theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

, quantum field theory (QFT) is a theoretical framework that combines

classical field theory A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...

special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...

, and

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

. QFT is used in

particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...

to construct

physical model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...

s of

subatomic particle In physical sciences, a subatomic particle is a particle that composes an atom. According to the Standard Model of particle physics, a subatomic particle can be either a composite particle, which is composed of other particles (for example, a p ...

s and in condensed matter physics to construct models of quasiparticles. QFT treats particles as

excited state In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). Excitation refers to ...

s (also called quanta) of their underlying quantum fields, which are more fundamental than the particles. The

equation of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (V ...

of the particle is determined by minimization of the Lagrangian, a functional of fields associated with the particle. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by Feynman diagrams according to perturbation theory in quantum mechanics.

History

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 Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...

and

electrons The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...

, 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 Weak may refer to: Songs * "Weak" (AJR song), 2016 * "Weak" (Melanie C song), 2011 * "Weak" (SWV song), 1993 * "Weak" (Skunk Anansie song), 1995 * "Weak", a song by Seether from '' Seether: 2002-2013'' Television episodes * "Weak" (''Fear t ...

and

strong interaction The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called th ...

s, to the point where some theorists called for the abandonment of the field theoretic approach. The development of

gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie grou ...

and the completion of the

Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...

in the 1970s led to a renaissance of quantum field theory.

Theoretical background

Quantum field theory results from the combination of

, and

. 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 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." It was not until the 18th century that mathematical physicists discovered a convenient description of gravity based on fields—a numerical quantity (a vector in the case of

gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational pheno ...

) 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. Fields began to take on an existence of their own with the development of

electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...

in the 19th century.

Michael Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic inducti ...

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. The theory of classical electromagnetism was completed in 1864 with Maxwell's equations, which described the relationship between the

electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...

, the

magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...

electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movi ...

, and

electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...

. 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 The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...

. Action-at-a-distance was thus conclusively refuted. Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in atomic spectra, nor for the distribution of blackbody radiation in different wavelengths. Max Planck's study of blackbody radiation marked the beginning of quantum mechanics. He treated atoms, which absorb and emit

electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...

, as tiny oscillators 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 oscillators. This process of restricting energies to discrete values is called quantization. Building on this idea,

Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...

proposed in 1905 an explanation for the photoelectric effect, that light is composed of individual packets of energy called

photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...

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. In 1913, Niels Bohr introduced the Bohr model of atomic structure, wherein

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 Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical physics, classical concepts "particle" or "wave" to fu ...

, that microscopic particles exhibit both wave-like and particle-like properties under different circumstances. Uniting these scattered ideas, a coherent discipline,

, was formulated between 1925 and 1926, with important contributions from Max Planck, Louis de Broglie,

Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a Über quantentheoretische Umdeutung kinematis ...

Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a ...

, Erwin Schrödinger,

Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...

, and Wolfgang Pauli. In the same year as his paper on the photoelectric effect, Einstein published his theory of

, built on Maxwell's electromagnetism. New rules, called

Lorentz transformations In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...

, 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. 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 The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...

underlying quantum mechanics could explain the

stimulated emission Stimulated emission is the process by which an incoming photon of a specific frequency can interact with an excited atomic electron (or other excited molecular state), causing it to drop to a lower energy level. The liberated energy transfers to th ...

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 Spontaneous emission is the process in which a quantum mechanical system (such as a molecule, an atom or a subatomic particle) transits from an excited energy state to a lower energy state (e.g., its ground state) and emits a quantized amount ...

, 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 operators.

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. 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 In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible. Historically, this was not quit ...

by treating the electromagnetic field as a set of quantum harmonic oscillators. With the exclusion of interactions, however, such a theory was yet incapable of making quantitative predictions about the real world. 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 In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional a ...

and the electromagnetic vector potential. Using first-order

perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...

, 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 A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...

, there remains an oscillating electromagnetic field having

zero-point energy Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty pri ...

. 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 Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...

of photons, resonance fluorescence and non-relativistic Compton scattering. Nonetheless, the application of higher-order perturbation theory was plagued with problematic infinities in calculations. In 1928, Dirac wrote down a

wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...

that described relativistic electrons—the Dirac equation. It had the following important consequences: the spin of an electron is 1/2; the electron ''g''-factor is 2; it led to the correct Sommerfeld formula for the

fine structure In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom ...

of the

hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen cons ...

; 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. The prevailing view at the time was that the world was composed of two very different ingredients: material particles (such as electrons) and

quantum fields In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...

(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

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 Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" an ...

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 In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar of that nuclide. For ...

known as

Fermi's interaction In particle physics, Fermi's interaction (also the Fermi theory of beta decay or the Fermi four-fermion interaction) is an explanation of the beta decay, proposed by Enrico Fermi in 1933. The theory posits four fermions directly interactin ...

Atomic nuclei The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron ...

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. 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 positrons was discovered in 1932 by Carl David Anderson in

cosmic ray Cosmic rays are high-energy particles or clusters of particles (primarily represented by protons or atomic nuclei) that move through space at nearly the speed of light. They originate from the Sun, from outside of the Solar System in our own ...

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. QFT naturally incorporated antiparticles in its formalism.

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 In quantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines self-energy \Sigma, and represents the contribution to the particle's energy, or effective mass, due to interactions between ...

and the vacuum zero-point energy of the electron and photon fields, suggesting that the computational methods at the time could not properly deal with interactions involving photons with extremely high momenta. 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. Unfortunately, such achievements were not understood and recognized by the theoretical community. 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 observables (''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 In physics, action at a distance is the concept that an object can be affected without being physically touched (as in mechanical contact) by another object. That is, it is the non-local interaction of objects that are separated in space. Non-c ...

as the mechanism of particle interactions. In 1947, Willis Lamb and Robert Retherford measured the minute difference in the ²''S''_1/2 and ²''P''_1/2 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. Subsequently, Norman Myles Kroll, Lamb,

James Bruce French James Bruce French (1921–2002) was a Canadian-American theoretical physicist, specializing in nuclear physics. J. Bruce French received in 1942 his bachelor's degree in physics from Dalhousie University and served during WWII in the Royal Canad ...

, 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. 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. As Tomonaga said in his Nobel lecture:

Since those parts of the modified mass and charge due to field reactions ecome 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 heAmericans'.

By applying the renormalization procedure, calculations were finally made to explain the electron's anomalous magnetic moment (the deviation of the electron ''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". At the same time, Feynman introduced the path integral formulation of quantum mechanics and Feynman diagrams. 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 In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.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. 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 physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...

, in which the series is expanded, must be a sufficiently small number. The coupling constant in QED is the fine-structure constant , 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

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. With these difficulties looming, many theorists began to turn away from QFT. Some focused on

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

principles and

conservation law In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...

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. Schwinger, however, took a different route. For more than a decade he and his students had been nearly the only exponents of field theory, but in 1966 he found a way around the problem of the infinities with a new method he called source theory. 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. 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. 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. 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. The neglect of source theory by the physics community was a major disappointment for Schwinger:

The lack of appreciation of these facts by others was depressing, but understandable. -J. Schwinger

Standard-Model

In 1954, Yang Chen-Ning and Robert Mills generalized the

local symmetry In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be ''continuou ...

of QED, leading to non-Abelian gauge theories (also known as Yang–Mills theories), which are based on more complicated local

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

s. 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 boson In particle physics, a gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of ga ...

s. Unlike photons, these gauge bosons themselves carry charge. Sheldon Glashow developed a non-Abelian gauge theory that unified the electromagnetic and weak interactions in 1960. In 1964,

Abdus Salam Mohammad Abdus Salam Salam adopted the forename "Mohammad" in 1974 in response to the anti-Ahmadiyya decrees in Pakistan, similarly he grew his beard. (; ; 29 January 192621 November 1996) was a Punjabis, Punjabi Pakistani theoretical physici ...

and John Clive Ward arrived at the same theory through a different path. This theory, nevertheless, was non-renormalizable.

Peter Higgs Peter Ware Higgs (born 29 May 1929) is a British theoretical physicist, Emeritus Professor in the University of Edinburgh,Griggs, Jessica (Summer 2008The Missing Piece ''Edit'' the University of Edinburgh Alumni Magazine, p. 17 and Nobel Prize ...

Robert Brout Robert Brout (; June 14, 1928 – May 3, 2011) was an American theoretical physicist who made significant contributions in elementary particle physics. He was a professor of physics at Université Libre de Bruxelles where he had created, together ...

, François Englert,

Gerald Guralnik Gerald Stanford "Gerry" Guralnik (; September 17, 1936 – April 26, 2014) was the Chancellor’s Professor of Physics at Brown University. In 1964 he co-discovered the Higgs mechanism and Higgs boson with C. R. Hagen and Tom Kibble (GHK). As par ...

, Carl Hagen, and

Tom Kibble Sir Thomas Walter Bannerman Kibble (; 23 December 1932 – 2 June 2016) was a British theoretical physicist, senior research investigator at the Blackett Laboratory and Emeritus Professor of Theoretical Physics at Imperial College London. His ...

proposed in their famous ''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. 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 The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Stan ...

. His theory was at first mostly ignored, until it was brought back to light in 1971 by

Gerard 't Hooft Gerardus (Gerard) 't Hooft (; born July 5, 1946) is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G. Veltman "for elucidating th ...

's proof that non-Abelian gauge theories are renormalizable. The electroweak theory of Weinberg and Salam was extended from leptons to

quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...

s in 1970 by Glashow, John Iliopoulos, and

Luciano Maiani Luciano Maiani (born 16 July 1941, in Rome) is a Sammarinese physicist best known for his prediction of the charm quark with Sheldon Glashow and John Iliopoulos (the "GIM mechanism"). Academic history In 1964 Luciano Maiani received his degree in ...

, marking its completion.

Harald Fritzsch Harald Fritzsch (born 10 February 1943 in Zwickau, Germany, died 16 August 2022 in München) was a German theoretical physicist known for his contributions to the theory of quarks, the development of Quantum Chromodynamics and the great unific ...

, Murray Gell-Mann, and Heinrich Leutwyler discovered in 1971 that certain phenomena involving the

could also be explained by non-Abelian gauge theory.

Quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...

(QCD) was born. In 1973, David Gross, Frank Wilczek, and

Hugh David Politzer Hugh David Politzer (; born August 31, 1949) is an American theoretical physicist and the Richard Chace Tolman Professor of Theoretical Physics at the California Institute of Technology. He shared the 2004 Nobel Prize in Physics with David Gro ...

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 previously, but they had been largely ignored.) 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. 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

of elementary particles. The Standard Model successfully describes all fundamental interactions except

gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...

, and its many predictions have been met with remarkable experimental confirmation in subsequent decades. The

, central to the mechanism of spontaneous symmetry breaking, was finally detected in 2012 at

CERN The European Organization for Nuclear Research, known as CERN (; ; ), is an intergovernmental organization that operates the largest particle physics laboratory in the world. Established in 1954, it is based in a northwestern suburb of Gen ...

, marking the complete verification of the existence of all constituents of the Standard Model.

Other developments

The 1970s saw the development of non-perturbative methods in non-Abelian gauge theories. The

't Hooft–Polyakov monopole __NOTOC__ In theoretical physics, the t Hooft–Polyakov monopole is a topological soliton similar to the Dirac monopole but without the Dirac string. It arises in the case of a Yang–Mills theory with a gauge group G, coupled to a Higgs field whi ...

was discovered theoretically by 't Hooft and Alexander Polyakov, flux tubes by Holger Bech Nielsen and

Poul Olesen Poul is a Danish masculine given name. It is the Danish cognate of the name Paul. Poul may refer to: People * Poul Andersen (1922–2006), Danish printer *Poul Anderson (1926–2001), American writer * Poul Erik Andreasen (born 1949), Danish foo ...

, and instantons by Polyakov and coauthors. These objects are inaccessible through perturbation theory.

Supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...

also appeared in the same period. The first supersymmetric QFT in four dimensions was built by

Yuri Golfand Yuri Abramovich Golfand (russian: Ю́рий Абра́мович Го́льфанд; January 10, 1922 – February 17, 1994) was a Russian and Israeli physicist known, in particular, for his 1971 paper (joint with his student Evgeny Likhtman) ...

and Evgeny Likhtman in 1970, but their result failed to garner widespread interest due to the Iron Curtain. Supersymmetry only took off in the theoretical community after the work of Julius Wess and Bruno Zumino in 1973. 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 In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

, itself a type of two-dimensional QFT with

conformal symmetry In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetr ...

. Joël Scherk and John Schwarz first proposed in 1974 that string theory could be ''the'' quantum theory of gravity.

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 systems 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 In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...

s in matter. Soon after the introduction of photons, Einstein performed the quantization procedure on vibrations in a crystal, leading to the first quasiparticle—

phonon In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechani ...

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. Gauge theory is used to describe the quantization of

magnetic flux In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the weber ...

in superconductors, the resistivity in the quantum Hall effect, as well as the relation between frequency and voltage in the AC Josephson effect.

Principles

For simplicity, natural units are used in the following sections, in which the reduced Planck constant and the

are both set to one.

Classical fields

A classical field is a function of spatial and time coordinates. Examples include the

Newtonian gravity Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...

and the

and

in classical electromagnetism. A classical field can be thought of as a numerical quantity assigned to every point in space that changes in time. Hence, it has infinitely many

degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

. 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 (

s), 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.

Canonical quantization In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible. Historically, this was not quit ...

and path integrals are two common formulations of QFT. To motivate the fundamentals of QFT, an overview of classical field theory follows. The simplest classical field is a real scalar field — a

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

at every point in space that changes in time. It is denoted as , where is the position vector, and is the time. Suppose the Lagrangian of the field,

L

, is :

L = \int d^3x\,\mathcal = \int d^3x\,\left frac 12 \dot\phi^2 - \frac 12 (\nabla\phi)^2 - \frac 12 m^2\phi^2\right

where

\mathcal

is the Lagrangian density,

\dot\phi

is the time-derivative of the field, is the gradient operator, and is a real parameter (the "mass" of the field). Applying the Euler–Lagrange equation on the Lagrangian: :

- \frac = 0,

we obtain the

equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (V ...

for the field, which describe the way it varies in time and space: :

\left(\frac - \nabla^2 + m^2\right)\phi = 0.

This is known as the Klein–Gordon equation. The Klein–Gordon equation is a

, so its solutions can be expressed as a sum of normal modes (obtained via

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...

) as follows: :

\phi(\mathbf, t) = \int \frac \frac\left(a_ e^ + a_^* e^\right),

where is a

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

(normalized by convention), denotes complex conjugation, and is the frequency of the normal mode: :

\omega_ = \sqrt.

Thus each normal mode corresponding to a single can be seen as a classical

harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive const ...

with frequency .

Canonical quantization

The quantization procedure for the above classical field to a quantum operator field is analogous to the promotion of a classical harmonic oscillator to a quantum harmonic oscillator. The displacement of a classical harmonic oscillator is described by :

x(t) = \frac a e^ + \frac a^* e^,

where is a complex number (normalized by convention), and is the oscillator's frequency. Note that is the displacement of a particle in simple harmonic motion from the equilibrium position, not to be confused with the spatial label of a quantum field. For a quantum harmonic oscillator, is promoted to a linear operator

\hat x(t)

: :

\hat x(t) = \frac \hat a e^ + \frac \hat a^\dagger e^.

Complex numbers and are replaced by the

annihilation operator Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...

\hat a

and the

creation operator Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...

\hat a^\dagger

, respectively, where denotes

Hermitian conjugation In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where ...

. The

commutation relation In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...

between the two is :

\left hat a, \hat a^\dagger\right = 1.

The Hamiltonian of the simple harmonic oscillator can be written as :

\hat H = \hbar\omega \hat^\dagger \hat +\frac\hbar\omega.

The vacuum state

, 0\rang

, which is the lowest energy state, is defined by :

\hat a, 0\rang = 0

and has energy

\frac12\hbar\omega

One can easily check that

\hbar\omega,

which implies that

\hat^\dagger

increases the energy of the simple harmonic oscillator by

\hbar\omega

. For example, the state

\hat^\dagger, 0\rang

is an eigenstate of energy

3\hbar\omega/2

. Any energy eigenstate state of a single harmonic oscillator can be obtained from

, 0\rang

by successively applying the creation operator

\hat a^\dagger

: and any state of the system can be expressed as a linear combination of the states :

, n\rang \propto \left(\hat a^\dagger\right)^n, 0\rang.

A similar procedure can be applied to the real scalar field , by promoting it to a quantum field operator

\hat\phi

, while the annihilation operator

\hat a_

, the creation operator

\hat a_^\dagger

and the angular frequency

w_\mathbf

are now for a particular : :

\hat \phi(\mathbf, t) = \int \frac \frac\left(\hat a_ e^ + \hat a_^\dagger e^\right).

Their commutation relations are: :

\left hat a_, \hat a_^\dagger\right = (2\pi)^3\delta(\mathbf - \mathbf),\quad \left hat a_, \hat a_\right = \left hat a_^\dagger, \hat a_^\dagger\right = 0,

where is the

Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...

. The vacuum state

, 0\rang

is defined by :

\hat a_, 0\rang = 0,\quad \text\mathbf p.

Any quantum state of the field can be obtained from

, 0\rang

by successively applying creation operators

\hat a_^\dagger

(or by a linear combination of such states), e.g. :

\left(\hat a_^\dagger\right)^3 \hat a_^\dagger \left(\hat a_^\dagger\right)^2 , 0\rang.

While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy levels of an arbitrary number of particles. The latter space is known as a Fock space, which can account for the fact that particle numbers are not fixed in relativistic quantum systems. The process of quantizing an arbitrary number of particles instead of a single particle is often also called

second quantization Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as t ...

. The foregoing procedure is a direct application of non-relativistic quantum mechanics and can be used to quantize (complex) scalar fields,

Dirac field In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of b ...

s, vector fields (''e.g.'' the electromagnetic field), and even strings. 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,

would be necessary. The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a quartic interaction term could be introduced to the Lagrangian of the real scalar field: :

\mathcal = \frac 12 (\partial_\mu\phi)\left(\partial^\mu\phi\right) - \frac 12 m^2\phi^2 - \frac\phi^4,

where is a spacetime index,

\partial_0 = \partial/\partial t,\ \partial_1 = \partial/\partial x^1

, etc. The summation over the index has been omitted following the

Einstein notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...

. If the parameter is sufficiently small, then the interacting theory described by the above Lagrangian can be considered as a small perturbation from the free theory.

Path integrals

The path integral formulation of QFT is concerned with the direct computation of the

, \phi_I\rang

at time to some final state

, \phi_F\rang

at , the total time is divided into small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let be the Hamiltonian (''i.e.'' generator of time evolution), then :

\lang \phi_F, e^, \phi_I\rang = \int d\phi_1\int d\phi_2\cdots\int d\phi_\,\lang \phi_F, e^, \phi_\rang\cdots\lang \phi_2, e^, \phi_1\rang\lang \phi_1, e^, \phi_I\rang.

Taking the limit , the above product of integrals becomes the Feynman path integral: :

\lang \phi_F, e^, \phi_I\rang = \int \mathcal\phi(t)\,\exp\left\,

where is the Lagrangian involving and its derivatives with respect to spatial and time coordinates, obtained from the Hamiltonian via Legendre transformation. The initial and final conditions of the path integral are respectively :

\phi(0) = \phi_I,\quad \phi(T) = \phi_F.

In other words, the overall amplitude is the sum over the amplitude of every possible path between the initial and final states, where the amplitude of a path is given by the exponential in the integrand.

Two-point correlation function

In calculations, one often encounters expression like

\lang 0, T\, 0\rang
\quad \text \quad
\lang \Omega , T\, \Omega \rang

in the free or interacting theory, respectively. Here,

x

and

y

are position four-vectors,

T

is the time ordering operator that shuffles its operands so the time-components

x^0

and

y^0

increase from right to left, and

, \Omega\rang

is the ground state (vacuum state) of the interacting theory, different from the free ground state

,  0 \rang

. This expression represents the probability amplitude for the field to propagate from to , and goes by multiple names, like the two-point

propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. ...

, two-point correlation function, two-point Green's function or two-point function for short. The free two-point function, also known as the

Feynman propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In ...

, can be found for the real scalar field by either canonical quantization or path integrals to be :

\lang 0, T\ , 0\rang \equiv D_F(x-y) = \lim_ \int\frac \frac e^.

In an interacting theory, where the Lagrangian or Hamiltonian contains terms

L_I(t)

H_I(t)

that describe interactions, the two-point function is more difficult to define. However, through both the canonical quantization formulation and the path integral formulation, it is possible to express it through an infinite perturbation series of the ''free'' two-point function. In canonical quantization, the two-point correlation function can be written as: :

\lang\Omega, T\, \Omega\rang = \lim_ \frac,

where is an

infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...

number and is the field operator under the free theory. Here, the exponential should be understood as its

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...

expansion. For example, in

\phi^4

-theory, the interacting term of the Hamiltonian is

H_I(t) = \int d^3 x\,\frac\phi_I(x)^4

, and the expansion of the two-point correlator in terms of

\lambda

becomes

\lang\Omega, T\, \Omega\rang = 
\frac.

This perturbation expansion expresses the interacting two-point function in terms of quantities

\lang 0 ,  \cdots ,  0 \rang

that are evaluated in the ''free'' theory. In the path integral formulation, the two-point correlation function can be written :

\lang\Omega, T\, \Omega\rang = \lim_ \frac,

where

\mathcal

is the Lagrangian density. As in the previous paragraph, the exponential can be expanded as a series in , reducing the interacting two-point function to quantities in the free theory. Wick's theorem further reduce any -point correlation function in the free theory to a sum of products of two-point correlation functions. For example, :

\begin
\lang 0, T\, 0\rang &= \lang 0, T\, 0\rang \lang 0, T\, 0\rang\\
&+ \lang 0, T\, 0\rang \lang 0, T\, 0\rang\\
&+ \lang 0, T\, 0\rang \lang 0, T\, 0\rang.
\end

Since interacting correlation functions can be expressed in terms of free correlation functions, only the latter need to be evaluated in order to calculate all physical quantities in the (perturbative) interacting theory. This makes the Feynman propagator one of the most important quantities in quantum field theory.

Feynman diagram

Correlation functions in the interacting theory can be written as a perturbation series. Each term in the series is a product of Feynman propagators in the free theory and can be represented visually by a Feynman diagram. For example, the term in the two-point correlation function in the theory is :

\frac\int d^4z\,\lang 0, T\, 0\rang.

After applying Wick's theorem, one of the terms is :

12\cdot \frac\int d^4z\, D_F(x-z)D_F(y-z)D_F(z-z).

This term can instead be obtained from the Feynman diagram : Phi-4 one-loop

. The diagram consists of * ''external vertices'' connected with one edge and represented by dots (here labeled

x

and

y

). * ''internal vertices'' connected with four edges and represented by dots (here labeled

z

). * ''edges'' connecting the vertices and represented by lines. Every vertex corresponds to a single

\phi

field factor at the corresponding point in spacetime, while the edges correspond to the propagators between the spacetime points. The term in the perturbation series corresponding to the diagram is obtained by writing down the expression that follows from the so-called Feynman rules: # For every internal vertex

z_i

, write down a factor

-i \lambda \int d^4 z_i

. # For every edge that connects two vertices

z_i

and

z_j

, write down a factor

D_F(z_i-z_j)

. # Divide by the symmetry factor of the diagram. With the symmetry factor

2

, following these rules yields exactly the expression above. By Fourier transforming the propagator, the Feynman rules can be reformulated from position space into momentum space. In order to compute the -point correlation function to the -th order, list all valid Feynman diagrams with external points and or fewer vertices, and then use Feynman rules to obtain the expression for each term. To be precise, :

\lang\Omega, T\, \Omega\rang

is equal to the sum of (expressions corresponding to) all connected diagrams with external points. (Connected diagrams are those in which every vertex is connected to an external point through lines. Components that are totally disconnected from external lines are sometimes called "vacuum bubbles".) In the interaction theory discussed above, every vertex must have four legs. In realistic applications, the scattering amplitude of a certain interaction or the

decay rate Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consid ...

of a particle can be computed from the S-matrix, which itself can be found using the Feynman diagram method. Feynman diagrams devoid of "loops" are called tree-level diagrams, which describe the lowest-order interaction processes; those containing loops are referred to as -loop diagrams, which describe higher-order contributions, or radiative corrections, to the interaction. Lines whose end points are vertices can be thought of as the propagation of

virtual particle A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle. The concept of virtual particles arises in the perturba ...

Renormalization

Feynman rules can be used to directly evaluate tree-level diagrams. However, naïve computation of loop diagrams such as the one shown above will result in divergent momentum integrals, which seems to imply that almost all terms in the perturbative expansion are infinite. The

renormalisation Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...

procedure is a systematic process for removing such infinities. Parameters appearing in the Lagrangian, such as the mass and the coupling constant , have no physical meaning — , , and the field strength are not experimentally measurable quantities and are referred to here as the bare mass, bare coupling constant, and bare field, respectively. The physical mass and coupling constant are measured in some interaction process and are generally different from the bare quantities. While computing physical quantities from this interaction process, one may limit the domain of divergent momentum integrals to be below some momentum cut-off , obtain expressions for the physical quantities, and then take the limit . This is an example of

regularization Regularization may refer to: * Regularization (linguistics) * Regularization (mathematics) * Regularization (physics) * Regularization (solid modeling) * Regularization Law, an Israeli law intended to retroactively legalize settlements See also ...

, a class of methods to treat divergences in QFT, with being the regulator. The approach illustrated above is called bare perturbation theory, as calculations involve only the bare quantities such as mass and coupling constant. A different approach, called renormalized perturbation theory, is to use physically meaningful quantities from the very beginning. In the case of theory, the field strength is first redefined: :

\phi = Z^\phi_r,

where is the bare field, is the renormalized field, and is a constant to be determined. The Lagrangian density becomes: :

\mathcal = \frac 12 (\partial_\mu\phi_r)(\partial^\mu\phi_r) - \frac 12 m_r^2\phi_r^2 - \frac\phi_r^4 + \frac 12 \delta_Z (\partial_\mu\phi_r)(\partial^\mu\phi_r) - \frac 12 \delta_m\phi_r^2 - \frac\phi_r^4,

where and are the experimentally measurable, renormalized, mass and coupling constant, respectively, and :

\delta_Z = Z-1,\quad \delta_m = m^2Z - m_r^2,\quad \delta_\lambda = \lambda Z^2 - \lambda_r

are constants to be determined. The first three terms are the Lagrangian density written in terms of the renormalized quantities, while the latter three terms are referred to as "counterterms". As the Lagrangian now contains more terms, so the Feynman diagrams should include additional elements, each with their own Feynman rules. The procedure is outlined as follows. First select a regularization scheme (such as the cut-off regularization introduced above or

dimensional regularization __NOTOC__ In theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini as well as – independently and more comprehensively – by 't Hooft and Veltman for regularizing integrals in the evaluation of ...

); call the regulator . Compute Feynman diagrams, in which divergent terms will depend on . Then, define , , and such that Feynman diagrams for the counterterms will exactly cancel the divergent terms in the normal Feynman diagrams when the limit is taken. In this way, meaningful finite quantities are obtained. It is only possible to eliminate all infinities to obtain a finite result in renormalizable theories, whereas in non-renormalizable theories infinities cannot be removed by the redefinition of a small number of parameters. The

of elementary particles is a renormalizable QFT, while quantum gravity is non-renormalizable.

Renormalization group

The

renormalization group In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in t ...

, developed by Kenneth Wilson, is a mathematical apparatus used to study the changes in physical parameters (coefficients in the Lagrangian) as the system is viewed at different scales. The way in which each parameter changes with scale is described by its ''β'' function. Correlation functions, which underlie quantitative physical predictions, change with scale according to the Callan–Symanzik equation. As an example, the coupling constant in QED, namely the

elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundam ...

, has the following ''β'' function: :

\beta(e) \equiv \frac\frac = \frac + O\mathord\left(e^5\right),

where is the energy scale under which the measurement of is performed. This

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...

implies that the observed elementary charge increases as the scale increases. The renormalized coupling constant, which changes with the energy scale, is also called the running coupling constant. The coupling constant in

quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...

, a non-Abelian gauge theory based on the symmetry group , has the following ''β'' function: :

\beta(g) \equiv \frac\frac = \frac\left(-11 + \frac 23 N_f\right) + O\mathord\left(g^5\right),

where is the number of

flavours. In the case where (the Standard Model has ), the coupling constant decreases as the energy scale increases. Hence, while the strong interaction is strong at low energies, it becomes very weak in high-energy interactions, a phenomenon known as asymptotic freedom. Conformal field theories (CFTs) are special QFTs that admit

. They are insensitive to changes in the scale, as all their coupling constants have vanishing ''β'' function. (The converse is not true, however — the vanishing of all ''β'' functions does not imply conformal symmetry of the theory.) Examples include

and supersymmetric Yang–Mills theory. According to Wilson's picture, every QFT is fundamentally accompanied by its energy cut-off , ''i.e.'' that the theory is no longer valid at energies higher than , and all degrees of freedom above the scale are to be omitted. For example, the cut-off could be the inverse of the atomic spacing in a condensed matter system, and in elementary particle physics it could be associated with the fundamental "graininess" of spacetime caused by quantum fluctuations in gravity. The cut-off scale of theories of particle interactions lies far beyond current experiments. Even if the theory were very complicated at that scale, as long as its couplings are sufficiently weak, it must be described at low energies by a renormalizable effective field theory. The difference between renormalizable and non-renormalizable theories is that the former are insensitive to details at high energies, whereas the latter do depend on them. According to this view, non-renormalizable theories are to be seen as low-energy effective theories of a more fundamental theory. The failure to remove the cut-off from calculations in such a theory merely indicates that new physical phenomena appear at scales above , where a new theory is necessary.

Other theories

The quantization and renormalization procedures outlined in the preceding sections are performed for the free theory and theory of the real scalar field. A similar process can be done for other types of fields, including the complex scalar field, the vector field, and the

, as well as other types of interaction terms, including the electromagnetic interaction and the Yukawa interaction. As an example, quantum electrodynamics contains a Dirac field representing the

electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...

field and a vector field representing the electromagnetic field (

field). (Despite its name, the quantum electromagnetic "field" actually corresponds to the classical electromagnetic four-potential, rather than the classical electric and magnetic fields.) The full QED Lagrangian density is: :

\mathcal = \bar\psi\left(i\gamma^\mu\partial_\mu - m\right)\psi - \frac 14 F_F^ - e\bar\psi\gamma^\mu\psi A_\mu,

where are

Dirac matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...

\bar\psi = \psi^\dagger\gamma^0

, and

F_ = \partial_\mu A_\nu - \partial_\nu A_\mu

is the electromagnetic field strength. The parameters in this theory are the (bare) electron mass and the (bare)

. The first and second terms in the Lagrangian density correspond to the free Dirac field and free vector fields, respectively. The last term describes the interaction between the electron and photon fields, which is treated as a perturbation from the free theories. ElectronPositronAnnihilation

Shown above is an example of a tree-level Feynman diagram in QED. It describes an electron and a positron annihilating, creating an off-shell photon, and then decaying into a new pair of electron and positron. Time runs from left to right. Arrows pointing forward in time represent the propagation of positrons, while those pointing backward in time represent the propagation of electrons. A wavy line represents the propagation of a photon. Each vertex in QED Feynman diagrams must have an incoming and an outgoing fermion (positron/electron) leg as well as a photon leg.

Gauge symmetry

If the following transformation to the fields is performed at every spacetime point (a local transformation), then the QED Lagrangian remains unchanged, or invariant: :

\psi(x) \to e^\psi(x),\quad A_\mu(x) \to A_\mu(x) + ie^ e^\partial_\mu e^,

where is any function of spacetime coordinates. If a theory's Lagrangian (or more precisely the action) is invariant under a certain local transformation, then the transformation is referred to as a gauge symmetry of the theory. Gauge symmetries form a group at every spacetime point. In the case of QED, the successive application of two different local symmetry transformations

e^

and

e^

is yet another symmetry transformation

e^

. For any ,

e^

is an element of the group, thus QED is said to have gauge symmetry. The photon field may be referred to as the

. is an

Abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...

, meaning that the result is the same regardless of the order in which its elements are applied. QFTs can also be built on non-Abelian groups, giving rise to non-Abelian gauge theories (also known as Yang–Mills theories).

, which describes the strong interaction, is a non-Abelian gauge theory with an gauge symmetry. It contains three Dirac fields representing

fields as well as eight vector fields representing

gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind ...

fields, which are the gauge bosons. The QCD Lagrangian density is: :

\mathcal = i\bar\psi^i \gamma^\mu (D_\mu)^ \psi^j - \frac 14 F_^aF^ - m\bar\psi^i \psi^i,

where is the gauge

covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differe ...

: :

D_\mu = \partial_\mu - igA_\mu^a t^a,

where is the coupling constant, are the eight generators of in the fundamental representation ( matrices), :

F_^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^A_\mu^b A_\nu^c,

and are the structure constants of . Repeated indices are implicitly summed over following Einstein notation. This Lagrangian is invariant under the transformation: :

^\dagger(x),

where is an element of at every spacetime point : :

U(x) = e^.

The preceding discussion of symmetries is on the level of the Lagrangian. In other words, these are "classical" symmetries. After quantization, some theories will no longer exhibit their classical symmetries, a phenomenon called anomaly. For instance, in the path integral formulation, despite the invariance of the Lagrangian density

\mathcal phi,\partial_\mu\phi /math> under a certain local transformation of the fields, the measure \int\mathcal D\phi of the path integral may change. For a theory describing nature to be consistent, it must not contain any anomaly in its gauge symmetry. The Standard Model of elementary particles is a gauge theory based on the group , in which all anomalies exactly cancel.

The theoretical foundation of

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

, the

equivalence principle In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (su ...

, can also be understood as a form of gauge symmetry, making general relativity a gauge theory based on the Lorentz group. Noether's theorem states that every continuous symmetry, ''i.e.'' the parameter in the symmetry transformation being continuous rather than discrete, leads to a corresponding

. For example, the symmetry of QED implies

charge conservation In physics, charge conservation is the principle that the total electric charge in an isolated system never changes. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is al ...

. Gauge-transformations do not relate distinct quantum states. Rather, it relates two equivalent mathematical descriptions of the same quantum state. As an example, the photon field , being a four-vector, has four apparent degrees of freedom, but the actual state of a photon is described by its two degrees of freedom corresponding to the polarization. The remaining two degrees of freedom are said to be "redundant" — apparently different ways of writing can be related to each other by a gauge transformation and in fact describe the same state of the photon field. In this sense, gauge invariance is not a "real" symmetry, but a reflection of the "redundancy" of the chosen mathematical description. To account for the gauge redundancy in the path integral formulation, one must perform the so-called Faddeev–Popov

gauge fixing In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...

procedure. In non-Abelian gauge theories, such a procedure introduces new fields called "ghosts". Particles corresponding to the ghost fields are called ghost particles, which cannot be detected externally. A more rigorous generalization of the Faddeev–Popov procedure is given by

BRST quantization In theoretical physics, the BRST formalism, or BRST quantization (where the ''BRST'' refers to the last names of Carlo Becchi, , Raymond Stora and Igor Tyutin) denotes a relatively rigorous mathematical approach to quantizing a field theory with ...

Spontaneous symmetry-breaking

Spontaneous symmetry breaking is a mechanism whereby the symmetry of the Lagrangian is violated by the system described by it. To illustrate the mechanism, consider a linear

sigma model In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or ...

containing real scalar fields, described by the Lagrangian density: :

\mathcal = \frac 12 \left(\partial_\mu\phi^i\right)\left(\partial^\mu\phi^i\right) + \frac 12 \mu^2 \phi^i\phi^i - \frac \left(\phi^i\phi^i\right)^2,

where and are real parameters. The theory admits an global symmetry: :

\phi^i \to R^\phi^j,\quad R\in\mathrm(N).

The lowest energy state (ground state or vacuum state) of the classical theory is any uniform field satisfying :

\phi_0^i \phi_0^i = \frac.

Without loss of generality, let the ground state be in the -th direction: :

\phi_0^i = \left(0,\cdots,0,\frac\right).

The original fields can be rewritten as: :

\phi^i(x) = \left(\pi^1(x),\cdots,\pi^(x),\frac + \sigma(x)\right),

and the original Lagrangian density as: :

\mathcal = \frac 12 \left(\partial_\mu\pi^k\right)\left(\partial^\mu\pi^k\right) + \frac 12 \left(\partial_\mu\sigma\right)\left(\partial^\mu\sigma\right) - \frac 12 \left(2\mu^2\right)\sigma^2 - \sqrt\mu\sigma^3 - \sqrt\mu\pi^k\pi^k\sigma - \frac \pi^k\pi^k\sigma^2 - \frac\left(\pi^k\pi^k\right)^2,

where . The original global symmetry is no longer manifest, leaving only the subgroup . The larger symmetry before spontaneous symmetry breaking is said to be "hidden" or spontaneously broken.

Goldstone's theorem In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu in part ...

states that under spontaneous symmetry breaking, every broken continuous global symmetry leads to a massless field called the Goldstone boson. In the above example, has continuous symmetries (the dimension of its

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

), while has . The number of broken symmetries is their difference, , which corresponds to the massless fields . On the other hand, when a gauge (as opposed to global) symmetry is spontaneously broken, the resulting Goldstone boson is "eaten" by the corresponding gauge boson by becoming an additional degree of freedom for the gauge boson. The Goldstone boson equivalence theorem states that at high energy, the amplitude for emission or absorption of a longitudinally polarized massive gauge boson becomes equal to the amplitude for emission or absorption of the Goldstone boson that was eaten by the gauge boson. In the QFT of ferromagnetism, spontaneous symmetry breaking can explain the alignment of magnetic dipoles at low temperatures. In the Standard Model of elementary particles, the

W and Z bosons In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...

, which would otherwise be massless as a result of gauge symmetry, acquire mass through spontaneous symmetry breaking of the

, a process called the

Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property " mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other b ...

Supersymmetry

All experimentally known symmetries in nature relate bosons to bosons and

fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...

s to fermions. Theorists have hypothesized the existence of a type of symmetry, called

supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...

, that relates bosons and fermions. The Standard Model obeys Poincaré symmetry, whose generators are the spacetime translations and the

. In addition to these generators, supersymmetry in (3+1)-dimensions includes additional generators , called supercharges, which themselves transform as Weyl fermions. The symmetry group generated by all these generators is known as the super-Poincaré group. In general there can be more than one set of supersymmetry generators, , which generate the corresponding supersymmetry, supersymmetry, and so on. Supersymmetry can also be constructed in other dimensions, most notably in (1+1) dimensions for its application in superstring theory. The Lagrangian of a supersymmetric theory must be invariant under the action of the super-Poincaré group. Examples of such theories include: Minimal Supersymmetric Standard Model (MSSM), supersymmetric Yang–Mills theory, and superstring theory. In a supersymmetric theory, every fermion has a bosonic superpartner and vice versa. If supersymmetry is promoted to a local symmetry, then the resultant gauge theory is an extension of general relativity called supergravity. Supersymmetry is a potential solution to many current problems in physics. For example, the

hierarchy problem In theoretical physics, the hierarchy problem is the problem concerning the large discrepancy between aspects of the weak force and gravity. There is no scientific consensus on why, for example, the weak force is 1024 times stronger than grav ...

of the Standard Model—why the mass of the Higgs boson is not radiatively corrected (under renormalization) to a very high scale such as the grand unified scale or the Planck scale—can be resolved by relating the

Higgs field The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the St ...

and its super-partner, the Higgsino. Radiative corrections due to Higgs boson loops in Feynman diagrams are cancelled by corresponding Higgsino loops. Supersymmetry also offers answers to the grand unification of all gauge coupling constants in the Standard Model as well as the nature of

dark matter Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not ...

. Nevertheless, , experiments have yet to provide evidence for the existence of supersymmetric particles. If supersymmetry were a true symmetry of nature, then it must be a broken symmetry, and the energy of symmetry breaking must be higher than those achievable by present-day experiments.

Other spacetimes

The theory, QED, QCD, as well as the whole Standard Model all assume a (3+1)-dimensional Minkowski space (3 spatial and 1 time dimensions) as the background on which the quantum fields are defined. However, QFT ''a priori'' imposes no restriction on the number of dimensions nor the geometry of spacetime. In condensed matter physics, QFT is used to describe (2+1)-dimensional electron gases. In

high-energy physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) a ...

is a type of (1+1)-dimensional QFT, while Kaluza–Klein theory uses gravity in extra dimensions to produce gauge theories in lower dimensions. In Minkowski space, the flat

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...

is used to raise and lower spacetime indices in the Lagrangian, ''e.g.'' :

A_\mu A^\mu = \eta_ A^\mu A^\nu,\quad \partial_\mu\phi \partial^\mu\phi = \eta^\partial_\mu\phi \partial_\nu\phi,

where is the inverse of satisfying . For QFTs in curved spacetime on the other hand, a general metric (such as the

Schwarzschild metric In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assump ...

describing a black hole) is used: :

A_\mu A^\mu = g_ A^\mu A^\nu,\quad \partial_\mu\phi \partial^\mu\phi = g^\partial_\mu\phi \partial_\nu\phi,

where is the inverse of . For a real scalar field, the Lagrangian density in a general spacetime background is :

\mathcal = \sqrt\left(\frac 12 g^ \nabla_\mu\phi \nabla_\nu\phi - \frac 12 m^2\phi^2\right),

where , and denotes the

. The Lagrangian of a QFT, hence its calculational results and physical predictions, depends on the geometry of the spacetime background.

Topological quantum field theory

The correlation functions and physical predictions of a QFT depend on the spacetime metric . For a special class of QFTs called topological quantum field theories (TQFTs), all correlation functions are independent of continuous changes in the spacetime metric. QFTs in curved spacetime generally change according to the ''geometry'' (local structure) of the spacetime background, while TQFTs are invariant under spacetime diffeomorphisms but are sensitive to the ''

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

'' (global structure) of spacetime. This means that all calculational results of TQFTs are topological invariants of the underlying spacetime.

Chern–Simons theory The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type developed by Edward Witten. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and Jam ...

is an example of TQFT and has been used to construct models of quantum gravity. Applications of TQFT include the

fractional quantum Hall effect The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of e^2/h. It is a property of a collective state in which elec ...

and topological quantum computers. The world line trajectory of fractionalized particles (known as anyons) can form a link configuration in the spacetime, which relates the braiding statistics of anyons in physics to the link invariants in mathematics. Topological quantum field theories (TQFTs) applicable to the frontier research of topological quantum matters include Chern-Simons-Witten gauge theories in 2+1 spacetime dimensions, other new exotic TQFTs in 3+1 spacetime dimensions and beyond.

Perturbative and non-perturbative methods

Using

, the total effect of a small interaction term can be approximated order by order by a series expansion in the number of

s participating in the interaction. Every term in the expansion may be understood as one possible way for (physical) particles to interact with each other via virtual particles, expressed visually using a Feynman diagram. The electromagnetic force between two electrons in QED is represented (to first order in perturbation theory) by the propagation of a virtual photon. In a similar manner, the

carry the weak interaction, while

s carry the strong interaction. The interpretation of an interaction as a sum of intermediate states involving the exchange of various virtual particles only makes sense in the framework of perturbation theory. In contrast, non-perturbative methods in QFT treat the interacting Lagrangian as a whole without any series expansion. Instead of particles that carry interactions, these methods have spawned such concepts as

domain wall A domain wall is a type of topological soliton that occurs whenever a discrete symmetry is spontaneously broken. Domain walls are also sometimes called kinks in analogy with closely related kink solution of the sine-Gordon model or models with pol ...

, flux tube, and instanton. Examples of QFTs that are completely solvable non-perturbatively include minimal models of conformal field theory and the Thirring model.

Mathematical rigor

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

of QFT, which underlies the entire Feynman diagram method, is fundamentally ill-defined. However, ''perturbative'' quantum field theory, which only requires that quantities be computable as a formal power series without any convergence requirements, can be given a rigorous mathematical treatment. In particular, Kevin Costello's monograph ''Renormalization and Effective Field Theory''Kevin Costello, ''Renormalization and Effective Field Theory'', Mathematical Surveys and Monographs Volume 170, American Mathematical Society, 2011, provides a rigorous formulation of perturbative renormalization that combines both the effective-field theory approaches of Kadanoff,

Wilson Wilson may refer to: People *Wilson (name) ** List of people with given name Wilson ** List of people with surname Wilson * Wilson (footballer, 1927–1998), Brazilian manager and defender * Wilson (footballer, born 1984), full name Wilson R ...

, and Polchinski, together with the Batalin-Vilkovisky approach to quantizing gauge theories. Furthermore, perturbative path-integral methods, typically understood as formal computational methods inspired from finite-dimensional integration theory,Gerald B. Folland, ''Quantum Field Theory: A Tourist Guide for Mathematicians'', Mathematical Surveys and Monographs Volume 149, American Mathematical Society, 2008, , chapter=8 can be given a sound mathematical interpretation from their finite-dimensional analogues. Since the 1950s, theoretical physicists and mathematicians have attempted to organize all QFTs into a set of

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s, in order to establish the existence of concrete models of relativistic QFT in a mathematically rigorous way and to study their properties. This line of study is called constructive quantum field theory, a subfield of

mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...

, which has led to such results as CPT theorem, spin–statistics theorem, and

, and also to mathematically rigorous constructions of many interacting QFTs in two and three spacetime dimensions, e.g. two-dimensional scalar field theories with arbitrary polynomial interactions, the three-dimensional scalar field theories with a quartic interaction, etc. Compared to ordinary QFT, topological quantum field theory and conformal field theory are better supported mathematically — both can be classified in the framework of representations of cobordisms.

Algebraic quantum field theory Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by . The axioms are stated in te ...

is another approach to the axiomatization of QFT, in which the fundamental objects are local operators and the algebraic relations between them. Axiomatic systems following this approach include Wightman axioms and Haag–Kastler axioms. One way to construct theories satisfying Wightman axioms is to use Osterwalder–Schrader axioms, which give the necessary and sufficient conditions for a real time theory to be obtained from an

imaginary time Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories. M ...

theory by

analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...

( Wick rotation). Yang–Mills existence and mass gap, one of the Millennium Prize Problems, concerns the well-defined existence of Yang–Mills theories as set out by the above axioms. The full problem statement is as follows.

References

;Bibliography * * *

External links

* * * ''

Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. E ...

'':
Quantum Field Theory
, by Meinard Kuhlmann. * Siegel, Warren, 2005.

' .
Quantum Field Theory
by P. J. Mulders {{DEFAULTSORT:Quantum Field Theory Quantum mechanics Mathematical physics

History

Theoretical background

Quantum electrodynamics

Infinities and renormalization

Standard-Model

Other developments

Condensed-matter-physics

Principles

Classical fields

Canonical quantization

Path integrals

Two-point correlation function

Feynman diagram

Renormalization

Renormalization group

Other theories

Gauge symmetry

Spontaneous symmetry-breaking

Supersymmetry

Other spacetimes

Topological quantum field theory

Perturbative and non-perturbative methods

Mathematical rigor

See also

References

Further reading

External links