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Encyclopedia > Gauge theory

In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... This article or section does not cite its references or sources. ... This article or section does not cite its references or sources. ... In quantum field theory, a global symmetry is any symmetry of a model which is not a gauge symmetry. ...

Many powerful theories in physics are described by Lagrangians which are invariant under certain symmetry transformation groups. When they are invariant under a transformation identically performed at every point in the space in which the physical processes occur; they are said to have a global symmetry. In a gauge theory the requirement of global transformations is relaxed such that the Lagrangian is required to have merely local symmetry. This can be viewed as a generalization of the equivalence principle of general relativity in which each point in spacetime is allowed a choice of local reference (coordinate) frame. As in that situation, gauge "symmetries" reflect a redundancy in the description of a system. Historically, these ideas were first noticed in the context of classical electromagnetism and later in general relativity. However, the modern importance of gauge symmetries appeared in relativistic quantum mechanics of electrons (see discussions below). Today, gauge theories are useful in condensed matter, nuclear and high energy physics among other subfields. A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... In physics, invariants are usually quantities conserved (unchanged) by the symmetries of the physical system. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... This article or section does not cite its references or sources. ... This article or section does not cite its references or sources. ... In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ...

Sometimes, the term 'gauge symmetry' is used in a more general sense to include any local symmetry, like for example, diffeomorphisms. This sense of the term will not be used in this article. In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

Yang-Mills theories are a particular example of gauge theories with non-abelian symmetry groups specified by the Yang-Mills action (Other gauge theories with a non-abelian gauge symmetry also exist, e.g., the Chern-Simons model). In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... The symmetry group of an object (e. ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... Chern-Simons theory is a topological gauge theory in three dimensions which describes knot and three-manifold invariants. ...

There is a certain inaccuracy in the way the term symmetry is used in part of the physics literature, especially in the more elementary books about elementary particles and field theory. In (quantum) physics, symmetry is a transformation between physical states that preserves the expectation values of all observables O (in particular the Hamiltonian). S: |φ> → |ψ> = S|φ>; |<ψ|O|ψ>|2=|<φ|O|φ>|2. The usual formulation of the physics theories uses fields, which sometimes are not physical quantities. Such are the gauge fields (fiber bundle connections for the mathematicians), which provide a redundant but convenient description of the physical degrees of freedom. The gauge (local) "symmetries" are a reflection of this redundancy. The physical quantities are certain equivalence classes of gauge fields. An analogy can be made with the construction of the real numbers. We can use sequences of rational numbers that have the same limit. Of course, each real number is represented by infinitely many such sequences. We can choose a particular well defined sequence to be a representative of the real number. This corresponds to the procedure of gauge fixing in gauge theories. The fact that gauge fields are not physical degrees of freedom becomes very clear when we try to quantize them. Then we are forced to work in one way or another with the physical quantities by removing the redundancy (the gauge symmetry). Another important illustration of the problem with the gauge “symmetries” is when we have anomalies. By definition these are symmetries which exist in the classical system, but not in its quantum counterpart. Anomalies are something quite usual and also an experimental fact — for example, the axial anomaly in the strong interactions (broken symmetries). However, because gauge symmetries are not symmetries, gauge anomalies are not something that just complicates a proposed quantum theory but something that kills it, i.e. there are no gauge "anomalies", because such theories don't exist. This is why having the exact relation between the number of flavours and quark colours in the Standard model is so important — otherwise there is a gauge anomaly and the theory does not exist. For the same reason, string theories are defined in 10 dimensions. Only then do the anomalies cancel.

## Importance

The importance of gauge theories for physics stems from the tremendous success of the mathematical formalism in providing a unified framework to describe the quantum field theories of electromagnetism, the weak force and the strong force. This theory, known as the Standard Model, accurately describes experimental predictions regarding three of the four fundamental forces of nature, and is a gauge theory with the gauge group SU(3) × SU(2) × U(1). Modern theories like string theory, as well as some formulations of general relativity, are, in one way or another, gauge theories. Quantum field theory (QFT) is the quantum theory of fields. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... The strong nuclear force or strong interaction (also called color force or colour force) is a fundamental force of nature which affects only quarks and antiquarks, and is mediated by gluons in a similar fashion to how the electromagnetic force is mediated by photons. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... In the scientific method, an experiment (Latin: ex- periri, of (or from) trying) is a set of observations performed in the context of solving a particular problem or question, to support or falsify a hypothesis or research concerning phenomena. ... A fundamental interaction is a mechanism by which particles interact with each other, and which cannot be explained by another more fundamental interaction. ... This Lie group is the formulation of the Standard Model as a gauge theory with the gauge group SU(3) × SU(2) × U(1) or with a couple of fermion fields and a Higgs field, which is a and/or a . ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory This box:      String theory is a model of fundamental physics, whose building blocks are one-dimensional extended objects called strings, rather than the zero... This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ...

## A brief history

In the 1950s, attempting to resolve some of the great confusion in elementary particle physics, Chen Ning Yang and Robert Mills introduced non-abelian gauge theories as models to understand the strong interaction holding together nucleons in atomic nuclei. (Ronald Shaw, working under Abdus Salam, independently introduced the same notion in his doctoral thesis.) Generalizing the gauge invariance of electromagnetism, they attempted to construct a theory based on the action of the (non-abelian) SU(2) symmetry group on the isospin doublet of protons and neutrons, similar to the action of the U(1) group on the spinor fields of quantum electrodynamics. In particle physics the emphasis was on using quantized gauge theories. Particle physics is a branch of physics that studies the elementary constituents of matter and radiation, and the interactions between them. ... Zhen-Ning Franklin Yang (Traditional Chinese: ; pinyin: ) (born 22 September, 1922) is a Chinese American physicist who worked on statistical mechanics and symmetry principles. ... Robert L. Mills (April 15, 1927 - October 27, 1999) was a physicist, specializing in quantum field theory, the theory of alloys, and many-body theory. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... The strong interaction or strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD). ... In physics a nucleon is a collective name for two baryons: the neutron and the proton. ... The nucleus of an atom is the very small dense region, of positive charge, in its centre consisting of nucleons (protons and neutrons). ... For other uses, see Abdus Salam (disambiguation). ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... This picture illustrates how the hours on a clock form a group under modular addition. ... Isospin (isotopic spin, isobaric spin) is a physical quantity which is mathematically analogous to spin. ... For other uses, see Proton (disambiguation). ... This article or section does not adequately cite its references or sources. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of nÃ—n unitary matrices, with the group operation that of matrix multiplication. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ...

This idea later found application in the quantum field theory of the weak force, and its unification with electromagnetism in the electroweak theory. Gauge theories became even more attractive when it was realized that non-abelian gauge theories reproduced a feature called asymptotic freedom, that was believed to be an important characteristic of strong interactions—thereby motivating the search for a gauge theory of the strong force. This theory, now known as quantum chromodynamics, is a gauge theory with the action of the SU(3) group on the color triplet of quarks. The Standard Model unifies the description of electromagnetism, weak interactions and strong interactions in the language of gauge theory. Quantum field theory (QFT) is the quantum theory of fields. ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ... In physics, asymptotic freedom is the property of some gauge theories in which the interaction between the particles, such as quarks, becomes arbitrarily weak at ever shorter distances, i. ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... In quantum chromodynamics (QCD), color or color charge refers to a certain property of the subatomic particles called quarks. ... For other uses of this term, see: Quark (disambiguation) 1974 discovery photograph of a possible charmed baryon, now identified as the &#931;c++ In particle physics, the quarks are subatomic particles thought to be elemental and indivisible. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ...

In the seventies, Sir Michael Atiyah began a program of studying the mathematics of solutions to the classical Yang-Mills equations. In 1983, Atiyah's student Simon Donaldson built on this work to show that the differentiable classification of smooth 4-manifolds is very different from their classification up to homeomorphism. Michael Freedman used Donaldson's work to exhibit exotic R4s, that is, exotic differentiable structures on Euclidean 4-dimensional space. This led to an increasing interest in gauge theory for its own sake, independent of its successes in fundamental physics. In 1994, Edward Witten and Nathan Seiberg invented gauge-theoretic techniques based on supersymmetry which enabled the calculation of certain topological invariants. These contributions to mathematics from gauge theory have led to a renewed interest in this area. The 1970s decade refers to the years from 1970 to 1979, also called The Seventies. ... Sir Michael Francis Atiyah, OM, FRS (b. ... Simon Kirwan Donaldson, born in Cambridge in 1957, is an English mathematician famous for his work on the topology of smooth (differentiable) four-dimensional manifolds. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... Michael Hartley Freedman (born 21 April 1951 in Los Angeles, California, USA) is a mathematician at Microsoft Research. ... In mathematics, an exotic or fake R4 is a differentiable manifold that is homeomorphic to the Euclidean space R4, but not diffeomorphic. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ... Year 1994 (MCMXCIV) The year 1994 was designated as the International Year of the Family and the International Year of the Sport and the Olympic Ideal by the United Nations. ... Edward Witten (born August 26, 1951) is an American theoretical physicist and professor at the Institute for Advanced Study. ... Nathan Seiberg at Harvard University Nathan Seiberg is an Israel-born theoretical physicist who works on string theory. ... This article or section is in need of attention from an expert on the subject. ... A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...

## A simple gauge symmetry example from electrodynamics

The definition of electrical ground in an electric circuit is an example of a gauge symmetry; when the electric potentials at all points in a circuit are raised by the same amount, the circuit would still operate identically; as the potential differences (voltages) in the circuit are unchanged. A common illustration of this fact is the sight of a small bird perched on a high voltage power line without electrocution, because the bird is insulated from the ground (as long as it doesn't complete the circuit by accidentally touching another wire or some grounded structure). The term ground (or earth) usually means a common return in circuits. ... An electrical network or electrical circuit is an interconnection of analog electrical elements such as resistors, inductors, capacitors, diodes, switches and transistors. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... This article does not cite any references or sources. ... International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ...

This is called a global gauge symmetry. The absolute value of the potential is immaterial; what matters to circuit operation is the potential differences across the components of the circuit. The definition of the ground point is arbitrary, but once that point is set, then that definition must be followed globally. The adjective global and adverb globally imply that the verb or noun to which they are applied applies to the entire Earth and all of its species and regions. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

In contrast, if some symmetry could be defined arbitrarily from one position to the next, that would be a local gauge symmetry. In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

• ^  James S. Trefil 1983, The moment of creation. Scribner, ISBN 0-684-17963-6 pages 92-93.

Year 1983 (MCMLXXXIII) was a common year starting on Saturday (link displays the 1983 Gregorian calendar). ...

## Classical gauge theory

This section requires some familiarity with classical or quantum field theory, and the use of Lagrangians. Quantum field theory (QFT) is the quantum theory of fields. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

Definitions in this section: gauge group, gauge field, interaction Lagrangian, gauge boson

### An example: Scalar O(n) gauge theory

The following illustrates how local gauge invariance can be "motivated" heuristically starting from global symmetry properties, and how it leads to an interaction between fields which were originally non-interacting.

Consider a set of n non-interacting scalar fields, with equal masses m. This system is described by an action which is the sum of the (usual) action for each scalar field φi The magnitude of an electric field surrounding two equally charged (repelling) particles. ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... $mathcal{S} = int , mathrm{d}^4 x sum_{i=1}^n left[ frac{1}{2} partial_mu varphi_i partial^mu varphi_i - frac{1}{2}m^2 varphi_i^2 right].$

The Lagrangian (density) can be compactly written as $mathcal{L} = frac{1}{2} (partial_mu Phi)^T partial^mu Phi - frac{1}{2}m^2 Phi^T Phi$

by introducing a vector of fields This article is about vectors that have a particular relation to the spatial coordinates. ... $Phi = ( varphi_1, varphi_2,ldots, varphi_n)^T .$

The term $partial_mu$ is sometimes confusing to those who have not seen it before. It is simply the use of Einstein notation to describe the partial derivative of Φ in each of the four dimensions. It is now transparent that the Lagrangian is invariant under the transformation This article or section does not adequately cite its references or sources. ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ... $Phi mapsto G Phi$

whenever G is a constant matrix belonging to the n-by-n orthogonal group O(n). This is the global symmetry of this particular Lagrangian, and the symmetry group is often called the gauge group; the mathematical term is structure group, especially in the theory of G-structures. Incidentally, Noether's theorem implies that invariance under this group of transformations leads to the conservation of the current In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In differential geometry, a G-structure on a n-manifold M, for a given structure group G (which is a Lie subgroup of the general linear group GL(n)) is a G-subbundle of the frame bundle on M. The notion of G-structures includes many other structures on manifolds... Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ... $J^{a}_{mu} = ipartial_mu Phi^T T^{a} Phi$

where the Ta matrices are generators of the SO(n) group. There is one conserved current for every generator. In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...

Now, demanding that this Lagrangian should have local O(n)-invariance requires that the G matrices (which were earlier constant) should be allowed to become functions of the space-time coordinates x. In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...

Unfortunately, the G matrices do not "pass through" the derivatives. When G = G(x), $partial_mu (G Phi)^T partial^mu (G Phi) neq partial_mu Phi^T partial^mu Phi.$

This suggests defining a "derivative" D with the property This article is about derivatives and differentiation in mathematical calculus. ... $D_mu (G(x) Phi(x)) = G(x) D_mu Phi.$

It can be checked that such a "derivative" (called a covariant derivative) is In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ... $D_mu = partial_mu + g A_mu(x)$

where the gauge field A(x) is defined to have the transformation law $A_{mu}(x) mapsto G(x)A_{mu}(x)G^{-1}(x) - frac{1}{g} partial_mu G(x) G^{-1}(x)$

and g is the coupling constant - a quantity defining the strength of an interaction.

The gauge field is an element of the Lie algebra, and can therefore be expanded as $A_{mu}(x)= sum_a A_{mu}^a (x) T^a$

There are therefore as many gauge fields as there are generators of the Lie algebra.

Finally, we now have a locally gauge invariant Lagrangian $mathcal{L}_mathrm{loc} = frac{1}{2} (D_mu Phi)^T D^mu Phi -frac{1}{2}m^2 Phi^T Phi.$

Pauli calls gauge transformation of the first type to the one applied to fields as Φ, while the compensating transformation in A is said to be a gauge transformation of the second type.  Feynman diagram of scalar bosons interacting via a gauge boson

The difference between this Lagrangian and the original globally gauge-invariant Lagrangian is seen to be the interaction Lagrangian Image File history File links Feynman-Diagram. ... Image File history File links Feynman-Diagram. ... In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ... $mathcal{L}_mathrm{int} = frac{g}{2} Phi^T A_{mu}^T partial^mu Phi + frac{g}{2} (partial_mu Phi)^T A^{mu} Phi + frac{g^2}{2} (A_mu Phi)^T A^mu Phi.$

This term introduces interactions between the n scalar fields just as a consequence of the demand for local gauge invariance. In the quantized version of this classical field theory, the quanta of the gauge field A(x) are called gauge bosons. The interpretation of the interaction Lagrangian in quantum field theory is of scalar bosons interacting by the exchange of these gauge bosons. For other uses, see Interaction (disambiguation). ... Generally, quantization is the state of being constrained to a set of discrete values, rather than varying continuously. ... Field theory (mathematics), the theory of the algebraic concept of field. ... In physics, a quantum (plural: quanta) is an indivisible entity of energy. ... Gauge bosons are bosonic particles which act as carriers of the fundamental forces of Nature. ... See scalar for an account of the broader concept also used in mathematics and computer science. ... In particle physics, bosons, named after Satyendra Nath Bose, are particles having integer spin. ...

### The Yang-Mills Lagrangian for the gauge field

Our picture of classical gauge theory is almost complete except for the fact that to define the covariant derivatives D, one needs to know the value of the gauge field A(x) at all space-time points. Instead of manually specifying the values of this field, it can be given as the solution to a field equation. Further requiring that the Lagrangian which generates this field equation is locally gauge invariant as well, one possible form for the gauge field Lagrangian is (conventionally) written as $mathcal{L}_mathrm{gf} = - frac{1}{2} operatorname{Tr}(F^{mu nu} F_{mu nu})$

with $F_{mu nu} = frac{1}{ig}[D_mu, D_nu]$

and the trace being taken over the vector space of the fields. This is called the Yang-Mills action. Other gauge invariant actions also exist (e.g. nonlinear electrodynamics, Born-Infeld action, Chern-Simons model, theta term etc.). In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... Nonlinear optics is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization P responds nonlinearly to the electric field E of the light. ... In physics, the Born-Infeld theory is a nonlinear generalization of electromagnetism (see nonlinear electrodynamics). ... Chern-Simons theory is a topological gauge theory in three dimensions which describes knot and three-manifold invariants. ... In particle physics, the strong CP problem is the puzzling question why Quantum Chromodynamics (QCD) does not seem to break the CP-symmetry. ...

Note that in this Lagrangian there is not a field Φ whose transformation counterweights the one of A. Invariance of this term under gauge transformations is a particular case of a prior classical (or geometrical, if you prefer) symmetry. This symmetry must be restricted in order to perform quantization, the procedure being denominated gauge fixing, but even after restriction, gauge transformations are possible (see Sakurai, Advanced Quantum Mechanics, sect 1-4). In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. ...

The complete Lagrangian for the O(n) gauge theory is now $mathcal{L} = mathcal{L}_mathrm{loc} + mathcal{L}_mathrm{gf} = mathcal{L}_mathrm{global} + mathcal{L}_mathrm{int} + mathcal{L}_mathrm{gf}$

### A simple example: Electrodynamics

As a simple application of the formalism developed in the previous sections, consider the case of electrodynamics, with only the electron field. The bare-bones action which generates the electron field's Dirac equation is Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ... For other uses, see Electron (disambiguation). ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-Â½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ... $mathcal{S} = int barpsi(i hbar c , gamma^mu partial_mu - m c^2 ) psi , mathrm{d}^4x.$

The global symmetry for this system is $psi mapsto e^{i theta} psi.$

The gauge group here is U(1), just the phase angle of the field, with a constant θ. In mathematics, the unitary group of degree n, denoted U(n), is the group of nÃ—n unitary matrices, with the group operation that of matrix multiplication. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

"Local"ising this symmetry implies the replacement of θ by θ(x).

An appropriate covariant derivative is then $D_mu = partial_mu - i frac{e}{hbar} A_mu.$

Identifying the "charge" e with the usual electric charge (this is the origin of the usage of the term in gauge theories), and the gauge field A(x) with the four-vector potential of electromagnetic field results in an interaction Lagrangian Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... In vector calculus, a vector potential is a vector field whose curl is a given vector field. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... $mathcal{L}_mathrm{int} = frac{e}{hbar}barpsi(x) gamma^mu psi(x) A_{mu}(x) = J^{mu}(x) A_{mu}(x).$

where Jμ(x) is the usual four vector electric current density. The gauge principle is therefore seen to introduce the so-called minimal coupling of the electromagnetic field to the electron field in a natural fashion. In relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space, whose components transform as the space and time coordinate differences, , under spatial translations, rotations, and boosts (a change by a constant velocity to another inertial reference frame). ...

Adding a Lagrangian for the gauge field Aμ(x) in terms of the field strength tensor exactly as in electrodynamics, one obtains the Lagrangian which is used as the starting point in quantum electrodynamics. For thermodynamic relations, see Maxwell relations. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... $mathcal{L}_{mathrm{QED}} = barpsi(ihbar c , gamma^mu D_mu - m c^2 )psi - frac{1}{4 mu_0}F_{munu}F^{munu}.$

See also: Dirac equation, Maxwell's equations, Quantum electrodynamics In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-Â½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ... For thermodynamic relations, see Maxwell relations. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ...

## Mathematical formalism

Gauge theories are usually discussed in the language of differential geometry. Mathematically, a gauge is just a choice of a (local) section of some principal bundle. A gauge transformation is just a transformation between two such sections. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... Look up section in Wiktionary, the free dictionary. ... In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X Ã— G of a space X with a group G. Analogous to the Cartesian product, a principal bundle P is equipped with An action of G on P, analogous to...

Although gauge theory is dominated by the study of connections (primarily because it's mainly studied by high-energy physicists), the idea of a connection is not central to gauge theory in general. In fact, a result in general gauge theory shows that affine representations (i.e. affine modules) of the gauge transformations can be classified as sections of a jet bundle satisfying certain properties. There are reps which transform covariantly pointwise (called by physicists gauge transformations of the first kind), reps which transform as a connection form (called by physicists gauge transformations of the second kind) (note that this is an affine rep) and other more general reps, such as the B field in BF theory. And of course, we can consider more general nonlinear reps (realizations), but that is extremely complicated. But still, nonlinear sigma models transform nonlinearly, so there are applications. In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ... Particle physics is a branch of physics that studies the elementary constituents of matter and radiation, and the interactions between them. ... An affine representation of a topological (Lie) group G is a continuous (smooth) homomorphism from G to the automorphism group of an affine space, A. An example is the action of the Euclidean group E(n) upon the Euclidean space En. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. ... In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ... The BF model is a topological field theory, which when quantized, becomes a topological quantum field theory. ... In quantum field theory, a nonlinear &#963; model is describes a scalar field &#931; which takes on values in a nonlinear manifold called the target manifold T. The target manifold is equipped with a Riemannian metric g. ...

If we have a principal bundle P whose base space is space or spacetime and structure group is a Lie group, then the sections of P form a principal homogeneous space of the group of gauge transformations. In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X Ã— G of a space X with a group G. Analogous to the Cartesian product, a principal bundle P is equipped with An action of G on P, analogous to... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... This article is about the idea of space. ... For other uses of this term, see Spacetime (disambiguation). ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. ...

We can define a connection (gauge connection) on this principal bundle, yielding a covariant derivative ∇ in each associated vector bundle. If we choose a local frame (a local basis of sections) then we can represent this covariant derivative by the connection form A, a Lie algebra-valued 1-form which is called the gauge potential in physics and which is evidently not an intrinsic but a frame-dependent quantity. From this connection form we can construct the curvature form F, a Lie algebra-valued 2-form which is an intrinsic quantity, by In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ... In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In differential geometry, the curvature form describes curvature of principal bundle with connection. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... $bold{F}=mathrm{d}bold{A}+bold{A}wedgebold{A}$

where d stands for the exterior derivative and $wedge$ stands for the wedge product. (Keep in mind that $bold{A}$ is an element of the vector space spanned by the generators Ta, and so the components of $bold{A}$ do not commute with one another. Hence the wedge product $bold{A}wedgebold{A}$ does not vanish.) In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...

Infinitesimal gauge transformations form a Lie algebra, which is characterized by a smooth Lie algebra valued scalar, ε. Under such an infinitesimal gauge transformation, In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... $delta_varepsilon bold{A}=[varepsilon,bold{A}]-mathrm{d}epsilon$

where $[cdot,cdot]$ is the Lie bracket. In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

One nice thing is that if $delta_varepsilon X=varepsilon X$, then $delta_varepsilon DX=varepsilon DX$ where D is the covariant derivative $DX stackrel{mathrm{def}}{=} mathrm{d}X+bold{A}X.$

Also, $delta_varepsilon bold{F}=varepsilon bold{F}$, which means F transforms covariantly.

One thing to note is that not all gauge transformations can be generated by infinitesimal gauge transformations in general; for example, when the base manifold is a compact manifold without boundary such that the homotopy class of mappings from that manifold to the Lie group is nontrivial. See instanton for an example. Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of... The two bold paths shown above are homotopic relative to their endpoints. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... This article or section is in need of attention from an expert on the subject. ...

The Yang-Mills action is now given by $frac{1}{4g^2}int operatorname{Tr}[*Fwedge F]$

where * stands for the Hodge dual and the integral is defined as in differential geometry. In mathematics, the Hodge star operator or Hodge dual is a signficant linear map introduced in general by W. V. D. Hodge. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

A quantity which is gauge-invariant i.e. invariant under gauge transformations is the Wilson loop, which is defined over any closed path, γ, as follows: Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share... In gauge theory, a Wilson loop is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop. ... $chi^{(rho)}left(mathcal{P}left{e^{int_gamma A}right}right)$

where χ is the character of a complex representation ρ and $mathcal{P}$ represents the path-ordered operator. Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...

## Quantization of gauge theories

Gauge theories may be quantized by specialization of methods which are applicable to any quantum field theory. However, because of the subtleties imposed by the gauge constraints (see section on Mathematical formalism, above) there are many technical problems to be solved which do not arise in other field theories. At the same time, the richer structure of gauge theories allow simplification of some computations: for example Ward identities connect different renormalization constants. Quantum field theory (QFT) is the quantum theory of fields. ... This article is about a formulation of quantum mechanics. ... Figure 1. ...

### Methods and aims

The first gauge theory to be quantized was quantum electrodynamics (QED). The first methods developed for this involved gauge fixing and then applying canonical quantization. The Gupta-Bleuler method was also developed to handle this problem. Non-abelian gauge theories are nowadays handled by a variety of means. Methods for quantization are covered in the article on quantization. Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... In physics, canonical quantization is one of many procedures for quantizing a classical theory. ... In quantum field theory, the Gupta-Bleuler formalism is a way of quantizing the electromagnetic field. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ...

The main point to quantization is to be able to compute quantum amplitudes for various processes allowed by the theory. Technically, they reduce to the computations of certain correlation functions in the vacuum state. This involves a renormalization of the theory. In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ... In statistics and probability, in particular in stochastic processes, the value of a correlation function r(s,t) is the correlation between random variables Xs and Xt, where the indices s and t are often either points in time or physical locations. ... In quantum field theory, the vacuum state, usually denoted , is the element of the Hilbert space with the lowest possible energy, and therefore containing no physical particles. ... Figure 1. ...

When the running coupling of the theory is small enough, then all required quantities may be computed in perturbation theory. Quantization schemes that are geared to simplify such computations (such as canonical quantization) may be called perturbative quantization schemes. At present some of these methods lead to the most precise experimental tests of gauge theories. In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. ... Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ...

However, in most gauge theories, there are many interesting questions which are non-perturbative. Quantization schemes which are geared to these problems (such as lattice gauge theory) may be called non-perturbative quantization schemes. Precise computations in such schemes often require supercomputing, and are therefore less well developed currently than other schemes. In physics, lattice gauge theory is the study of the behaviour of lattice model gauge theories. ... A supercomputer is a device for turning compute-bound problems into I/O-bound problems. ...

### Anomalies

Some of the classical symmetries of the theory are then seen not to hold in the quantum theory — a phenomenon called an anomaly. Among the most well known are: In physics, an anomaly is a classical symmetry â€” a symmetry of the Lagrangian â€” that is broken in quantum field theories. ...

Conformal anomaly is an anomaly a quantum phenomenon that breaks the conformal symmetry of the classical theory. ... In physics, Landau pole is the energy scale (or the precise value of the energy) where a coupling constant (the strength of an interaction) of a quantum field theory becomes infinite. ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... In physics, asymptotic freedom is the property of some gauge theories in which the interaction between the particles, such as quarks, becomes arbitrarily weak at ever shorter distances, i. ... A chiral anomaly is the anomalous nonconservation of a chiral current. ... A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... This article or section is in need of attention from an expert on the subject. ... The initialism QCD can mean: Quantum chromodynamics Quintessential Player, formerly known as Quintessential CD Quality, Cost, Delivery, A three-letter acronym used in lean manufacturing This page concerning a three-letter acronym or abbreviation is a disambiguation page â€” a navigational aid which lists other pages that might otherwise share the... In particle physics, pion (short for pi meson) is the collective name for three subatomic particles: Ï€0, Ï€+ and Ï€âˆ’. Pions are the lightest mesons and play an important role in explaining low-energy properties of the strong nuclear force. ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... In 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. ... In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ... For other uses, see Quark (disambiguation). ... In physics, a lepton is a particle with spin-1/2 (a fermion) that does not experience the strong interaction (that is, the strong nuclear force). ... Results from FactBites:

 NationMaster.com - Encyclopedia: Gauge theory (7878 words) The importance of gauge theories for physics stems from the tremendous success of the mathematical formalism in providing a unified framework to describe the quantum field theories of electromagnetism, the weak force and the strong force. Gauge theories became even more attractive when it was realized that non-abelian gauge theories reproduced a feature called asymptotic freedom, that was believed to be an important characteristic of strong interactions—thereby motivating the search for a gauge theory of the strong force. Note that although gauge theory is dominated by the study of connections (primarily because it's mainly studied by high-energy physicists), the idea of a connection is not essential or central to gauge theory in general.
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