal of antimatter. Carl David Anderson discovered evidence for positrons in cosmic rays in 1932. With enough energy from absorbing photons, electron-positron pairs could be created through pair production. The reverse process called annihilation occurred with photon emission. This showed particle numbers need not remain fixed during interactions.
Historically, physicists initially viewed positrons as holes in an infinite electron sea rather than new particles. This concept was referred to as Dirac hole theory. Quantum field theory naturally incorporated antiparticles into its formalism. Between 1928 and 1930, 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, each type of particle had its corresponding quantum field like an electron field or proton field.
In 1954, Yang Chen-Ning and Robert Mills generalized local symmetry of quantum electrodynamics leading to non-Abelian gauge theories known as Yang-Mills theories. These theories rely on more complicated local symmetry groups. In quantum electrodynamics, electrically charged particles interact via exchange of photons while in non-Abelian gauge theory, particles carrying a new charge interact via massless gauge bosons. Unlike photons, these gauge bosons themselves carry charge.
Sheldon Glashow developed a non-Abelian gauge theory unifying electromagnetic and weak interactions in 1960. Abdus Salam and John Clive Ward arrived at the same theory through different paths in 1964. This early version was non-renormalizable until Gerard 't Hooft proved non-Abelian gauge theories were renormalizable in 1971. Steven Weinberg wrote down a theory describing electroweak interactions between all leptons and effects of the Higgs boson in 1967. The electroweak theory extended from leptons to quarks in 1970 by Glashow, John Iliopoulos, and Luciano Maiani marking its completion.
Harald Fritzsch, Murray Gell-Mann, and Heinrich Leutwyler discovered phenomena involving strong interaction could be explained by non-Abelian gauge theory in 1971. Quantum chromodynamics emerged as a result. David Gross, Frank Wilczek, and Hugh David Politzer showed non-Abelian gauge theories are asymptotically free in 1973. Their work demonstrated that coupling constants decrease as interaction energy increases making quantitative predictions possible for strong interactions.
Peter Higgs, Robert Brout, François Englert, Gerald Guralnik, Carl Hagen, and Tom Kibble proposed mechanisms called spontaneous symmetry breaking through which originally massless gauge bosons could acquire mass. This mechanism became central to understanding particle masses within the Standard Model. The full theory including electroweak theory and chromodynamics is referred to today as the Standard Model of elementary particles.
The Standard Model successfully describes all fundamental interactions except gravity. Its many predictions have been met with remarkable experimental confirmation over subsequent decades. The Higgs boson, central to the mechanism of spontaneous symmetry breaking, was finally detected in 2012 at CERN. This detection marked complete verification of existence for all constituents of the Standard Model. Experimental evidence continues to support the theoretical framework despite ongoing challenges regarding mathematical rigor and incorporation of gravitational forces.
Supersymmetry appeared in the same period as other developments but has not yet been widely accepted due to lack of experimental evidence. Experiments have yet to provide proof for existence of supersymmetric particles. If supersymmetry were true symmetry of nature it must be broken symmetry with energy higher than achievable by present-day experiments. String theory remains a potential solution to quantum gravity problems though it lacks direct experimental confirmation.
Although quantum field theory arose from study of interactions between elementary particles it has been successfully applied to condensed matter physics systems. Yoichiro Nambu applied superconductor theory to elementary particles resulting in the Higgs mechanism concept. Renormalization concepts emerged from studying second-order phase transitions in matter soon after introduction of photons.
Einstein performed quantization procedure on vibrations in crystals leading to first quasiparticle called phonons. Lev Landau claimed low-energy excitations in many condensed matter systems could be described through interactions between sets of quasiparticles. Feynman diagram method proved well suited for analyzing various phenomena in condensed matter systems. Gauge theory describes quantization of magnetic flux in superconductors resistivity in quantum Hall effect and relation between frequency and voltage in AC Josephson effect.
In condensed matter physics quantum field theory describes two-dimensional electron gases while string theory represents one-dimensional quantum field theory. Topological quantum field theories applicable to frontier research include Chern-Simons-Witten gauge theories in three spacetime dimensions. These theories help construct models of topological quantum matters and fractionalized particles known as anyons whose world line trajectories form link configurations relating braiding statistics to mathematical link invariants.
Despite overwhelming success in particle physics and condensed matter physics quantum field theory lacks formal mathematical foundation. According to Haag's theorem there does not exist a well-defined interaction picture for quantum field theory implying perturbation theory underlying entire Feynman diagram method is fundamentally ill-defined. Theoretical physicists and mathematicians have attempted since 1950s to organize all quantum field theories into set of axioms establishing existence of concrete models in mathematically rigorous way.
This line of study called constructive quantum field theory has led to results like CPT theorem spin-statistics theorem and Goldstone's theorem. It also produced mathematically rigorous constructions of many interacting quantum field theories in two and three spacetime dimensions including two-dimensional scalar field theories with arbitrary polynomial interactions. Compared to ordinary quantum field theory topological quantum field theory and conformal field theory are better supported mathematically both classifiable within framework of representations of cobordisms.
Yang-Mills existence and mass gap remains one of Millennium Prize Problems concerning well-defined existence of Yang-Mills theories as set out by above axioms. Algebraic quantum field theory offers another approach where fundamental objects are local operators and algebraic relations between them. Axiomatic systems following this approach include Wightman axioms and Haag-Kastler axioms. One way to construct theories satisfying Wightman axioms uses Osterwalder-Schrader axioms giving necessary conditions for real time theory obtained from imaginary time theory by analytic continuation.
Common questions
When did Robert Oppenheimer publish his paper on infinite quantities in quantum electrodynamics?
Robert Oppenheimer published a paper showing that calculations in quantum electrodynamics always resulted in infinite quantities in 1930. These infinities appeared as electron self-energy and vacuum zero-point energy for both electrons and photons.
Who developed the renormalization procedure around 1950 to resolve infinite values in quantum field theory?
Julian Schwinger, Richard Feynman, Freeman Dyson, and Shinichiro Tomonaga developed a systematic procedure called renormalization around 1950. This method replaced calculated values of mass and charge which were infinite with their finite measured values.
What year did Carl David Anderson discover evidence for positrons in cosmic rays?
Carl David Anderson discovered evidence for positrons in cosmic rays in 1932. This discovery confirmed Dirac's 1929 proposal that particles with the same mass as electrons but opposite electric charge existed.
Which physicists proved non-Abelian gauge theories were renormalizable in 1971?
Gerard 't Hooft proved non-Abelian gauge theories were renormalizable in 1971. This proof resolved issues with earlier versions of the theory developed by Sheldon Glashow, Abdus Salam, and John Clive Ward.
When was the Higgs boson detected at CERN to complete verification of the Standard Model?
The Higgs boson was finally detected in 2012 at CERN. This detection marked complete verification of existence for all constituents of the Standard Model including electroweak theory and chromodynamics.