**Quantum field theory** provides a theoretical framework for constructing

quantum mechanicalQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

models of systems classically parametrized (represented) by an infinite number of dynamical degrees of freedom, that is,

fieldsIn physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...

and (in a condensed matter context)

many-body systemsThe many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of a large number of interacting particles. Microscopic here implies that quantum mechanics has to be used to provide an accurate description of the system...

. It is the natural and quantitative language of

particle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

and

condensed matter physicsCondensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...

. Most theories in modern particle physics, including the

Standard ModelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

of elementary particles and their interactions, are formulated as

relativisticSpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

quantum field theories. Quantum field theories are used in many contexts, and are especially vital in

elementary particle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, where the particle count/number may change over the course of a reaction. They are also used in the description of

critical phenomenaIn physics, critical phenomena is the collective name associated with thephysics of critical points. Most of them stem from the divergence of thecorrelation length, but also the dynamics slows down...

and

quantum phase transitionsIn physics, a quantum phase transition is a phase transition between different quantum phases . Contrary to classical phase transitions, quantum phase transitions can only be accessed by varying a physical parameter - such as magnetic field or pressure - at absolute zero temperature...

, such as in the

BCS theoryBCS theory — proposed by Bardeen, Cooper, and Schrieffer in 1957 — is the first microscopic theory of superconductivity since its discovery in 1911. The theory describes superconductivity as a microscopic effect caused by a "condensation" of pairs of electrons into a boson-like state...

of

superconductivitySuperconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...

. Quantum field theory is thought by many to be the unique and correct outcome of combining the rules of quantum mechanics with special relativity.

In

perturbativeIn quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...

quantum field theory, the forces between particles are mediated by other particles. The electromagnetic force between two electrons is caused by an exchange of photons. Intermediate vector bosons mediate the weak force and gluons mediate the strong force. There is currently no complete quantum theory of the remaining fundamental force, gravity, but many of the

proposed theoriesQuantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

postulate the existence of a

gravitonIn physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2...

particle that mediates it. These force-carrying particles are virtual particles and, by definition, cannot be detected while carrying the force, because such detection will imply that the force is not being carried. In addition, the notion of "force mediating particle" comes from perturbation theory, and thus does not make sense in a context of bound states.

In QFT photons are not thought of as 'little billiard balls', they are considered to be field quanta – necessarily chunked ripples in a field, or "excitations", that 'look like' particles. Fermions, like the electron, can also be described as ripples/excitations in a field, where each kind of fermion has its own field. In summary, the classical visualisation of "everything is particles and field", in quantum field theory, resolves into "everything is particles", which then resolves into "everything is fields". In the end, particles are regarded as excited states of a field (field quanta). The

gravitational fieldThe gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

and the

electromagnetic fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

are the only two fundamental fields in Nature that have infinite range and a corresponding classical low-energy limit, which greatly diminishes and hides their "particle-like" excitations.

Albert EinsteinAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

, in 1905, attributed "particle-like" and discrete exchanges of momenta and energy, characteristic of "field quanta", to the electromagnetic field. Originally, his principal motivation was to explain the thermodynamics of radiation. Although it is often claimed that the

photoelectricIn the photoelectric effect, electrons are emitted from matter as a consequence of their absorption of energy from electromagnetic radiation of very short wavelength, such as visible or ultraviolet light. Electrons emitted in this manner may be referred to as photoelectrons...

and

Compton effectsIn physics, Compton scattering is a type of scattering that X-rays and gamma rays undergo in matter. The inelastic scattering of photons in matter results in a decrease in energy of an X-ray or gamma ray photon, called the Compton effect...

require a quantum description of the EM field, this is now understood to be untrue, and proper proof of the quantum nature of radiation is now taken up into modern

quantum opticsQuantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter.- History of quantum optics :...

as in the antibunching effect. The word "photon" was coined in 1926 by the great physical chemist Gilbert Newton Lewis (see also the articles

photon antibunchingPhoton antibunching generally refers to a light field with photons more equally spaced than a coherent laser field and a signal at detectors is anticorrelated. More specifically, it can refer to sub-Poisson photon statistics, that is a photon number distribution for which the variance is less than...

and

laserA laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

).

In the "low-energy limit", the quantum field-theoretic description of the electromagnetic field,

quantum electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

, does not exactly reduce to

James Clerk MaxwellJames Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

's 1864 theory of classical electrodynamics. Small quantum corrections due to virtual electron positron pairs give rise to

small non-linear corrections to the Maxwell equationsIn physics, the Euler-Heisenberg Lagrangian describes the non-linear dynamics of electromagnetic fields in vacuum. It takes into account vacuum polarization to one loop, and it is valid for electromagnetic fields that change slowly compared to the inverse electron mass...

, although the "classical limit" of quantum electrodynamics has not been as widely explored as that of quantum mechanics.

Presumably, the as yet unknown correct quantum field-theoretic treatment of the gravitational field will become and "look exactly like" Einstein's general theory of relativity in the "low-energy limit". Indeed, quantum field theory itself is possibly the low-energy-effective-field-theory limit of a more fundamental theory such as

superstring theorySuperstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...

. Compare in this context the article

effective field theoryIn physics, an effective field theory is, as any effective theory, an approximate theory, that includes appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, while ignoring substructure and degrees of freedom at shorter distances .-The renormalization...

.

## History

Quantum field theory originated in the 1920s from the problem of creating a

quantum mechanical theoryQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

of the

electromagnetic fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

. In particlular de Broglie in 1924 introduced the idea of a wave description of elementary systems in the following way: "we proceed in this work from the assumption of existence of a certain periodic phenomenon of a yet to be determined character, which is to be attributed to each and every isolated energy parcel" .

In 1925,

Werner HeisenbergWerner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

,

Max BornMax Born was a German-born 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 number of notable physicists in the 1920s and 30s...

, and

Pascual Jordan

constructed such a theory by expressing the field's internal

degrees of freedomA degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

as an infinite set of

harmonic oscillatorIn 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 constant....

s and by employing the

canonical quantizationIn physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...

procedure to those oscillators. This theory assumed that no electric charges or currents were present and today would be called a free field theory. The first reasonably complete theory of

quantum electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

, which included both the electromagnetic field and electrically charged matter (specifically,

electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s) as quantum mechanical objects, was created by

Paul DiracPaul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

in 1927. This quantum field theory could be used to model important processes such as the emission of a

photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

by an electron dropping into a

quantum state of lower energy, a process in which the

*number of particles changes*—one atom in the initial state becomes an atom plus a

photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

in the final state. It is now understood that the ability to describe such processes is one of the most important features of quantum field theory.

It was evident from the beginning that a proper quantum treatment of the electromagnetic field had to somehow incorporate

Einstein'sAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

relativity theoryThe theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

, which had grown out of the study of

classical electromagnetismClassical electromagnetism is a branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents...

. This need to put together relativity and quantum mechanics was the second major motivation in the development of quantum field theory.

Pascual Jordan

and

Wolfgang PauliWolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...

showed in 1928 that quantum fields could be made to behave in the way predicted by

special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

during coordinate transformations (specifically, they showed that the field

commutatorIn 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:...

s were Lorentz invariant). A further boost for quantum field theory came with the discovery of the

Dirac equationThe Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...

, which was originally formulated and interpreted as a single-particle equation analogous to the

Schrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

, but unlike the Schrödinger equation, the Dirac equation satisfies both the Lorentz invariance, that is, the requirements of special relativity, and the rules of quantum mechanics. The Dirac equation accommodated the spin-1/2 value of the electron and accounted for its magnetic moment as well as giving accurate predictions for the spectra of hydrogen. The attempted interpretation of the Dirac equation as a single-particle equation could not be maintained long, however, and finally it was shown that several of its undesirable properties (such as negative-energy states) could be made sense of by reformulating and reinterpreting the Dirac equation as a true field equation, in this case for the quantized "Dirac field" or the "electron field", with the "negative-energy solutions" pointing to the existence of anti-particles. This work was performed first by Dirac himself with the invention of hole theory in 1930 and by Wendell Furry,

Robert OppenheimerJulius Robert Oppenheimer was an American theoretical physicist and professor of physics at the University of California, Berkeley. Along with Enrico Fermi, he is often called the "father of the atomic bomb" for his role in the Manhattan Project, the World War II project that developed the first...

,

Vladimir FockVladimir Aleksandrovich Fock was a Soviet physicist, who did foundational work on quantum mechanics and quantum electrodynamics....

, and others. Schrödinger, during the same period that he discovered his famous equation in 1926, also independently found the relativistic generalization of it known as the

Klein-Gordon equationThe Klein–Gordon equation is a relativistic version of the Schrödinger equation....

but dismissed it since, without

spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

, it predicted impossible properties for the hydrogen spectrum. (See

Oskar KleinOskar Benjamin Klein was a Swedish theoretical physicist.Klein was born in Danderyd outside Stockholm, son of the chief rabbi of Stockholm, Dr. Gottlieb Klein from Homonna in Hungary and Antonie Levy...

and

Walter GordonWalter Gordon may refer to:*Jamaa Fanaka, African-American filmmaker . He is sometimes credited as either Walt Gordon or Walter Gordon....

.) All relativistic wave equations that describe spin-zero particles are said to be of the Klein-Gordon type.

Of great importance are the studies of soviet physicists, Viktor Ambartsumian and

Dmitri IvanenkoDmitri Ivanenko , Professor of Moscow State University , made a great contribution to the physical science of the twentieth century, especially to nuclear physics, field theory , and gravitation theory.His outstanding achievements include:* the Fock-Ivanenko coefficients of parallel...

, in particular the Ambarzumian-Ivanenko hypothesis of creation of massive particles (published in 1930) which is the cornerstone of the contemporary quantum field theory. The idea is that not only the quanta of the electromagnetic field, photons, but also other particles (including particles having nonzero rest mass) may be born and disappear as a result of their interaction with other particles. This idea of Ambartsumian and Ivanenko formed the basis of modern quantum field theory and theory of elementary particles.

A subtle and careful analysis in 1933 and later in 1950 by

Niels BohrNiels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...

and

Leon RosenfeldLéon Rosenfeld was a Belgian physicist. He obtained a PhD at the University of Liège in 1926, and he was a collaborator of the physicist Niels Bohr. He did early work in quantum electrodynamics that predates by two decades the work by Dirac and Bergmann. He coined the name lepton...

showed that there is a fundamental limitation on the ability to simultaneously measure the electric and magnetic field strengths that enter into the description of charges in interaction with radiation, imposed by the

uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

, which must apply to all canonically conjugate quantities. This limitation is crucial for the successful formulation and interpretation of a quantum field theory of photons and electrons(quantum electrodynamics), and indeed, any perturbative quantum field theory. The analysis of Bohr and Rosenfeld explains fluctuations in the values of the electromagnetic field that differ from the classically "allowed" values distant from the sources of the field. Their analysis was crucial to showing that the limitations and physical implications of the uncertainty principle apply to all dynamical systems, whether fields or material particles. Their analysis also convinced most people that any notion of returning to a fundamental description of nature based on classical field theory, such as what Einstein aimed at with his numerous and failed attempts at a classical

unified field theoryIn physics, a unified field theory, occasionally referred to as a uniform field theory, is a type of field theory that allows all that is usually thought of as fundamental forces and elementary particles to be written in terms of a single field. There is no accepted unified field theory, and thus...

, was simply out of the question.

The third thread in the development of quantum field theory was the need to handle the statistics of many-particle systems consistently and with ease. In 1927, Jordan tried to extend the canonical quantization of fields to the many-body wave functions of

identical particlesIdentical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...

, a procedure that is sometimes called second quantization. In 1928, Jordan and Eugene Wigner found that the quantum field describing electrons, or other

fermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s, had to be expanded using anti-commuting creation and annihilation operators due to the

Pauli exclusion principleThe Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

. This thread of development was incorporated into

many-body theoryThe many-body theory is an area of physics which provides the framework for understanding the collective behavior of vast assemblies of interacting particles. In general terms, the many-body theory deals with effects that manifest themselves only in systems containing large numbers of constituents...

and strongly influenced

condensed matter physicsCondensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...

and

nuclear physicsNuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...

.

Despite its early successes quantum field theory was plagued by several serious theoretical difficulties. Basic physical quantities, such as the self-energy of the electron, the energy shift of electron states due to the presence of the electromagnetic field, gave infinite, divergent contributions—a nonsensical result—when computed using the perturbative techniques available in the 1930s and most of the 1940s. The electron self-energy problem was already a serious issue in the classical electromagnetic field theory, where the attempt to attribute to the electron a finite size or extent (the classical electron-radius) led immediately to the question of what non-electromagnetic stresses would need to be invoked, which would presumably hold the electron together against the Coulomb repulsion of its finite-sized "parts". The situation was dire, and had certain features that reminded many of the

"Rayleigh-Jeans difficulty"The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was a prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power....

. What made the situation in the 1940s so desperate and gloomy, however, was the fact that the correct ingredients (the second-quantized Maxwell-Dirac field equations) for the theoretical description of interacting photons and electrons were well in place, and no major conceptual change was needed analogous to that which was necessitated by a finite and physically sensible account of the radiative behavior of hot objects, as provided by the Planck radiation law.

This "divergence problem" was solved in the case of quantum electrodynamics during the late 1940s and early 1950s by

Hans BetheHans Albrecht Bethe was a German-American nuclear physicist, and Nobel laureate in physics for his work on the theory of stellar nucleosynthesis. A versatile theoretical physicist, Bethe also made important contributions to quantum electrodynamics, nuclear physics, solid-state physics and...

,

Tomonagawas a Japanese physicist, influential in the development of quantum electrodynamics, work for which he was jointly awarded the Nobel Prize in Physics in 1965 along with Richard Feynman and Julian Schwinger.-Biography:...

,

SchwingerJulian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...

,

FeynmanRichard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

, and

DysonFreeman John Dyson FRS is a British-born American theoretical physicist and mathematician, famous for his work in quantum field theory, solid-state physics, astronomy and nuclear engineering. Dyson is a member of the Board of Sponsors of the Bulletin of the Atomic Scientists...

, through the procedure known as

renormalizationIn quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....

. Great progress was made after realizing that ALL infinities in quantum electrodynamics are related to two effects: the self-energy of the electron/positron and vacuum polarization. Renormalization concerns the business of paying very careful attention to just what is meant by, for example, the very concepts "charge" and "mass" as they occur in the pure, non-interacting field-equations. The "vacuum" is itself polarizable and, hence, populated by

virtual particleIn physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...

(

on shell and off shellIn physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell....

) pairs, and, hence, is a seething and busy dynamical system in its own right. This was a critical step in identifying the source of "infinities" and "divergences". The "bare mass" and the "bare charge" of a particle, the values that appear in the free-field equations (non-interacting case), are abstractions that are simply not realized in experiment (in interaction). What we measure, and hence, what we must take account of with our equations, and what the solutions must account for, are the "renormalized mass" and the "renormalized charge" of a particle. That is to say, the "shifted" or "dressed" values these quantities must have when due care is taken to include all deviations from their "bare values" is dictated by the very nature of quantum fields themselves.

The first approach that bore fruit is known as the "interaction representation", (see the article

Interaction pictureIn quantum mechanics, the Interaction picture is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of...

) a Lorentz covariant and gauge-invariant generalization of time-dependent perturbation theory used in ordinary quantum mechanics, and developed by Tomonaga and Schwinger, generalizing earlier efforts of Dirac, Fock and Podolsky. Tomonaga and Schwinger invented a relativistically covariant scheme for representing field commutators and field operators intermediate between the two main representations of a quantum system, the Schrödinger and the Heisenberg representations (see the article on

quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

). Within this scheme, field commutators at separated points can be evaluated in terms of "bare" field creation and annihilation operators. This allows for keeping track of the time-evolution of both the "bare" and "renormalized", or perturbed, values of the

HamiltonianHamiltonian may refer toIn mathematics :* Hamiltonian system* Hamiltonian path, in graph theory** Hamiltonian cycle, a special case of a Hamiltonian path* Hamiltonian group, in group theory* Hamiltonian...

and expresses everything in terms of the coupled, gauge invariant "bare" field-equations. Schwinger gave the most elegant formulation of this approach. The next and most famous development is due to

FeynmanFeynman may refer to:* Richard Feynman , physicist** Feynman diagram** Feynman graph** Feynman–Kac formula** The Feynman Lectures on Physics** Feynman integral, see Path integral formulation** Feynman parametrization...

, who, with his brilliant rules for assigning a "graph"/"diagram" to the terms in the scattering matrix (See S-Matrix Feynman diagrams). These directly corresponded (through the

Schwinger-Dyson equationThe Schwinger–Dyson equation , also Dyson-Schwinger equations, named after Julian Schwinger and Freeman Dyson, are general relations between Green functions in quantum field theories...

) to the measurable physical processes (cross sections, probability amplitudes, decay widths and lifetimes of excited states) one needs to be able to calculate. This revolutionized how quantum field theory calculations are carried-out in practice.

Two classic text-books from the 1960s, J.D. Bjorken and S.D. Drell,

*Relativistic Quantum Mechanics* (1964) and J.J. Sakurai,

*Advanced Quantum Mechanics* (1967), thoroughly developed the Feynman graph expansion techniques using physically intuitive and practical methods following from the

correspondence principleIn physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....

, without worrying about the technicalities involved in deriving the Feynman rules from the superstructure of quantum field theory itself. Although both Feynman's heuristic and pictorial style of dealing with the infinities, as well as the formal methods of Tomonaga and Schwinger, worked extremely well, and gave spectacularly accurate answers, the true analytical nature of the question of "renormalizability", that is, whether ANY theory formulated as a "quantum field theory" would give finite answers, was not worked-out till much later, when the urgency of trying to formulate finite theories for the strong and electro-weak (and gravitational interactions) demanded its solution.

Renormalization in the case of QED was largely fortuitous due to the smallness of the coupling constant, the fact that the coupling has no dimensions involving mass, the so-called fine structure constant, and also the zero-mass of the gauge boson involved, the photon, rendered the small-distance/high-energy behavior of QED manageable. Also, electromagnetic processess are very "clean" in the sense that they are not badly suppressed/damped and/or hidden by the other gauge interactions. By 1958

Sidney DrellSidney David Drell is an American theoretical physicist and arms control expert. He is a professor emeritus at the Stanford Linear Accelerator Center and a senior fellow at Stanford University's Hoover Institution. Drell is a noted contributor in the field of quantum electrodynamics and particle...

observed: "Quantum electrodynamics (QED) has achieved a status of peaceful coexistence with its divergences...".

The unification of the electromagnetic force with the weak force encountered initial difficulties due to the lack of accelerator energies high enough to reveal processes beyond the

Fermi interactionIn particle physics, Fermi's interaction also known as Fermi coupling, is an old explanation of the weak force, proposed by Enrico Fermi, in which four fermions directly interact with one another at one vertex...

range. Additionally, a satisfactory theoretical understanding of hadron substructure had to be developed, culminating in the

quark modelIn physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons....

.

In the case of the strong interactions, progress concerning their short-distance/high-energy behavior was much slower and more frustrating. For strong interactions with the electro-weak fields, there were difficult issues regarding the strength of coupling, the mass generation of the force carriers as well as their non-linear, self interactions. Although there has been theoretical progress toward a grand unified quantum field theory incorporating the electro-magnetic force, the weak force and the strong force, empirical verification is still pending.

SuperunificationA theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle....

, incorporating the gravitational force, is still very speculative, and is under intensive investigation by many of the best minds in contemporary theoretical physics. Gravitation is a

tensor fieldIn mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...

description of a spin-2 gauge-boson, the "graviton", and is further discussed in the articles on

general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

and

quantum gravityQuantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

.

From the point of view of the techniques of (four-dimensional) quantum field theory, and as the numerous and heroic efforts to formulate a consistent quantum gravity theory by some very able minds attests, gravitational quantization was, and is still, the reigning champion for bad behavior. There are problems and frustrations stemming from the fact that the gravitational coupling constant has dimensions involving inverse powers of mass, and as a simple consequence, it is plagued by badly behaved (in the sense of perturbation theory) non-linear and violent self-interactions. Gravity, basically, gravitates, which in turn...gravitates...and so on, (i.e., gravity is itself a source of gravity,...,) thus creating a nightmare at all orders of perturbation theory. Also, gravity couples to all energy equally strongly, as per the

equivalence principleIn the physics of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body is actually...

, so this makes the notion of ever really "switching-off", "cutting-off" or separating, the gravitational interaction from other interactions ambiguous and impossible since, with gravitation, we are dealing with the very structure of space-time itself. (See

general covarianceIn theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations...

and, for a modest, yet highly non-trivial and significant interplay between (QFT) and gravitation (spacetime), see the article

Hawking radiationHawking radiation is a thermal radiation with a black body spectrum predicted to be emitted by black holes due to quantum effects. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974, and sometimes also after the physicist Jacob Bekenstein...

and references cited therein. Also

quantum field theory in curved spacetimeQuantum field theory in curved spacetime is an extension of standard, Minkowski-space quantum field theory to curved spacetime. A general prediction of this theory is that particles can be created by time dependent gravitational fields , or by time independent gravitational fields that contain...

).

Thanks to the somewhat brute-force, clanky and heuristic methods of Feynman, and the elegant and abstract methods of Tomonaga/Schwinger, from the period of early renormalization, we do have the modern theory of

quantum electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

(QED). It is still the most accurate physical theory known, the prototype of a successful quantum field theory. Beginning in the 1950s with the work of

Yang and

MillsRobert L. Mills was a physicist, specializing in quantum field theory, the theory of alloys, and many-body theory. While sharing an office at Brookhaven National Laboratory, in 1954, Chen Ning Yang and Mills proposed a tensor equation for what are now called Yang-Mills fields...

, as well as Ryoyu Utiyama, following the previous lead of Weyl and Pauli, deep explorations illuminated the types of symmetries and invariances any field theory must satisfy. QED, and indeed, all field theories, were generalized to a class of quantum field theories known as

gauge theoriesIn physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

. Quantum electrodynamics is the most famous example of what is known as an

AbelianIn mathematics, Abelian refers to any of number of different mathematical concepts named after Niels Henrik Abel:- Group theory :*Abelian group, a group in which the binary operation is commutative...

gauge theory. It relies on the symmetry group U(1) and has one massless gauge field, the U(1) gauge symmetry, dictating the form of the interactions involving the electromagnetic field, with the photon being the gauge boson. That symmetries dictate, limit and necessitate the form of interaction between particles is the essence of the "gauge theory revolution". Yang and Mills formulated the first explicit example of a non-Abelian gauge theory, Yang-Mills theory, with an attempted explanation of the strong interactions in mind. The strong interactions were then (incorrectly) understood in the mid-1950s, to be mediated by the pi-mesons, the particles predicted by

Hideki Yukawané , was a Japanese theoretical physicist and the first Japanese Nobel laureate.-Biography:Yukawa was born in Tokyo and grew up in Kyoto. In 1929, after receiving his degree from Kyoto Imperial University, he stayed on as a lecturer for four years. After graduation, he was interested in...

in 1935, based on his profound reflections concerning the reciprocal connection between the mass of any force-mediating particle and the range of the force it mediates. This was allowed by the

uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

. The 1960s and 1970s saw the formulation of a gauge theory now known as the

Standard ModelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

of

particle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, which systematically describes the elementary particles and the interactions between them.

The electroweak interaction part of the standard model was formulated by Sheldon Glashow in the years 1958-60 with his discovery of the SU(2)xU(1) group structure of the theory.

Steven WeinbergSteven Weinberg is an American theoretical physicist and Nobel laureate in Physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic interaction between elementary particles....

and

Abdus SalamMohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk (Urdu: محمد عبد السلام, pronounced , (January 29, 1926– November 21, 1996) was a Pakistani theoretical physicist and Nobel laureate in Physics for his work on the electroweak unification of the...

brilliantly invoked the

Anderson-Higgs mechanismIn particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....

for the generation of the W's and Z masses (the intermediate

vector bosonIn particle physics, a vector boson is a boson with the spin quantum number equal to 1.The vector bosons considered to be elementary particles in the Standard Model are the gauge bosons or, the force carriers of fundamental interactions: the photon of electromagnetism, the W and Z bosons of the...

(s) responsible for the weak interactions and neutral-currents) and keeping the mass of the photon zero. The Goldstone/Higgs idea for generating mass in gauge theories was sparked in the late 1950s and early 1960s when a number of theoreticians (including

Yoichiro Nambuis a Japanese-born American physicist, currently a professor at the University of Chicago. Known for his contributions to the field of theoretical physics, he was awarded a one-half share of the Nobel Prize in Physics in 2008 for the discovery of the mechanism of spontaneous broken symmetry in...

,

Steven WeinbergSteven Weinberg is an American theoretical physicist and Nobel laureate in Physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic interaction between elementary particles....

,

Jeffrey GoldstoneJeffrey Goldstone is a British-born theoretical physicist and an emeritus physics faculty at MIT Center for Theoretical Physics.He worked at the University of Cambridge until 1977....

,

François EnglertFrançois Englert is a Belgian theoretical physicist. He was awarded the 2010 J. J. Sakurai Prize for Theoretical Particle Physics , the Wolf Prize in Physics in 2004 and the High Energy and Particle Prize of the European Physical Society François Englert (born 6 November 1932) is a...

,

Robert BroutRobert Brout was an American-Belgian theoretical physicist who has made significant contributions in elementary particle physics...

, G. S. Guralnik,

C. R. HagenCarl Richard Hagen is a professor of particle physics at the University of Rochester. He is most noted for his contributions to the Standard Model and Symmetry breaking as well as the co-discovery of the Higgs mechanism and Higgs boson with Gerald Guralnik and Tom Kibble...

, Tom Kibble and

Philip Warren AndersonPhilip Warren Anderson is an American physicist and Nobel laureate. Anderson has made contributions to the theories of localization, antiferromagnetism and high-temperature superconductivity.- Biography :...

) noticed a possibly useful analogy to the (spontaneous) breaking of the U(1) symmetry of electromagnetism in the formation of the BCS ground-state of a superconductor. The gauge boson involved in this situation, the photon, behaves as though it has acquired a finite mass. There is a further possibility that the physical vacuum (ground-state) does not respect the symmetries implied by the "unbroken" electroweak

LagrangianThe Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

(see the article

Electroweak interactionIn particle physics, the electroweak interaction is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different...

for more details) from which one arrives at the field equations. The electroweak theory of Weinberg and Salam was shown to be renormalizable (finite) and hence consistent by

Gerardus 't HooftGerardus 't Hooft 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...

and Martinus Veltman. The Glashow-Weinberg-Salam theory (GWS-Theory) is a triumph and, in certain applications, gives an accuracy on a par with quantum electrodynamics.

Also during the 1970s, parallel developments in the study of phase transitions in

condensed matter physicsCondensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...

led

Leo KadanoffLeo Philip Kadanoff is an American physicist. He is a professor of physics at the University of Chicago and a former President of the American Physical Society . He has contributed to the fields of statistical physics, chaos theory, and theoretical condensed matter physics.-Biography:Kadanoff...

,

Michael FisherMichael Ellis Fisher is an English physicist, as well as chemist and mathematician, known for his many seminal contributions...

and

Kenneth WilsonKenneth Geddes Wilson is an American theoretical physicist and Nobel Prize winner.As an undergraduate at Harvard, he was a Putnam Fellow. He earned his PhD from Caltech in 1961, studying under Murray Gell-Mann....

(extending work of

Ernst StueckelbergErnst Carl Gerlach Stueckelberg was a Swiss mathematician and physicist.- Career :In 1927 Stueckelberg got his Ph. D. at the University of Basel under August Hagenbach...

, Andre Peterman,

Murray Gell-MannMurray Gell-Mann is an American physicist and linguist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles...

, and Francis Low) to a set of ideas and methods known as the

renormalization groupIn theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...

. By providing a better physical understanding of the renormalization procedure invented in the 1940s, the renormalization group sparked what has been called the "grand synthesis" of theoretical physics, uniting the quantum field theoretical techniques used in particle physics and condensed matter physics into a single theoretical framework.

### Classical fields and quantum fields

Quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, in its most general formulation, is a theory of abstract

operatorsIn physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....

(observables) acting on an abstract state space (

Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

), where the observables represent physically observable quantities and the state space represents the possible states of the system under study. Furthermore, each observable

correspondsIn physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....

, in a technical sense, to the classical idea of a

degree of freedomA degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

. For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the

positionIn quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. Consider, for example, the case of a spinless particle moving on a line. The state space for such a particle is L2, the Hilbert space of complex-valued and square-integrable ...

and

momentum operatorIn quantum mechanics, momentum is defined as an operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once...

s

and

. Ordinary quantum mechanics deals with systems such as this, which possess a small set of degrees of freedom.

(It is important to note, at this point, that this article does not use the word "particle" in the context of

wave–particle dualityWave–particle duality postulates that all particles exhibit both wave and particle properties. A central concept of quantum mechanics, this duality addresses the inability of classical concepts like "particle" and "wave" to fully describe the behavior of quantum-scale objects...

. In quantum field theory, "particle" is a generic term for any discrete quantum mechanical entity, such as an electron or photon, which can behave like

classical particlesIn particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

or

classical wavesIn physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

under different experimental conditions, such that one could say 'this "particle" can behave like a wave or a particle'.)

A

**quantum field** is a quantum mechanical system containing a large, and possibly

infiniteInfinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

, number of degrees of freedom. A

classical fieldIn physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...

contains a set of degrees of freedom at each point of space; for instance, the classical

electromagnetic fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

defines two vectors — the

electric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

and the

magnetic fieldA magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

— that can in principle take on distinct values for each position

. When the field

*as a whole* is considered as a quantum mechanical system, its observables form an infinite (in fact

uncountableIn mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.-Characterizations:There...

) set, because

is continuous.

Furthermore, the degrees of freedom in a quantum field are arranged in "repeated" sets. For example, the degrees of freedom in an electromagnetic field can be grouped according to the position

, with exactly two vectors for each

. Note that

is an ordinary number that "indexes" the observables; it is not to be confused with the position operator

encountered in ordinary quantum mechanics, which is an observable. (Thus, ordinary quantum mechanics is sometimes referred to as "zero-dimensional quantum field theory", because it contains only a single set of observables.)

It is also important to note that there is nothing special about

because, as it turns out, there is generally more than one way of indexing the degrees of freedom in the field.

In the following sections, we will show how these ideas can be used to construct a quantum mechanical theory with the desired properties. We will begin by discussing single-particle quantum mechanics and the associated theory of many-particle quantum mechanics. Then, by finding a way to index the degrees of freedom in the many-particle problem, we will construct a quantum field and study its implications.

### Single-particle and many-particle quantum mechanics

In quantum mechanics, the time-dependent

Schrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

for a single particle is

where

is the particle's

massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

,

is the applied

potential*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...

, and

denotes the wavefunction.

We wish to consider how this problem generalizes to

particles. There are two motivations for studying the many-particle problem. The first is a straightforward need in

condensed matter physicsCondensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...

, where typically the number of particles is on the order of

Avogadro's numberIn chemistry and physics, the Avogadro constant is defined as the ratio of the number of constituent particles N in a sample to the amount of substance n through the relationship NA = N/n. Thus, it is the proportionality factor that relates the molar mass of an entity, i.e...

(6.0221415 x 10

^{23}). The second motivation for the many-particle problem arises from

particle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

and the desire to incorporate the effects of

special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

. If one attempts to include the relativistic rest energy into the above equation (in quantum mechanics where position is an observable), the result is either the

Klein-Gordon equationThe Klein–Gordon equation is a relativistic version of the Schrödinger equation....

or the

Dirac equationThe Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...

. However, these equations have many unsatisfactory qualities; for instance, they possess energy eigenvalues that extend to –∞, so that there seems to be no easy definition of a

ground stateThe ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...

. It turns out that such inconsistencies arise from relativistic wavefunctions having a probabilistic interpretation in position space, as probability conservation is not a

relativistically covariantIn standard physics, Lorentz symmetry is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space"...

concept. In quantum field theory, unlike in quantum mechanics, position is not an observable, and thus, one does not need the concept of a position-space probability density. For quantum fields whose interaction can be treated perturbatively, this is equivalent to neglecting the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativistic quantum theory.

Einstein'sAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

famous mass-energy relation allows for the possibility that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. For example, an electron and a

positronThe positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...

can annihilate each other to create

photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s. This suggests that a consistent relativistic quantum theory should be able to describe many-particle dynamics.

Furthermore, we will assume that the

particles are

indistinguishableIdentical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...

. As described in the article on

identical particlesIdentical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...

, this implies that the state of the entire system must be either symmetric (

bosonIn particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s) or antisymmetric (

fermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s) when the coordinates of its constituent particles are exchanged. These multi-particle states are rather complicated to write. For example, the general quantum state of a system of

bosons is written as

where

are the single-particle states,

is the number of particles occupying state

, and the sum is taken over all possible

permutationIn mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

s

acting on

elements. In general, this is a sum of

(

factorialIn mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

) distinct terms, which quickly becomes unmanageable as

increases. The way to simplify this problem is to turn it into a quantum field theory.

### Second quantization

In this section, we will describe a method for constructing a quantum field theory called

**second quantization**. This basically involves choosing a way to index the quantum mechanical degrees of freedom in the space of multiple identical-particle states. It is based on the

HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

formulation of quantum mechanics; several other approaches exist, such as the Feynman path integral, which uses a

LagrangianThe Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

formulation. For an overview, see the article on

quantizationIn physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...

.

#### Second quantization of bosons

For simplicity, we will first discuss second quantization for

bosonIn particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s, which form perfectly symmetric quantum states. Let us denote the mutually orthogonal single-particle states by

and so on. For example, the 3-particle state with one particle in state

and two in state

is

The first step in second quantization is to express such quantum states in terms of

**occupation numbers**, by listing the number of particles occupying each of the single-particle states

etc. This is simply another way of labelling the states. For instance, the above 3-particle state is denoted as

The next step is to expand the

-particle state space to include the state spaces for all possible values of

. This extended state space, known as a

Fock spaceThe Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...

, is composed of the state space of a system with no particles (the so-called

vacuum stateIn quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...

), plus the state space of a 1-particle system, plus the state space of a 2-particle system, and so forth. It is easy to see that there is a one-to-one correspondence between the occupation number representation and valid boson states in the Fock space.

At this point, the quantum mechanical system has become a quantum field in the sense we described above. The field's elementary degrees of freedom are the occupation numbers, and each occupation number is indexed by a number

, indicating which of the single-particle states

it refers to.

The properties of this quantum field can be explored by defining

creation and annihilation operatorsCreation 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 lowers the number of particles in a given state by one...

, which add and subtract particles. They are analogous to "ladder operators" in the

quantum harmonic oscillatorThe quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

problem, which added and subtracted energy quanta. However, these operators literally create and annihilate particles of a given quantum state. The bosonic annihilation operator

and creation operator

have the following effects:

It can be shown that these are operators in the usual quantum mechanical sense, i.e. linear operators acting on the Fock space. Furthermore, they are indeed

Hermitian conjugatesIn mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator.Adjoints of operators generalize conjugate transposes of square matrices to infinite-dimensional situations...

, which justifies the way we have written them. They can be shown to obey the

commutation relationIn 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:...

where

stands for the

Kronecker delta. These are precisely the relations obeyed by the ladder operators for an infinite set of independent

quantum harmonic oscillatorThe quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

s, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator.

The

HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

of the quantum field (which, through the

Schrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

, determines its dynamics) can be written in terms of creation and annihilation operators. For instance, the Hamiltonian of a field of free (non-interacting) bosons is

where

is the energy of the

-th single-particle energy eigenstate. Note that

Hence,

is known as the number operator for the

-th eigenstate.

#### Second quantization of fermions

It turns out that a different definition of creation and annihilation must be used for describing

fermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s. According to the

Pauli exclusion principleThe Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

, fermions cannot share quantum states, so their occupation numbers

can only take on the value 0 or 1. The fermionic annihilation operators

and creation operators

are defined by their actions on a Fock state thus

These obey an anticommutation relation:

One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the exclusion principle.

#### Field operators

We have previously mentioned that there can be more than one way of indexing the degrees of freedom in a quantum field. Second quantization indexes the field by enumerating the single-particle quantum states. However, as we have discussed, it is more natural to think about a "field", such as the electromagnetic field, as a set of degrees of freedom indexed by position.

To this end, we can define

*field operators* that create or destroy a particle at a particular point in space. In particle physics, these operators turn out to be more convenient to work with, because they make it easier to formulate theories that satisfy the demands of relativity.

Single-particle states are usually enumerated in terms of their

momentaIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

(as in the

particle in a boxIn quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems...

problem.) We can construct field operators by applying the

Fourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator

is

The bosonic field operators obey the commutation relation

where

stands for the

Dirac delta functionThe Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.

The field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is a quantum-mechanical amplitude for finding a particle in some position. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say

where the indices

and

run over all particles, then the field theory Hamiltonian (in the non-relativistic limit and for negligible self-interactions) is

This looks remarkably like an expression for the expectation value of the energy, with

playing the role of the wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from space-projected Hamiltonians.

#### Unification of fields and particles

The "second quantization" procedure that we have outlined in the previous section takes a set of single-particle quantum states as a starting point. Sometimes, it is impossible to define such single-particle states, and one must proceed directly to quantum field theory. For example, a quantum theory of the

electromagnetic fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

*must* be a quantum field theory, because it is impossible (for various reasons) to define a

wavefunctionNot to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...

for a single

photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

. In such situations, the quantum field theory can be constructed by examining the mechanical properties of the classical field and guessing the

correspondingIn physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....

quantum theory. For free (non-interacting) quantum fields, the quantum field theories obtained in this way have the same properties as those obtained using second quantization, such as well-defined creation and annihilation operators obeying commutation or anticommutation relations.

Quantum field theory thus provides a unified framework for describing "field-like" objects (such as the electromagnetic field, whose excitations are photons) and "particle-like" objects (such as electrons, which are treated as excitations of an underlying electron field), so long as one can treat interactions as "perturbations" of free fields. There are still unsolved problems relating to the more general case of interacting fields that may or may not be adequately described by perturbation theory. For more on this topic, see

Haag's theoremRudolf Haag postulatedthat the interaction picture does not exist in an interacting, relativistic quantum field theory , something now commonly known as Haag's Theorem...

.

#### Physical meaning of particle indistinguishability

The second quantization procedure relies crucially on the particles being

identicalIdentical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...

. We would not have been able to construct a quantum field theory from a distinguishable many-particle system, because there would have been no way of separating and indexing the degrees of freedom.

Many physicists prefer to take the converse interpretation, which is that

*quantum field theory explains what identical particles are*. In ordinary quantum mechanics, there is not much theoretical motivation for using symmetric (bosonic) or antisymmetric (fermionic) states, and the need for such states is simply regarded as an empirical fact. From the point of view of quantum field theory, particles are identical

if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

they are excitations of the same underlying quantum field. Thus, the question "why are all electrons identical?" arises from mistakenly regarding individual electrons as fundamental objects, when in fact it is only the electron field that is fundamental.

#### Particle conservation and non-conservation

During second quantization, we started with a Hamiltonian and state space describing a fixed

number of particlesThe particle number of a thermodynamic system, conventionally indicated with the letter N, is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is conjugate to the chemical potential. Unlike most physical quantities, particle...

(

), and ended with a Hamiltonian and state space for an arbitrary number of particles. Of course, in many common situations

is an important and perfectly well-defined quantity, e.g. if we are describing a gas of

atoms sealed in a boxIn quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems...

. From the point of view of quantum field theory, such situations are described by quantum states that are eigenstates of the

number operatorIn quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles.The number operator acts on Fock space...

, which measures the total number of particles present. As with any quantum mechanical observable,

is conserved if it commutes with the Hamiltonian. In that case, the quantum state is trapped in the

-particle

subspaceIn linear algebra, a Euclidean subspace is a set of vectors that is closed under addition and scalar multiplication. Geometrically, a subspace is a flat in n-dimensional Euclidean space that passes through the origin...

of the total Fock space, and the situation could equally well be described by ordinary

-particle quantum mechanics. (Strictly speaking, this is only true in the noninteracting case or in the low energy density limit of renormalized quantum field theories)

For example, we can see that the free-boson Hamiltonian described above conserves particle number. Whenever the Hamiltonian operates on a state, each particle destroyed by an annihilation operator

is immediately put back by the creation operator

.

On the other hand, it is possible, and indeed common, to encounter quantum states that are

*not* eigenstates of

, which do not have well-defined particle numbers. Such states are difficult or impossible to handle using ordinary quantum mechanics, but they can be easily described in quantum field theory as

quantum superpositionQuantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...

s of states having different values of

. For example, suppose we have a bosonic field whose particles can be created or destroyed by interactions with a fermionic field. The Hamiltonian of the combined system would be given by the Hamiltonians of the free boson and free fermion fields, plus a "potential energy" term such as

where

and

denotes the bosonic creation and annihilation operators,

and

denotes the fermionic creation and annihilation operators, and

is a parameter that describes the strength of the interaction. This "interaction term" describes processes in which a fermion in state

either absorbs or emits a boson, thereby being kicked into a different eigenstate

. (In fact, this type of Hamiltonian is used to describe interaction between conduction electrons and

phononIn physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...

s in

metalA metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...

s. The interaction between electrons and

photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s is treated in a similar way, but is a little more complicated because the role of

spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

must be taken into account.) One thing to notice here is that even if we start out with a fixed number of bosons, we will typically end up with a superposition of states with different numbers of bosons at later times. The number of fermions, however, is conserved in this case.

In

condensed matter physicsCondensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...

, states with ill-defined particle numbers are particularly important for describing the various

superfluidSuperfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...

s. Many of the defining characteristics of a superfluid arise from the notion that its quantum state is a superposition of states with different particle numbers. In addition, the concept of a coherent state (used to model the laser and the BCS ground state) refers to a state with an ill-defined particle number but a well-defined phase.

### Axiomatic approaches

The preceding description of quantum field theory follows the spirit in which most

physicistA physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

s approach the subject. However, it is not

mathematically rigorousRigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism...

. Over the past several decades, there have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of

axiomIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s for it. These attempts fall into two broad classes.

The first class of axioms, first proposed during the 1950s, include the

WightmanIn physics the Wightman axioms are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s but they were first published only in 1964, after Haag-Ruelle scattering theory affirmed their significance.The axioms exist in...

, Osterwalder-Schrader, and Haag-Kastler systems. They attempted to formalize the physicists' notion of an "operator-valued field" within the context of

functional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

, and enjoyed limited success. It was possible to prove that any quantum field theory satisfying these axioms satisfied certain general theorems, such as the

spin-statistics theoremIn quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum...

and the

CPT theoremCPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity, and time simultaneously.-History:...

. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory, including the

Standard ModelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

, satisfied these axioms. Most of the theories that could be treated with these analytic axioms were physically trivial, being restricted to low-dimensions and lacking interesting dynamics. The construction of theories satisfying one of these sets of axioms falls in the field of

constructive quantum field theoryIn mathematical physics, constructive quantum field theory is the field devoted to showing that quantum theory is mathematically compatible with special relativity. This demonstration requires new mathematics, in a sense analogous to Newton developing calculus in order to understand planetary...

. Important work was done in this area in the 1970s by Segal, Glimm, Jaffe and others.

During the 1980s, a second set of axioms based on

geometricGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

ideas was proposed. This line of investigation, which restricts its attention to a particular class of quantum field theories known as

topological quantum field theoriesA topological quantum field theory is a quantum field theory which computes topological invariants....

, is associated most closely with

Michael AtiyahSir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...

and

Graeme SegalGraeme Bryce Segal is a British mathematician, and professor at the University of Oxford.Segal was educated at the University of Sydney, where he received his BSc degree in 1961. He went on to receive his D.Phil...

, and was notably expanded upon by

Edward WittenEdward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....

,

Richard BorcherdsRichard Ewen Borcherds is a British mathematician specializing in lattices, number theory, group theory, and infinite-dimensional algebras. He was awarded the Fields Medal in 1998.- Personal life :...

, and

Maxim KontsevichMaxim Lvovich Kontsevich is a Russian mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami...

. However, most of the physically relevant quantum field theories, such as the

Standard ModelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

, are not topological quantum field theories; the quantum field theory of the

fractional quantum Hall effect is a notable exception. The main impact of axiomatic topological quantum field theory has been on mathematics, with important applications in

representation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

,

algebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

, and differential geometry.

Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. One of the

Millennium Prize ProblemsThe Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...

—proving the existence of a mass gap in Yang-Mills theory—is linked to this issue.

## Phenomena associated with quantum field theory

In the previous part of the article, we described the most general properties of quantum field theories. Some of the quantum field theories studied in various fields of theoretical physics possess additional special properties, such as renormalizability, gauge symmetry, and supersymmetry. These are described in the following sections.

### Renormalization

Early in the history of quantum field theory, it was found that many

seemingly innocuous calculations, such as the

perturbativeIn quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...

shift in the energy of an electron due to the presence of the electromagnetic field, give infinite results. The reason is that the perturbation theory for the shift in an energy involves a sum over all other energy levels, and there are infinitely many levels at short distances that each give a finite contribution.

Many of these problems are related to failures in classical electrodynamics that were identified but unsolved in the 19th century, and they basically stem from the fact that many of the supposedly "intrinsic" properties of an electron are tied to the electromagnetic field that it carries around with it. The energy carried by a single electron—its self energy—is not simply the bare value, but also includes the energy contained in its electromagnetic field, its attendant cloud of photons. The energy in a field of a spherical source diverges in both classical and quantum mechanics, but as discovered by Weisskopf with help from Wendell Furry, in quantum mechanics the divergence is much milder, going only as the logarithm of the radius of the sphere.

The solution to the problem, presciently suggested by

StueckelbergErnst Carl Gerlach Stueckelberg was a Swiss mathematician and physicist.- Career :In 1927 Stueckelberg got his Ph. D. at the University of Basel under August Hagenbach...

, independently by

BetheHans Albrecht Bethe was a German-American nuclear physicist, and Nobel laureate in physics for his work on the theory of stellar nucleosynthesis. A versatile theoretical physicist, Bethe also made important contributions to quantum electrodynamics, nuclear physics, solid-state physics and...

after the crucial experiment by

LambWillis Eugene Lamb, Jr. was an American physicist who won the Nobel Prize in Physics in 1955 together with Polykarp Kusch "for his discoveries concerning the fine structure of the hydrogen spectrum". Lamb and Kusch were able to precisely determine certain electromagnetic properties of the electron...

, implemented at one loop by

SchwingerJulian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...

, and systematically extended to all loops by

FeynmanRichard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

and

DysonFreeman John Dyson FRS is a British-born American theoretical physicist and mathematician, famous for his work in quantum field theory, solid-state physics, astronomy and nuclear engineering. Dyson is a member of the Board of Sponsors of the Bulletin of the Atomic Scientists...

, with converging work by

Tomonagawas a Japanese physicist, influential in the development of quantum electrodynamics, work for which he was jointly awarded the Nobel Prize in Physics in 1965 along with Richard Feynman and Julian Schwinger.-Biography:...

in isolated postwar Japan, comes from recognizing that all the infinities in the interactions of photons and electrons can be isolated into redefining a finite number of quantities in the equations by replacing them with the observed values: specifically the electron 's mass and charge: this is called

renormalizationIn quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....

. The technique of renormalization recognizes that the problem is essentially purely mathematical, that extremely short distances are at fault. In order to define a theory on a continuum, first place a

cutoffIn theoretical physics, cutoff is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculation...

on the fields, by postulating that quanta cannot have energies above some extremely high value. This has the effect of replacing continuous space by a structure where very short wavelengths do not exist, as on a lattice. Lattices break rotational symmetry, and one of the crucial contributions made by Feynman, Pauli and Villars, and modernized by

't HooftGerardus 't Hooft 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...

and Veltman, is a symmetry-preserving cutoff for perturbation theory (this process is called

regularizationRegularization may refer to:* Regularization ** Regularization * Regularization * Regularization * Regularization...

). There is no known symmetrical cutoff outside of perturbation theory, so for rigorous or numerical work people often use an actual lattice.

On a lattice, every quantity is finite but depends on the spacing. When taking the limit of zero spacing, we make sure that the physically observable quantities like the observed electron mass stay fixed, which means that the constants in the Lagrangian defining the theory depend on the spacing. Hopefully, by allowing the constants to vary with the lattice spacing, all the results at long distances become insensitive to the lattice, defining a continuum limit.

The renormalization procedure only works for a certain class of quantum field theories, called

**renormalizable quantum field theories**. A theory is

**perturbatively renormalizable** when the constants in the Lagrangian only diverge at worst as logarithms of the lattice spacing for very short spacings. The continuum limit is then well defined in perturbation theory, and even if it is not fully well defined non-perturbatively, the problems only show up at distance scales that are exponentially small in the inverse coupling for weak couplings. The

Standard ModelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

of

particle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

is perturbatively renormalizable, and so are its component theories (

quantum electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

/

electroweak theoryIn particle physics, the electroweak interaction is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different...

and

quantum chromodynamicsIn theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

). Of the three components, quantum electrodynamics is believed to not have a continuum limit, while the

asymptotically freeIn physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become arbitrarily weak at energy scales that become arbitrarily large, or, equivalently, at length scales that become arbitrarily small .Asymptotic freedom is a feature of quantum...

SU(2) and SU(3) weak hypercharge and strong color interactions are nonperturbatively well defined.

The

renormalization groupIn theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...

describes how renormalizable theories emerge as the long distance low-energy

effective field theoryIn physics, an effective field theory is, as any effective theory, an approximate theory, that includes appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, while ignoring substructure and degrees of freedom at shorter distances .-The renormalization...

for any given high-energy theory. Because of this, renormalizable theories are insensitive to the precise nature of the underlying high-energy short-distance phenomena. This is a blessing because it allows physicists to formulate low energy theories without knowing the details of high energy phenomenon. It is also a curse, because once a renormalizable theory like the standard model is found to work, it gives very few clues to higher energy processes. The only way high energy processes can be seen in the standard model is when they allow otherwise forbidden events, or if they predict quantitative relations between the coupling constants.

### Gauge freedom

A

gauge theoryIn physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

is a theory that admits a

symmetryIn physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation...

with a local parameter. For example, in every

quantumQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

theory the global

phasePhase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

of the wave function is arbitrary and does not represent something physical. Consequently, the theory is invariant under a global change of phases (adding a constant to the phase of all wave functions, everywhere); this is a

global symmetryA global symmetry is a symmetry that holds at all points in the spacetime under consideration, as opposed to a local symmetry which varies from point to point.Global symmetries require conservation laws, but not forces, in physics.-See also:...

. In

quantum electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

, the theory is also invariant under a

*local* change of phase, that is – one may shift the phase of all wave functions so that the shift may be different at every point in space-time. This is a

*local* symmetry. However, in order for a well-defined

derivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

operator to exist, one must introduce a new

fieldIn physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...

, the gauge field, which also transforms in order for the local change of variables (the phase in our example) not to affect the derivative. In

quantum electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

this gauge field is the

electromagnetic fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

. The change of local gauge of variables is termed gauge transformation.

In quantum field theory the excitations of fields represent

particlesIn particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

. The particle associated with excitations of the gauge field is the

gauge bosonIn particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental forces of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles.-...

, which is the

photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

in the case of

quantum electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

.

The

degrees of freedomA degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

in quantum field theory are local fluctuations of the fields. The existence of a gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero by gauge transformations, so they are equivalent to having no fluctuations at all, and they therefore have no physical meaning. Such fluctuations are usually called "non-physical degrees of freedom" or

*gauge artifacts*; usually some of them have a negative

normIn linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

, making them inadequate for a consistent theory. Therefore, if a classical field theory has a gauge symmetry, then its quantized version (i.e. the corresponding quantum field theory) will have this symmetry as well. In other words, a gauge symmetry cannot have a quantum

anomalyIn quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physics an anomaly is the failure of a symmetry to be restored in the limit in which the symmetry-breaking...

. If a gauge symmetry is

anomalousIn theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e...

(i.e. not kept in the quantum theory) then the theory is non-consistent: for example, in

quantum electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

, had there been a

gauge anomalyIn theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e...

, this would require the appearance of

photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s with

longitudinalLongitudinal waves, as known as "l-waves", are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also...

polarizationPhoton polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. Individual photons are completely polarized...

and polarization in the time direction, the latter having a negative

normIn linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

, rendering the theory inconsistent; another possibility would be for these photons to appear only in intermediate processes but not in the final products of any interaction, making the theory non-unitary and again inconsistent (see

optical theoremIn physics, the optical theorem is a general law of wave scattering theory, which relates the forward scattering amplitude to the total cross section of the scatterer...

).

In general, the gauge transformations of a theory consist of several different transformations, which may not be commutative. These transformations are together described by a mathematical object known as a gauge group.

InfinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

gauge transformations are the gauge group generators. Therefore the number of

gauge bosonIn particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental forces of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles.-...

s is the group

dimensionIn mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...

(i.e. number of generators forming a

basisIn linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

).

All the

fundamental interactionIn particle physics, fundamental interactions are the ways that elementary particles interact with one another...

s in nature are described by

gauge theoriesIn physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

. These are:

- Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

, whose gauge transformation is a local change of phase, so that the gauge group is U(1). The gauge bosonIn particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental forces of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles.-...

is the photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

.
- Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

, whose gauge group is SU(3). The gauge bosonIn particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental forces of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles.-...

s are eight gluonGluons are elementary particles which act as the exchange particles for the color force between quarks, analogous to the exchange of photons in the electromagnetic force between two charged particles....

s.
- The electroweak theory
Weak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...

, whose gauge group is , (a direct productIn the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...

of U(1) and SU(2)).
- Gravity, whose classical theory is general relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, admits the equivalence principleIn the physics of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body is actually...

, which is a form of gauge symmetry. However, it is explicitly non-renormalizable.

### Multivalued Gauge Transformations

The gauge transformations

which leave the theory invariant

involve by definition only single-valued gauge functions

which satisfy the

Schwarz integrability criterionIn mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a functionfof n variables...

An interesting extension of gauge transformations arises if

the gauge functions

are allowed to be multivalued functions which violate the integrability criterion.

These are capable of changing the physical field strengths

and are therefore no proper symmetry transformations.

Nevertheless, the transformed field equations describe correctly

the physical laws in the presence of the newly generated field strengths.

See the textbook by

H. KleinertHagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany , at theWest University of Timişoara, at thein Bishkek. He is also of the...

cited below

for the applications to phenomena in physics.

### Supersymmetry

SupersymmetryIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

assumes that every fundamental

fermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

has a superpartner that is a

bosonIn particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

and vice versa. It was introduced in order to solve the so-called

Hierarchy ProblemIn theoretical physics, a hierarchy problem occurs when the fundamental parameters of some Lagrangian are vastly different than the parameters measured by experiment. This can happen because measured parameters are related to the fundamental parameters by a prescription known as renormalization...

, that is, to explain why particles not protected by any symmetry (like the

Higgs bosonThe Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...

) do not receive radiative corrections to its mass driving it to the larger scales (GUT, Planck...). It was soon realized that supersymmetry has other interesting properties: its gauged version is an extension of general relativity (

SupergravityIn theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry...

), and it is a key ingredient for the consistency of

string theoryString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

.

The way supersymmetry protects the hierarchies is the following: since for every particle there is a superpartner with the same mass, any loop in a radiative correction is cancelled by the loop corresponding to its superpartner, rendering the theory UV finite.

Since no superpartners have yet been observed, if supersymmetry exists it must be broken (through a so-called soft term, which breaks supersymmetry without ruining its helpful features). The simplest models of this breaking require that the energy of the superpartners not be too high; in these cases, supersymmetry is expected to be observed by experiments at the

Large Hadron ColliderThe Large Hadron Collider is the world's largest and highest-energy particle accelerator. It is expected to address some of the most fundamental questions of physics, advancing the understanding of the deepest laws of nature....

.

## See also

- Abraham-Lorentz force
In the physics of electromagnetism, the Abraham–Lorentz force is the recoil force on an accelerating charged particle caused by the particle emitting electromagnetic radiation. It is also called the radiation reaction force....

- Basic concepts of quantum mechanics
Quantum mechanics explains the behaviour of matter and energy on the scale of atoms and subatomic particles.Classical physics explains matter and energy at the macroscopic level of the scale familiar to human experience, including the behavior of astronomical bodies...

- Common integrals in quantum field theory
There are common integrals in quantum field theory that appear repeatedly. These integrals are all variations and generalizations of gaussian integrals to the complex plane and to multiple dimensions. Other integrals can be approximated by versions of the gaussian integral...

- Constructive quantum field theory
In mathematical physics, constructive quantum field theory is the field devoted to showing that quantum theory is mathematically compatible with special relativity. This demonstration requires new mathematics, in a sense analogous to Newton developing calculus in order to understand planetary...

- Einstein-Maxwell-Dirac equations
Einstein-Maxwell-Dirac equations are related to quantum field theory. The current Big Bang Model is a quantum field theory in a curved spacetime. Unfortunately, no such theory is mathematically well-defined; in spite of this, theoreticians claim to extract information from this hypothetical theory...

- Feynman path integral
- Form factor
- Green–Kubo relations
- Green's function (many-body theory)
In many-body theory, the term Green's function is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators....

- Invariance mechanics
In physics, invariance mechanics, in its simplest form, is the rewriting of the laws of quantum field theory in terms of invariant quantities only. For example, the positions of a set of particles in a particular coordinate system is not invariant under translations of the system...

- List of quantum field theories
- Photon polarization
Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. Individual photons are completely polarized...

- Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

- Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

- Quantum flavordynamics
In quantum mechanics, Quantum Flavordynamics is a mathematical model used to describe the interaction of flavored particles through the exchange of intermediate vector bosons . In general it refers to the theory of weak interactions under the standard model...

- Quantum geometrodynamics
In theoretical physics, geometrodynamics generally denotes a program of reformulation and unification which was enthusiastically promoted by John Archibald Wheeler in the 1960s.-Einstein's geometrodynamics:...

- Quantum hydrodynamics
Quantum hydrodynamics is most generally the study of hydrodynamic systems which demonstrate behavior implicit in quantum subsystems . They arise in semiclassical mechanics in the study of semiconductor devices, in which case being derived from the Wigner-Boltzmann equation...

- Quantum magnetodynamics
- Quantum triviality
In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. Ifthe only allowed value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting...

- Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
This article relates the Schrödinger equation with the path integral formulation of quantum mechanics using a simple nonrelativistic one-dimensional single-particle Hamiltonian composed of kinetic and potential energy.-Schrödinger's equation:...

- Relationship between string theory and quantum field theory
Many first principles in quantum field theory are explained, or get further insight, in string theory:* Emission and absorption: one of the most basic building blocks of quantum field theory, is the notion that particles can emit and absorb other particles...

- Schwinger-Dyson equation
The Schwinger–Dyson equation , also Dyson-Schwinger equations, named after Julian Schwinger and Freeman Dyson, are general relations between Green functions in quantum field theories...

- Static forces and virtual-particle exchange
Static force fields are fields, such as a simple electric, magnetic or gravitational fields, that exist without excitations. The most common approximation method that physicists use for scattering calculations can be interpreted as static forces arising from the interactions between two bodies...

- Theoretical and experimental justification for the Schrödinger equation
The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles...

- Ward–Takahashi identity
In quantum field theory, a Ward-Takahashi identity is an identity between correlation functions that follows from the global or gauged symmetries of the theory, and which remains valid after renormalization....

- Wheeler-Feynman absorber theory
- Wigner's classification
In mathematics and theoretical physics, Wigner's classificationis a classification of the nonnegative energy irreducible unitary representations of the Poincaré group, which have sharp mass eigenvalues...

- Wigner's theorem
Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT act on the Hilbert space of states....

## Further reading

**General readers**:

- Weinberg, S. Quantum Field Theory, Vols. I to III, 2000, Cambridge University Press: Cambridge, UK.
- Schumm, Bruce A. (2004)
*Deep Down Things*. Johns Hopkins Univ. Press. Chpt. 4.

**Introductory texts**:

**Advanced texts**:

**Articles**:

- Gerard 't Hooft (2007) "The Conceptual Basis of Quantum Field Theory" in Butterfield, J., and John Earman
John Earman is a philosopher of physics. He is currently an emeritus professor in the History and Philosophy of Science department at the University of Pittsburgh. He has also taught at UCLA, the Rockefeller University, and the University of Minnesota, and was president of the Philosophy of...

, eds., *Philosophy of Physics, Part A*. Elsevier: 661-730.
- Frank Wilczek
Frank Anthony Wilczek is a theoretical physicist from the United States and a Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology ....

(1999) "Quantum field theory", *Reviews of Modern Physics* 71: S83-S95. Also doi=10.1103/Rev. Mod. Phys. 71.

## External links

- Stanford Encyclopedia of Philosophy
The Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...

: "Quantum Field Theory", by Meinard Kuhlmann.
- Siegel, Warren, 2005.
*Fields.* A free text, also available from arXiv:hep-th/9912205.
- Quantum Field Theory by P. J. Mulders