Quantum field theory
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.[8]:1
Quantum fiction
Quantum fiction brings quantum theory forward as the explanation behind the concept of life imitating art and art imitating life via substantiation of literary plot developments, time sequence, character experiences and other literary elements based on quantum mechanics. The term Quantum fiction is etymologically based on the discovery of Max Planck, who first used the word quantum to describe the minute forces at play in the realm of physics. The field was pioneered by quantum physicists Erwin Schrödinger, Werner Heisenberg, Wolfgang Pauli, Niels Bohr, and Eugene Wigner, as well as contentions of Louis DeBroglie, Max von Laue and Albert Einstein. One contention, among others, is that quantum mechanics is a statistical approximation to a deeper reality which behaves predictably via the observer being an inextricable part of reality (Observer effect (physics)). Canonical quantization
Later the electromagnetic field was also quantized, and even the particles themselves became represented through quantized fields, resulting in the development of quantum electrodynamics (QED) and quantum field theory in general. Thus, by convention, the original form of particle quantum mechanics is denoted first quantization, while quantum field theory is formulated in the language of second quantization. Group field theory
Group field theory (GFT) is a quantum field theory in which the base manifold is taken to be a Lie group. It is closely related to background independent quantum gravity approaches such as loop quantum gravity, the spin foam formalism and causal dynamical triangulation. It can be shown that its perturbative expansion can be interpreted as spin foams and simplicial pseudo-manifolds (depending on the representation of the fields). Thus, its partition function defines a non-perturbative sum over all simplicial topologies and geometries, giving a path integral formulation of quantum spacetime. Reeh–Schlieder theorem
The Reeh–Schlieder theorem is a result in relativistic local quantum field theory published by Helmut Reeh and Siegfried Schlieder (1918-2003) in 1961. Quantum field theory
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 "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 prior, but they had been largely ignored.) [9]: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.[3]:32 Quantum field theory in curved spacetime
In particle physics, quantum field theory in curved spacetime is an extension of standard, Minkowski space quantum field theory to curved spacetime. A general prediction of this theory is that particles can be created by time-dependent gravitational fields (multigraviton pair production), or by time-independent gravitational fields that contain horizons. History of quantum field theory
The final crucial step was Enrico Fermi's theory of "β"-decay (1934). In it, fermion species nonconservation was shown to follow from second quantization: creation and annihilation of fermions came to the fore and quantum field theory was seen to describe particle decays. (Fermi's breakthrough was somewhat foreshadowed in the abstract studies of Soviet physicists, Viktor Ambartsumian and Dmitri Ivanenko, in particular the Ambarzumian–Ivanenko hypothesis of creation of massive particles (1930). The idea was that not only the quanta of the electromagnetic field, photons, but also other particles might emerge and disappear as a result of their interaction with other particles.) Quantum field theory in curved spacetime
Since the end of the eighties, the local quantum field theory approach due to Rudolf Haag and Daniel Kastler has been implemented in order to include an algebraic version of quantum field theory in curved spacetime. Indeed, the viewpoint of local quantum physics is suitable to generalize the renormalization procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in the presence of a black hole have been obtained. In particular the algebraic approach allows one to deal with the problems, above mentioned, arising from the absence of a preferred reference vacuum state, the absence of a natural notion of particle and the appearance of unitarily inequivalent representations of the algebra of observables. (See these lecture notes  Mathematical physics
Another revolutionary development of the 20th century was quantum theory, which emerged from the seminal contributions of Max Planck (1856–1947) (on black body radiation) and Einstein's work on the photoelectric effect. This was, at first, followed by a heuristic framework devised by Arnold Sommerfeld (1868–1951) and Niels Bohr (1885–1962), but this was soon replaced by the quantum mechanics developed by Max Born (1882–1970), Werner Heisenberg (1901–1976), Paul Dirac (1902–1984), Erwin Schrödinger (1887–1961), Satyendra Nath Bose (1894–1974), and Wolfgang Pauli (1900–1958). This revolutionary theoretical framework is based on a probabilistic interpretation of states, and evolution and measurements in terms of self-adjoint operators on an infinite dimensional vector space. That is called Hilbert space, introduced in its elementary form by David Hilbert (1862–1943) and Frigyes Riesz (1880-1956), and rigorously defined within the axiomatic modern version by John von Neumann in his celebrated book Mathematical Foundations of Quantum Mechanics, where he built up a relevant part of modern functional analysis on Hilbert spaces, the spectral theory in particular. Paul Dirac used algebraic constructions to produce a relativistic model for the electron, predicting its magnetic moment and the existence of its antiparticle, the positron. History of quantum mechanics
The field of quantum chemistry was pioneered by physicists Walter Heitler and Fritz London, who published a study of the covalent bond of the hydrogen molecule in 1927. Quantum chemistry was subsequently developed by a large number of workers, including the American theoretical chemist Linus Pauling at Caltech, and John C. Slater into various theories such as Molecular Orbital Theory or Valence Theory. Arthur Jaffe
With James Glimm, he founded the subject called constructive quantum field theory. Their major achievement was to establish existence theorems for two- and three-dimensional examples of non-linear, relativistic quantum fields. Quantum triviality
The first evidence of possible triviality of quantum field theories was obtained by Landau, Abrikosov, and Khalatnikov by finding the following relation of the observable charge with the “bare” charge ₀, 
where is the mass of the particle, and is the momentum cut-off. If ₀ is finite, then tends to zero in the limit of infinite cut-off . Quantum field theory
The preceding procedure is a direct application of non-relativistic quantum mechanics and can be used to quantise (complex) scalar fields, Dirac fields,[1]:52 vector fields (e.g. the electromagnetic field), and even strings.[21] However, creation and annihilation operators are only well defined in the simplest theories that contain no interactions (so-called free theory). In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes. To perform calculations on any realistic interacting theory, perturbation theory would be necessary. Quantum field theory
Any quantum state of the field can be obtained from | 0 ⟩ {\displaystyle |0\rangle } by successively applying creation operators a ^ p † {\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }} , e.g.[1]:22 Quantization (physics)
The first method to be developed for quantization of field theories was canonical quantization. While this is extremely easy to implement on sufficiently simple theories, there are many situations where other methods of quantization yield more efficient procedures for computing quantum amplitudes. However, the use of canonical quantization has left its mark on the language and interpretation of quantum field theory. Vladimir Krivchenkov
Quantum field theory is a generalization of quantum mechanics. Krivchenkov believed that people need to understand at least the non–relativistic theory. He almost asked students to excuse him for the field theory, where only perturbation theory can be offered. Not only do the perturbation theory series diverge, but each perturbational term also in some sense is infinite, and a special renormalization of the interaction constant is required to give the result the physical sense.
One of the colleagues of Krivchenkov, Yurii Shirokov tried to construct quantum field theory in terms of wave packets, without divergences, using the algebra of generalized functions, but even now this approach is not sufficiently developed. Krivchenkov told students that they are supposed to build up a "true" theory, not just a perturbation, that always gives a divergent series. This statement (dogma) applies not only to Field Theory and quantum mechanics, but to any perturbation theory with respect to any distributed system; the perturbation series always diverges. Krivchenkov had realized this and brought this knowledge to students. Many problems in his books stress this property of perturbation theory. Quantum spacetime
In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebra still varies from theory to theory.
As a result of this change some variables that are usually continuous may become discrete.
Often only such discrete variables are called "quantized"; usage varies.
The idea of quantum spacetime was proposed in the early days of quantum theory by Heisenberg and Ivanenko
as a way to eliminate infinities from quantum field theory.
The germ of the idea passed from Heisenberg to Rudolf Peierls, who noted that electrons in a magnetic field
can be regarded as moving in a quantum space-time, and to Robert Oppenheimer, who carried it
to Hartland Snyder,
who published the first concrete example.
Snyder's Lie algebra was made simple by C. N. Yang in the same year. Quantum field theory
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,[6] suggesting that the computational methods at the time could not properly deal with interactions involving photons with extremely high momenta.[3]:25 It was not until 20 years later that a systematic approach to remove such infinities was developed. Electromagnetic field
This quantum picture of the electromagnetic field (which treats it as analogous to harmonic oscillators) has proven very successful, giving rise to quantum electrodynamics, a quantum field theory describing the interaction of electromagnetic radiation with charged matter. It also gives rise to quantum optics, which is different from quantum electrodynamics in that the matter itself is modelled using quantum mechanics rather than quantum field theory. History of physics
In 1905, Einstein used the quantum theory to explain the photoelectric effect, and in 1913 the Danish physicist Niels Bohr used the same constant to explain the stability of Rutherford's atom as well as the frequencies of light emitted by hydrogen gas. The quantized theory of the atom gave way to a full-scale quantum mechanics in the 1920s. New principles of a "quantum" rather than a "classical" mechanics, formulated in matrix-form by Werner Heisenberg, Max Born, and Pascual Jordan in 1925, were based on the probabilistic relationship between discrete "states" and denied the possibility of causality. Quantum mechanics was extensively developed by Heisenberg, Wolfgang Pauli, Paul Dirac, and Erwin Schrödinger, who established an equivalent theory based on waves in 1926; but Heisenberg's 1927 "uncertainty principle" (indicating the impossibility of precisely and simultaneously measuring position and momentum) and the "Copenhagen interpretation" of quantum mechanics (named after Bohr's home city) continued to deny the possibility of fundamental causality, though opponents such as Einstein would metaphorically assert that "God does not play dice with the universe". The new quantum mechanics became an indispensable tool in the investigation and explanation of phenomena at the atomic level. Also in the 1920s, the Indian scientist Satyendra Nath Bose's work on photons and quantum mechanics provided the foundation for Bose–Einstein statistics, the theory of the Bose–Einstein condensate. Noncommutative quantum field theory
In mathematical physics, noncommutative quantum field theory (or quantum field theory on noncommutative spacetime) is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functions are noncommutative. One commonly studied version of such theories has the "canonical" commutation relation: Axiomatic quantum field theory
The first set of axioms for quantum field theories, known as the Wightman axioms, were proposed by Arthur Wightman in the early 1950s. These axioms attempt to describe QFTs on flat Minkowski spacetime by regarding quantum fields as operator-valued distributions acting on a Hilbert space. In practice, one often uses the Wightman reconstruction theorem, which guarantees that the operator-valued distributions and the Hilbert space can be recovered from the collection of correlation functions. Mathai–Quillen formalism
In mathematics, the Mathai–Quillen formalism is an approach to topological quantum field theory introduced by , based on the Mathai–Quillen form constructed in . String theory
The application of quantum mechanics to physical objects such as the electromagnetic field, which are extended in space and time, is known as quantum field theory. In particle physics, quantum field theories form the basis for our understanding of elementary particles, which are modeled as excitations in the fundamental fields. Quantum electrodynamics
Dirac described the quantization of the electromagnetic field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators of particles. In the following years, with contributions from Wolfgang Pauli, Eugene Wigner, Pascual Jordan, Werner Heisenberg and an elegant formulation of quantum electrodynamics due to Enrico Fermi, physicists came to believe that, in principle, it would be possible to perform any computation for any physical process involving photons and charged particles. However, further studies by Felix Bloch with Arnold Nordsieck, and Victor Weisskopf, in 1937 and 1939, revealed that such computations were reliable only at a first order of perturbation theory, a problem already pointed out by Robert Oppenheimer. At higher orders in the series infinities emerged, making such computations meaningless and casting serious doubts on the internal consistency of the theory itself. With no solution for this problem known at the time, it appeared that a fundamental incompatibility existed between special relativity and quantum mechanics.
Difficulties with the theory increased through the end of the 1940s. Improvements in microwave technology made it possible to take more precise measurements of the shift of the levels of a hydrogen atom, now known as the Lamb shift and magnetic moment of the electron. These experiments exposed discrepancies which the theory was unable to explain. Quantum field theory
Sheldon Glashow developed a non-Abelian gauge theory that unified the electromagnetic and weak interactions in 1960. In 1964, Abdul Salam and John Clive Ward arrived at the same theory through a different path. This theory, nevertheless, was non-renormalizable.[11] Timeline of quantum computing
2000 Arun K. Pati and Samuel L. Braunstein proved the quantum no-deleting theorem. This is dual to the no-cloning theorem which shows that one cannot delete a copy of an unknown qubit. Together with the stronger no-cloning theorem, the no-deleting theorem has important implication, i.e., quantum information can neither be created nor be destroyed. First working 5-qubit NMR computer demonstrated at the Technical University of Munich. First execution of order finding (part of Shor's algorithm) at IBM's Almaden Research Center and Stanford University. First working 7-qubit NMR computer demonstrated at the Los Alamos National Laboratory. The standard textbook, Quantum Computation and Quantum Information, by Michael Nielsen and Isaac Chuang is published. 2001 First execution of Shor's algorithm at IBM's Almaden Research Center and Stanford University. The number 15 was factored using 1018 identical molecules, each containing seven active nuclear spins. Noah Linden and Sandu Popescu proved that the presence of entanglement is a necessary condition for a large class of quantum protocols. This, coupled with Braunstein's result (see 1999 above), called the validity of NMR quantum computation into question.[17] Emanuel Knill, Raymond Laflamme, and Gerard Milburn show that optical quantum computing is possible with single photon sources, linear optical elements, and single photon detectors, launching the field of linear optical quantum computing. Robert Raussendorf and Hans J. Briegel propose measurement-based quantum computation.[18] 2002 The Quantum Information Science and Technology Roadmapping Project, involving some of the main participants in the field, laid out the . The Institute for Quantum Computing was established at the University of Waterloo in Waterloo, Ontario by Mike Lazaridis, Raymond Laflamme and Michele Mosca.[19] 2003 Todd D. Pittman and collaborators at Johns Hopkins University, Applied Physics Laboratory and independently Jeremy L. O'Brien and collaborators at the University of Queensland, demonstrate quantum controlled-not gates using only linear optical elements.[20][21] DARPA Quantum Network becomes fully operational on October 23, 2003. 2004 First working pure state NMR quantum computer (based on parahydrogen) demonstrated at Oxford University and University of York. First five-photon entanglement demonstrated by Jian-Wei Pan's group at the University of Science and Technology of China, the minimal number of qubits required for universal quantum error correction.[22] Steven Weinberg
In 1979 he pioneered the modern view on the renormalization aspect of quantum field theory that considers all quantum field theories as effective field theories and changed the viewpoint of previous work (including his own in his 1967 paper) that a sensible quantum field theory must be renormalizable. This approach allowed the development of effective theory of quantum gravity, low energy QCD, heavy quark effective field theory and other developments, and it is a topic of considerable interest in current research. Quantum field theory
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 transformation, 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.[3]: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. Axiomatic quantum field theory
Axiomatic quantum field theory is a mathematical discipline which aims to describe quantum field theory in terms of rigorous axioms. It is strongly associated with functional analysis and operator algebras, but has also been studied in recent years from a more geometric and functorial perspective. History of quantum field theory
Two classic text-books from the 1960s, James D. Bjorken, Sidney David Drell, "Relativistic Quantum Mechanics" (1964) and J. J. Sakurai, "Advanced Quantum Mechanics" (1967), thoroughly developed the Feynman graph expansion techniques using physically intuitive and practical methods following from the correspondence principle, without worrying about the technicalities involved in deriving the Feynman rules from the superstructure of quantum field theory itself. Although both Feynman's heuristic and pictorial style of dealing with the infinities, as well as the formal methods of Tomonaga and Schwinger, worked extremely well, and gave spectacularly accurate answers, the true analytical nature of the question of "renormalizability", that is, whether ANY theory formulated as a "quantum field theory" would give finite answers, was not worked-out until much later, when the urgency of trying to formulate finite theories for the strong and electro-weak (and gravitational interactions) demanded its solution. Quantum gravity
String theory can be seen as a generalization of quantum field theory where instead of point particles, string-like objects propagate in a fixed spacetime background, although the interactions among closed strings give rise to space-time in a dynamical way.
Although string theory had its origins in the study of quark confinement and not of quantum gravity, it was soon discovered that the string spectrum contains the graviton, and that "condensation" of certain vibration modes of strings is equivalent to a modification of the original background. In this sense, string perturbation theory exhibits exactly the features one would expect of a perturbation theory that may exhibit a strong dependence on asymptotics (as seen, for example, in the AdS/CFT correspondence) which is a weak form of background dependence. History of quantum field theory
Parallel breakthroughs in the understanding of phase transitions in condensed matter physics led to novel insights based on the renormalization group. They involved the work of Leo Kadanoff (1966) and Kenneth Geddes Wilson & Michael Fisher (1972)—extending the work of Ernst Stueckelberg–André Petermann (1953) and Murray Gell-Mann–Francis Low (1954)—which led to the seminal reformulation of quantum field theory by Kenneth Geddes Wilson in 1975. Introduction to gauge theory
With the advent of quantum mechanics in the 1920s, and with successive advances in quantum field theory, the importance of gauge transformations has steadily grown. Gauge theories constrain the laws of physics, because all the changes induced by a gauge transformation have to cancel each other out when written in terms of observable quantities. Over the course of the 20th century, physicists gradually realized that all forces (fundamental interactions) arise from the constraints imposed by "local" gauge symmetries, in which case the transformations vary from point to point in space and time. Perturbative quantum field theory (usually employed for scattering theory) describes forces in terms of force-mediating particles called gauge bosons. The nature of these particles is determined by the nature of the gauge transformations. The culmination of these efforts is the Standard Model, a quantum field theory that accurately predicts all of the fundamental interactions except gravity. Field (physics)
The development of the independent concept of a field truly began in the nineteenth century with the development of the theory of electromagnetism. In the early stages, André-Marie Ampère and Charles-Augustin de Coulomb could manage with Newton-style laws that expressed the forces between pairs of electric charges or electric currents. However, it became much more natural to take the field approach and express these laws in terms of electric and magnetic fields; in 1849 Michael Faraday became the first to coin the term "field". History of quantum field theory
In particle physics, the history of quantum field theory starts with its creation by Paul Dirac, when he attempted to quantize the electromagnetic field in the late 1920s. Major advances in the theory were made in the 1940s and 1950s, and led to the introduction of renormalized quantum electrodynamics (QED). QED was so successful and accurately predictive that efforts were made to apply the same basic concepts for the other forces of nature. By the late 1970s, these efforts successfully utilized gauge theory in the strong nuclear force and weak nuclear force, producing the modern standard model of particle physics. History of the philosophy of field theory
Fields become the fundamental object of study in quantum field theory. Mathematically, quantum fields are formalized as operator-valued distributions. Although there is no direct method of measuring the fields themselves, the framework asserts that all particles are "excitations" of these fields. For example: whereas Maxwell's theory of classical electromagnetism describes light as a self-propagating wave in the electromagnetic field, in quantum electrodynamics light is the massless gauge boson particle called the "photon." Furthermore, the number of particles in an isolated system need not be conserved; an example of a process for which this is the case is bremsstrahlung. A heuristic for suggesting particles may be created and destroyed is found in Albert Einstein's famous equation formula_1, which asserts that energy and matter may in principle be exchanged. A more detailed understanding of the framework is obtained by studying the Lagrangian density of a field theory which encodes the information of its allowed particle interactions. Timeline of quantum computing
Subwavelength waveguide developed for light.[41] Single photon emitter for optical fibers developed.[42] Six-photon one-way quantum computer is created in lab.[43] New material proposed for quantum computing.[44] Single atom single photon server devised.[45] First use of Deutsch's Algorithm in a cluster state quantum computer.[46] University of Cambridge develops electron quantum pump.[47] Superior method of qubit coupling developed.[48] Successful demonstration of controllably coupled qubits.[49] Breakthrough in applying spin-based electronics to silicon.[50] Scientists demonstrate quantum state exchange between light and matter.[51] Diamond quantum register developed.[52] Controlled-NOT quantum gates on a pair of superconducting quantum bits realized.[53] Scientists contain, study hundreds of individual atoms in 3D array.[54] Nitrogen in buckyball molecule used in quantum computing.[55] Large number of electrons quantum coupled.[56] Spin-orbit interaction of electrons measured.[57] Atoms quantum manipulated in laser light.[58] Light pulses used to control electron spins.[59] Quantum effects demonstrated across tens of nanometers.[60] Light pulses used to accelerate quantum computing development.[61] Quantum RAM blueprint unveiled.[62] Model of quantum transistor developed.[63] Long distance entanglement demonstrated.[64] Photonic quantum computing used to factor number by two independent labs.[65] Quantum bus developed by two independent labs.[66] Superconducting quantum cable developed.[67] Transmission of qubits demonstrated.[68] Superior qubit material devised.[69] Single electron qubit memory.[70] Bose-Einstein condensate quantum memory developed.[71] D-Wave Systems demonstrates use of a 28-qubit quantum annealing computer.[72] New cryonic method reduces decoherence and increases interaction distance, and thus quantum computing speed.[73] Photonic quantum computer demonstrated.[74] Graphene quantum dot spin qubits proposed.[75] Timeline of quantum computing
1980 Paul Benioff describes quantum mechanical Hamiltonian models of computers[2] Yuri Manin briefly motivates the idea of quantum computing[3] 1981 Richard Feynman observes in his talk at the First Conference on the Physics of Computation, held at MIT in May, that it appeared to be impossible in general to simulate an evolution of a quantum system on a classical computer in an efficient way. He proposes a basic model for a quantum computer that would be capable of such simulations[4] Paul Benioff gives talk at the same conference with the title "Quantum mechanical Hamiltonian models of discrete processes that erase their own histories: application to Turing machines". Tommaso Toffoli introduces the reversible Toffoli gate, which, together with the NOT and XOR gates provides a universal set for reversible classical computation. 1982 Paul Benioff proposes the first recognisable theoretical framework for a quantum computer[5] William Wootters and Wojciech Zurek,[6] and independently Dennis Dieks[7] prove the no-cloning theorem. 1984 Charles Bennett and Gilles Brassard employ Wiesner's conjugate coding for distribution of cryptographic keys. 1985 David Deutsch, at the University of Oxford, describes the first universal quantum computer. Just as a Universal Turing machine can simulate any other Turing machine efficiently (Church-Turing thesis), so the universal quantum computer is able to simulate any other quantum computer with at most a polynomial slowdown. 1989 Bikas K. Chakrabarti & collaborators from Saha Institute of Nuclear Physics, Kolkata, proposes the idea that quantum fluctuations could help explore rough energy landscapes by escaping from local minima of glassy stems having tall but thin barriers by tunneling (instead of climbing over using thermal excitations), suggesting the effectiveness of quantum annealing over classical simulated annealing. Quantum field theory
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 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."[2]: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)) 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.[3]:18 Quantum field theory
As a successful theoretical framework today, 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 and strong interactions, 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. History of quantum field theory
A subtle and careful analysis in 1933 by Niels Bohr and Léon Rosenfeld showed that there is a fundamental limitation on the ability to simultaneously measure the electric and magnetic field strengths that enter into the description of charges in interaction with radiation, imposed by the uncertainty principle, which must apply to all canonically conjugate quantities. This limitation is crucial for the successful formulation and interpretation of a quantum field theory of photons and electrons (quantum electrodynamics), and indeed, any perturbative quantum field theory. The analysis of Bohr and Rosenfeld explains fluctuations in the values of the electromagnetic field that differ from the classically "allowed" values distant from the sources of the field. Cosmological constant problem
After the development of quantum field theory in the 1940s, the first to address contributions of quantum fluctuations to the cosmological constant was Zel’dovich (1967, 1968). In quantum mechanics, the vacuum itself should experience quantum fluctuations. In general relativity, those quantum fluctuations constitute energy that would add to the cosmological constant. However, this calculated vacuum energy density is many orders of magnitude bigger than the observed cosmological constant. Original estimates of the degree of mismatch were as high as 120 orders of magnitude; however, modern research suggests that, when Lorentz invariance is taken into account, the degree of mismatch is closer to 60 orders of magnitude. Paul Dirac
The necessity of fermions (matter) being created and destroyed in Enrico Fermi's 1934 theory of beta decay led to a reinterpretation of Dirac's equation as a "classical" field equation for any point particle of spin "ħ"/2, itself subject to quantisation conditions involving anti-commutators. Thus reinterpreted, in 1934 by Werner Heisenberg, as a (quantum) field equation accurately describing all elementary matter particles – today quarks and leptons – this Dirac field equation is as central to theoretical physics as the Maxwell, Yang–Mills and Einstein field equations. Dirac is regarded as the founder of quantum electrodynamics, being the first to use that term. He also introduced the idea of vacuum polarisation in the early 1930s. This work was key to the development of quantum mechanics by the next generation of theorists, in particular Schwinger, Feynman, Sin-Itiro Tomonaga and Dyson in their formulation of quantum electrodynamics. Quantum field theory
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 interactions between all leptons and the effects of the Higgs boson. His theory was at first mostly ignored,[11][9]: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 quarks in 1970 by Glashow, John Iliopoulos, and Luciano Maiani, marking its completion.[11] Canonical quantization
When it was first developed, quantum physics dealt only with the quantization of the motion of particles, leaving the electromagnetic field classical, hence the name quantum mechanics. Planck's law
Planck did not propose that light propagating in free space is quantized. The idea of quantization of the free electromagnetic field was developed later, and eventually incorporated into what we now know as quantum field theory. Quantum field theory
Soon after the introduction of photons, Einstein performed the quantization procedure on vibrations in a crystal, leading to the first quasiparticle — phonons. 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.[17] History of quantum field theory
Efforts to describe gravity using the same techniques have, to date, failed. The study of quantum field theory is still flourishing, as are applications of its methods to many physical problems. It remains one of the most vital areas of theoretical physics today, providing a common language to several different branches of physics. Quantum field theory
In 1954, Yang Chen-Ning and Robert Mills generalised the local symmetry of QED, leading to non-Abelian gauge theories (also known as Yang-Mills theories), which are based on more complicated local symmetry groups.[9]: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" interact via the exchange of massless gauge bosons. Unlike photons, these gauge bosons themselves carry charge.[3]:32[10] Quantum optics
As laser science needed good theoretical foundations, and also because research into these soon proved very fruitful, interest in quantum optics rose. Following the work of Dirac in quantum field theory, George Sudarshan, Roy J. Glauber, and Leonard Mandel applied quantum theory to the electromagnetic field in the 1950s and 1960s to gain a more detailed understanding of photodetection and the statistics of light (see degree of coherence). This led to the introduction of the coherent state as a concept which addressed variations between laser light, thermal light, exotic squeezed states, etc. as it became understood that light cannot be fully described just referring to the electromagnetic fields describing the waves in the classical picture. In 1977, Kimble et al. demonstrated a single atom emitting one photon at a time, further compelling evidence that light consists of photons. Previously unknown quantum states of light with characteristics unlike classical states, such as squeezed light were subsequently discovered. Quantum biology
Quantum biology is an emerging field; most of the current research is theoretical and subject to questions that require further experimentation. Though the field has only recently received an influx of attention, it has been conceptualized by physicists all throughout the 20th century. Early pioneers of quantum physics saw applications of quantum mechanics in biological problems. Erwin Schrödinger's 1944 book "What is Life?" discussed applications of quantum mechanics in biology. Schrödinger introduced the idea of an "aperiodic crystal" that contained genetic information in its configuration of covalent chemical bonds. He further suggested that mutations are introduced by "quantum leaps". Other pioneers Niels Bohr, Pascual Jordan, and Max Delbruck argued that the quantum idea of complementarity was fundamental to the life sciences. In 1963, Per-Olov Löwdin published proton tunneling as another mechanism for DNA mutation. In his paper, he stated that there is a new field of study called "quantum biology". Quantum chromodynamics
In 1973 the concept of color as the source of a "strong field" was developed into the theory of QCD by physicists Harald Fritzsch and Heinrich Leutwyler, together with physicist Murray Gell-Mann. In particular, they employed the general field theory developed in 1954 by Chen Ning Yang and Robert Mills (see Yang–Mills theory), in which the carrier particles of a force can themselves radiate further carrier particles. (This is different from QED, where the photons that carry the electromagnetic force do not radiate further photons.) Quantum field theory
In spite of its overwhelming success in particle physics and condensed matter physics, QFT itself lacks a formal mathematical foundation. For example, according to Haag's theorem, there does not exist a well-defined interaction picture for QFT, which implies that perturbation theory of QFT, which underlies the entire Feynman diagram method, is fundamentally not rigorous.[36] Asymptotic freedom
The discovery was instrumental in "rehabilitating" quantum field theory. Prior to 1973, many theorists suspected that field theory was fundamentally inconsistent because the interactions become infinitely strong at short distances. This phenomenon is usually called a Landau pole, and it defines the smallest length scale that a theory can describe. This problem was discovered in field theories of interacting scalars and spinors, including quantum electrodynamics (QED), and Lehman positivity led many to suspect that it is unavoidable. 
Asymptotically free theories become weak at short distances, there is no Landau pole, and these quantum field theories are believed to be completely consistent down to any length scale. Zero-point energy
In 1926 Pascual Jordan published the first attempt to quantize the electromagnetic field. In a joint paper with Max Born and Werner Heisenberg he considered the field inside a cavity as a superposition of quantum harmonic oscillators. In his calculation he found that in addition to the "thermal energy" of the oscillators there also had to exist infinite zero-point energy term. He was able to obtain the same fluctuation formula that Einstein had obtained in 1909. However, Jordan did not think that his infinite zero-point energy term was "real", writing to Einstein that "it is just a quantity of the calculation having no direct physical meaning" Jordan found a way to get rid of the infinite term, publishing a joint work with Pauli in 1928, performing what has been called "the first infinite subtraction, or renormalisation, in quantum field theory"
Building on the work of Heisenberg and others Paul Dirac's theory of emission and absorption (1927) was the first application of the quantum theory of radiation. Dirac's work was seen as crucially important to the emerging field of quantum mechanics; it dealt directly with the process in which "particles" are actually created: spontaneous emission. Dirac described the quantization of the electromagnetic field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators of particles. The theory showed that spontaneous emission depends upon the zero-point energy fluctuations of the electromagnetic field in order to get started. In a process in which a photon is annihilated (absorbed), the photon can be thought of as making a transition into the vacuum state. Similarly, when a photon is created (emitted), it is occasionally useful to imagine that the photon has made a transition out of the vacuum state. In the words of Dirac: Quantum field theory
It was with the invention of the renormalization procedure and Feynman diagrams that QFT finally arose as a complete theoretical framework.[8]:2 Quantum field theory in curved spacetime
The theory of quantum field theory in curved spacetime can be considered as a first approximation to quantum gravity. A second step towards that theory would be semiclassical gravity, which would include the influence of particles created by a strong gravitational field on the spacetime (which is still considered classical and the equivalence principle still holds). However gravity is not renormalizable in QFT, so merely formulating QFT in curved spacetime is not a theory of quantum gravity. Quantum field theory
Abraham–Lorentz force AdS/CFT correspondence Introduction to quantum mechanics Common integrals in quantum field theory Einstein–Maxwell–Dirac equations Form factor (quantum field theory) Green–Kubo relations Green's function (many-body theory) List of quantum field theories Quantization of a field Quantum electrodynamics Quantum field theory in curved spacetime Quantum flavordynamics Quantum hydrodynamics Quantum triviality Relation between Schrödinger's equation and the path integral formulation of quantum mechanics Relationship between string theory and quantum field theory Schwinger–Dyson equation Static forces and virtual-particle exchange Symmetry in quantum mechanics Theoretical and experimental justification for the Schrödinger equation Ward–Takahashi identity Wheeler–Feynman absorber theory Wigner's classification Wigner's theorem M-theory
The application of quantum mechanics to physical objects such as the electromagnetic field, which are extended in space and time, is known as quantum field theory. In particle physics, quantum field theories form the basis for our understanding of elementary particles, which are modeled as excitations in the fundamental fields. Quantum field theories are also used throughout condensed matter physics to model particle-like objects called quasiparticles. Introduction to quantum mechanics
In the 1960s [[physicist]]s realized that QED broke down at extremely high energies. From this inconsistency the [[Standard Model]] of particle physics was discovered, which remedied the higher energy breakdown in theory. It is another extended quantum field theory that unifies the electromagnetic and [[weak interaction]]s into one theory. This is called the [[electroweak theory]]. Field (physics)
It is now believed that quantum mechanics should underlie all physical phenomena, so that a classical field theory should, at least in principle, permit a recasting in quantum mechanical terms; success yields the corresponding quantum field theory. For example, quantizing classical electrodynamics gives quantum electrodynamics. Quantum electrodynamics is arguably the most successful scientific theory; experimental data confirm its predictions to a higher precision (to more significant digits) than any other theory. The two other fundamental quantum field theories are quantum chromodynamics and the electroweak theory. Quantum chemistry
Some view the birth of quantum chemistry as starting with the discovery of the Schrödinger equation and its application to the hydrogen atom in 1926. However, the 1927 article of Walter Heitler (1904-1981) and Fritz London, is often recognized as the first milestone in the history of quantum chemistry. This is the first application of quantum mechanics to the diatomic hydrogen molecule, and thus to the phenomenon of the chemical bond. In the following years much progress was accomplished by Edward Teller, Robert S. Mulliken, Max Born, J. Robert Oppenheimer, Linus Pauling, Erich Hückel, Douglas Hartree, Vladimir Fock, to cite a few. The history of quantum chemistry also goes through the 1838 discovery of cathode rays by Michael Faraday, the 1859 statement of the black body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system could be discrete, and the 1900 quantum hypothesis by Max Planck that any energy radiating atomic system can theoretically be divided into a number of discrete energy elements "ε" such that each of these energy elements is proportional to the frequency "ν" with which they each individually radiate energy and a numerical value called Planck’s Constant. Then, in 1905, to explain the photoelectric effect (1839), i.e., that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated, based on Planck’s quantum hypothesis, that light itself consists of individual quantum particles, which later came to be called photons (1926). In the years to follow, this theoretical basis slowly began to be applied to chemical structure, reactivity, and bonding. Probably the greatest contribution to the field was made by Linus Pauling. Newton–Wigner localization
Apparently these notions were discovered when attempting to define a self adjoint operator in the relativistic setting that resembled the 
position operator in basic quantum mechanics in the sense that at low momenta it 
approximately agreed with that operator. It also has several famous strange behaviors, one of
which is seen as the motivation for having to introduce quantum field theory. Introduction to quantum mechanics
The idea of quantum field theory began in the late 1920s with British physicist [[Paul Dirac]], when he attempted to [[quantization (physics)|quantise]] the [[electromagnetic field]] – a procedure for constructing a quantum theory starting from a classical theory. Quantum field theory
A series of papers were 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. Unfortunately, such achievements were not understood and recognized by the theoretical community.[6] String theory
In quantum field theory, one typically computes the probabilities of various physical events using the techniques of perturbation theory. Developed by Richard Feynman and others in the first half of the twentieth century, perturbative quantum field theory uses special diagrams called Feynman diagrams to organize computations. One imagines that these diagrams depict the paths of point-like particles and their interactions. Quantum gravity
One attempt to overcome these limitations is to replace ordinary quantum field theory, which is based on the classical concept of a point particle, with a quantum theory of one-dimensional extended objects: string theory. At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different modes of oscillation of one and the same type of fundamental string appear as particles with different (electric and other) charges. In this way, string theory promises to be a unified description of all particles and interactions. The theory is successful in that one mode will always correspond to a graviton, the messenger particle of gravity; however, the price of this success are unusual features such as six extra dimensions of space in addition to the usual three for space and one for time. Werner Heisenberg
In early 1929, Heisenberg and Pauli submitted the first of two papers laying the foundation for relativistic quantum field theory. Also in 1929, Heisenberg went on a lecture tour of China, Japan, India, and the United States. In the spring of 1929, he was a visiting lecturer at the University of Chicago, where he lectured on quantum mechanics. History of quantum field theory
Beginning in the 1950s with the work of Yang and Mills, following the previous lead of Weyl and Pauli, deep explorations illuminated the types of symmetries and invariances any field theory must satisfy. QED, and indeed, all field theories, were generalized to a class of quantum field theories known as gauge theories. That symmetries dictate, limit and necessitate the form of interaction between particles is the essence of the "gauge theory revolution". Yang and Mills formulated the first explicit example of a non-abelian gauge theory, Yang–Mills theory, with an attempted explanation of the strong interactions in mind. The strong interactions were then (incorrectly) understood in the mid-1950s, to be mediated by the pi-mesons, the particles predicted by Hideki Yukawa in 1935, based on his profound reflections concerning the reciprocal connection between the mass of any force-mediating particle and the range of the force it mediates. This was allowed by the uncertainty principle. In the absence of dynamical information, Murray Gell-Mann pioneered the extraction of physical predictions from sheer non-abelian symmetry considerations, and introduced non-abelian Lie groups to current algebra and so the gauge theories that came to supersede it. Classical field theory
Further development of quantum field theory changed the focus of searching for unified field theory from classical to quantum description. Because of that, many theoretical physicists gave up looking for a classical unified field theory. Quantum field theory would include unification of other two fundamental forces of nature, strong and weak nuclear force which act on subatomic level. History of quantum field theory
Despite its early successes quantum field theory was plagued by several serious theoretical difficulties. Basic physical quantities, such as the self-energy of the electron, the energy shift of electron states due to the presence of the electromagnetic field, gave infinite, divergent contributions—a nonsensical result—when computed using the perturbative techniques available in the 1930s and most of the 1940s. The electron self-energy problem was already a serious issue in the classical electromagnetic field theory, where the attempt to attribute to the electron a finite size or extent (the classical electron-radius) led immediately to the question of what non-electromagnetic stresses would need to be invoked, which would presumably hold the electron together against the Coulomb repulsion of its finite-sized "parts". The situation was dire, and had certain features that reminded many of the "Rayleigh–Jeans catastrophe". What made the situation in the 1940s so desperate and gloomy, however, was the fact that the correct ingredients (the second-quantized Maxwell–Dirac field equations) for the theoretical description of interacting photons and electrons were well in place, and no major conceptual change was needed analogous to that which was necessitated by a finite and physically sensible account of the radiative behavior of hot objects, as provided by the Planck radiation law. Quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics[1]:xi and is used to construct physical models of subatomic particles (in particle physics) and quasiparticles (in condensed matter physics). Nathan Rosen
At the beginning of the 20th century science was progressing quickly and the inner workings of the atom were just beginning to be discovered. In 1900, Max Planck proposed the quantum theory, the idea that all energy moves in discrete amounts called quanta. In 1905, Albert Einstein published his theory of special relativity, which would be instrumental in the progression of physics and the understanding of the universe. Around 1927, Niels Bohr and Werner Heisenberg, collaborating with many other physicists, developed the Copenhagen interpretation of quantum theory, determining the probabilities of the movement of particles. These breakthroughs provided the model for the structure and workings of the atom and drove the revolution that would sweep up Nathan Rosen. Noncommutative quantum field theory
The particle physics community became interested in the noncommutative approach because of a paper by Nathan Seiberg and Edward Witten. They argued in the context of string theory that the coordinate functions of the endpoints of open strings constrained to a D-brane in the presence of a constant Neveu-Schwarz B-field—equivalent to a constant magnetic field on the brane—would satisfy the noncommutative algebra set out above. The implication is that a quantum field theory on noncommutative spacetime can be interpreted as a low energy limit of the theory of open strings. Vacuum energy
Quantum field theory states that all fundamental fields, such as the electromagnetic field, must be quantized at each and every point in space. A field in physics may be envisioned as if space were filled with interconnected vibrating balls and springs, and the strength of the field is like the displacement of a ball from its rest position. The theory requires "vibrations" in, or more accurately changes in the strength of, such a field to propagate as per the appropriate wave equation for the particular field in question. The second quantization of quantum field theory requires that each such ball–spring combination be quantized, that is, that the strength of the field be quantized at each point in space. Canonically, if the field at each point in space is a simple harmonic oscillator, its quantization places a quantum harmonic oscillator at each point. Excitations of the field correspond to the elementary particles of particle physics. Thus, according to the theory, even the vacuum has a vastly complex structure and all calculations of quantum field theory must be made in relation to this model of the vacuum. History of quantum mechanics
The history of quantum mechanics is a fundamental part of the history of modern physics. Quantum mechanics' history, as it interlaces with the history of quantum chemistry, began essentially with a number of different scientific discoveries: the 1838 discovery of cathode rays by Michael Faraday; the 1859–60 winter statement of the black-body radiation problem by Gustav Kirchhoff; the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system could be "discrete"; the discovery of the photoelectric effect by Heinrich Hertz in 1887; and the 1900 quantum hypothesis by Max Planck that any energy-radiating atomic system can theoretically be divided into a number of discrete "energy elements" "ε" (epsilon) such that each of these energy elements is proportional to the frequency "ν" with which each of them individually radiate energy, as defined by the following formula:
where "h" is a numerical value called Planck's constant. Quantum field theory
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.[6] Uniting these scattered ideas, a coherent discipline, quantum mechanics, was formulated between 1925 and 1926, with important contributions from de Broglie, Werner Heisenberg, Max Born, Erwin Schrödinger, Paul Dirac, and Wolfgang Pauli.[3]:22-23 History of the philosophy of field theory
Field theory had its origins in the 18th century in a mathematical formulation of Newtonian mechanics, but it was seen as deficient as it implied action at a distance. In 1852, Michael Faraday treated the magnetic field as a physical object, reasoning about lines of force. James Clerk Maxwell used Faraday's conceptualisation to help formulate his unification of electricity and magnetism in his electromagnetic theory. Quantum mechanics
Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles. Quantum field theory
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,[8]:6 itself a type of two-dimensional QFT with conformal symmetry.[14] Joël Scherk and John Schwarz first proposed in 1974 that string theory could be the quantum theory of gravity.[15] Quantum field theory
Many phenomena exhibiting quantum mechanical properties cannot be explained by classical fields alone. Phenomena such as the photoelectric effect are best explained by discrete particles (photons), rather than a spatially continuous field. The goal of quantum field theory is to describe various quantum mechanical phenomena using a modified concept of fields. Relationship between string theory and quantum field theory
Many first principles in quantum field theory are explained, or get further insight, in string theory. History of subatomic physics
This culminated in the formulation of ideas of a quantum field theory. The first (and the only mathematically complete) of these theories, quantum electrodynamics, allowed to explain thoroughly the structure of atoms, including the Periodic Table and atomic spectra. Ideas of quantum mechanics and quantum field theory were applied to nuclear physics too. For example, α decay was explained as a quantum tunneling through nuclear potential, nucleons' fermionic statistics explained the nucleon pairing, and Hideki Yukawa proposed certain virtual particles (now knows as π-mesons) as an explanation of the nuclear force. Quantum field theory
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 systems in condensed matter physics. Quantum field theory
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.[2][4]:301[5]:2 Gauge theory
The earliest field theory having a gauge symmetry was Maxwell's formulation, in 1864–65, of electrodynamics ("A Dynamical Theory of the Electromagnetic Field") which stated that any function whose curl vanishes—and can therefore normally be written as a gradient—could be added to the vector potential without affecting the magnetic field. The importance of this symmetry remained unnoticed in the earliest formulations. Similarly unnoticed, Hilbert had derived the Einstein field equations by postulating the invariance of the action under a general coordinate transformation. Later Hermann Weyl, in an attempt to unify general relativity and electromagnetism, conjectured that "Eichinvarianz" or invariance under the change of scale (or "gauge") might also be a local symmetry of general relativity. After the development of quantum mechanics, Weyl, Vladimir Fock and Fritz London modified gauge by replacing the scale factor with a complex quantity and turned the scale transformation into a change of phase, which is a U(1) gauge symmetry. This explained the electromagnetic field effect on the wave function of a charged quantum mechanical particle. This was the first widely recognised gauge theory, popularised by Pauli in 1941. John Moffat (physicist)
In 1990, Moffat proposed a finite, non-local quantum field theory. The theory was developed extensively by Evens, Moffat, Kleppe and Woodard in 1991. In subsequent work, Moffat proposed this theory as an alternative to the standard electroweak unification of electromagnetism and the weak nuclear interactions. Moffat's theory is a quantum field theory with a non-local term in the field Lagrangian. The theory is gauge invariant and it is finite to all orders of perturbation theory. For the standard model it can solve the Higgs boson mass hierarchy naturalness problem. It also leads to a finite quantum gravity theory.University of Toronto press releases re Moffat: Deductive-nomological model
In 1925, Werner Heisenberg as well as Erwin Schrödinger independently formalized quantum mechanics (QM). Despite clashing explanations, the two theories made identical predictions. Paul Dirac's 1928 model of the electron was set to special relativity, launching QM into the first quantum field theory (QFT), quantum electrodynamics (QED). From it, Dirac interpreted and predicted the electron's antiparticle, soon discovered and termed "positron", but the QED failed electrodynamics at high energies. Elsewhere and otherwise, strong nuclear force and weak nuclear force were discovered. History of quantum field theory
Moreover, it has not been established that a theory of quantum gravity is necessary (see Quantum field theory in curved spacetime). History of quantum mechanics
The theory of quantum chromodynamics was formulated beginning in the early 1960s. The theory as we know it today was formulated by Politzer, Gross and Wilczek in 1975. World line
Quantum field theory, the framework in which all of modern particle physics is described, is usually described as a theory of quantized fields. However, although not widely appreciated, it has been known since Feynman that many quantum field theories may equivalently be described in terms of world lines. The world line formulation of quantum field theory has proved particularly fruitful for various calculations in gauge theories and in describing nonlinear effects of electromagnetic fields . Quantum field theory
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 oscillators.[8]:1 With the exclusion of interactions, however, such a theory was yet incapable of making quantitative predictions about the real world.[3]:22 Field (physics)
In the late 1920s, the new rules of quantum mechanics were first applied to the electromagnetic fields. In 1927, Paul Dirac used quantum fields to successfully explain how the decay of an atom to a lower quantum state lead to the spontaneous emission of a photon, the quantum of the electromagnetic field. This was soon followed by the realization (following the work of Pascual Jordan, Eugene Wigner, Werner Heisenberg, and Wolfgang Pauli) that all particles, including electrons and protons, could be understood as the quanta of some quantum field, elevating fields to the status of the most fundamental objects in nature. That said, John Wheeler and Richard Feynman seriously considered Newton's pre-field concept of action at a distance (although they set it aside because of the ongoing utility of the field concept for research in general relativity and quantum electrodynamics). AdS/CFT correspondence
In quantum field theory, one typically computes the probabilities of various physical events using the techniques of perturbation theory. Developed by Richard Feynman and others in the first half of the twentieth century, perturbative quantum field theory uses special diagrams called Feynman diagrams to organize computations. One imagines that these diagrams depict the paths of point-like particles and their interactions. Although this formalism is extremely useful for making predictions, these predictions are only possible when the strength of the interactions, the coupling constant, is small enough to reliably describe the theory as being close to a theory without interactions. History of electromagnetic theory
The first formulation of a quantum theory describing radiation and matter interaction is due to Paul Dirac, who, during 1920, was first able to compute the coefficient of spontaneous emission of an atom. Paul Dirac described the quantization of the electromagnetic field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators of particles. In the following years, with contributions from Wolfgang Pauli, Eugene Wigner, Pascual Jordan, Werner Heisenberg and an elegant formulation of quantum electrodynamics due to Enrico Fermi, physicists came to believe that, in principle, it would be possible to perform any computation for any physical process involving photons and charged particles. However, further studies by Felix Bloch with Arnold Nordsieck, and Victor Weisskopf, in 1937 and 1939, revealed that such computations were reliable only at a first order of perturbation theory, a problem already pointed out by Robert Oppenheimer. At higher orders in the series infinities emerged, making such computations meaningless and casting serious doubts on the internal consistency of the theory itself. With no solution for this problem known at the time, it appeared that a fundamental incompatibility existed between special relativity and quantum mechanics. Quantum thermodynamics
Quantum thermodynamics is the study of the relations between two independent physical theories: thermodynamics and quantum mechanics. The two independent theories address the physical phenomena of light and matter.
In 1905 Einstein argued that the requirement of consistency between thermodynamics and electromagnetism leads to the conclusion that light is quantized obtaining the relation formula_1. This paper is the dawn of quantum theory. In a few decades 
quantum theory became established with an independent set of rules. Currently quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. 
It differs from quantum statistical mechanics in the emphasis on dynamical processes out of equilibrium. 
In addition there is a quest for the theory to be relevant for a single individual quantum system. Timeline of quantum computing
University of Illinois at Urbana–Champaign scientists demonstrate quantum entanglement of multiple characteristics, potentially allowing multiple qubits per particle. Two teams of physicists measured the capacitance of a Josephson junction for the first time. The methods could be used to measure the state of quantum bits in a quantum computer without disturbing the state.[23] In December, the first quantum byte, or qubyte, is announced to have been created by scientists at the Institute of Quantum Optics and Quantum Information at the University of Innsbruck in Austria.[24] Harvard University and Georgia Institute of Technology researchers succeeded in transferring quantum information between "quantum memories" – from atoms to photons and back again. Qubit field theory
A qubit field theory is a quantum field theory in which the canonical commutation relations involved in the quantisation of pairs of observables are relaxed. Specifically it is a quantum field theory in which, unlike most other quantum field theories, the pair of observables is not required to always commute. Axiomatic quantum field theory
There are two main challenges in this discipline. First, one must propose a set of axioms which describe the general properties of any mathematical object that deserves to be called a "quantum field theory". Then, one gives rigorous mathematical constructions of examples satisfying these axioms.