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{{more footnotes|date=September 2016}}
{{Short description|Physical theory with fields invariant under the action of local "gauge" Lie groups}}{{See introduction}}{{Hatnote|This article discusses the physics of gauge theories. For the mathematical field of gauge theory, see [[Gauge theory (mathematics)]].}}{{Quantum field theory}}

In [[physics]], a '''gauge theory''' is a type of [[Field theory (physics)|field theory]] in which the [[Lagrangian (field theory)|Lagrangian]], and hence the dynamics of the system itself, does not change under [[local symmetry|local transformations]] according to certain smooth families of operations ([[Lie group]]s). Formally, the Lagrangian is [[Invariant (physics)|invariant]] under these transformations. 

The term "gauge" refers to any specific mathematical formalism to regulate redundant [[Degrees of freedom (physics and chemistry)|degrees of freedom]] in the Lagrangian of a physical system. The transformations between possible gauges, called '''gauge transformations''', form a Lie group—referred to as the ''[[symmetry group]]'' or the '''gauge group''' of the theory. Associated with any Lie group is the [[Lie algebra]] of [[group generator]]s. For each group generator there necessarily arises a corresponding [[Field (physics)|field]] (usually a [[vector field]]) called the '''gauge field'''. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called '''gauge invariance'''). When such a theory is [[Canonical quantization|quantized]], the [[Quantum|quanta]] of the gauge fields are called ''[[gauge boson]]s''. If the symmetry group is non-commutative, then the gauge theory is referred to as ''[[non-Abelian group|non-abelian]] gauge theory'', the usual example being the [[Yang–Mills theory]].

Many powerful theories in physics are described by Lagrangians that are invariant under some symmetry transformation groups. When they are invariant under a transformation identically performed at ''every'' [[Point (geometry)|point]] in the [[spacetime]] in which the physical processes occur, they are said to have a [[global symmetry]]. [[Local symmetry]], the cornerstone of gauge theories, is a stronger constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in spacetime (the same way a constant value can be understood as a function of a certain parameter, the output of which is always the same).

Gauge theories are important as the successful field theories explaining the dynamics of [[elementary particles]]. [[Quantum electrodynamics]] is an [[abelian group|abelian]] gauge theory with the symmetry group [[Unitary group|U(1)]] and has one gauge field, the [[electromagnetic four-potential]], with the [[photon]] being the gauge boson. The [[Standard Model]] is a non-abelian gauge theory with the symmetry group U(1) × [[Special unitary group#The group SU(2)|SU(2)]] × [[Special unitary group#SU(3)|SU(3)]] and has a total of twelve gauge bosons: the [[photon]], three [[weak boson]]s and eight [[gluons]].

Gauge theories are also important in explaining [[gravitation]] in the theory of [[general relativity]]. Its case is somewhat unusual in that the gauge field is a [[tensor]], the [[Lanczos tensor]]. Theories of [[quantum gravity]], beginning with [[gauge gravitation theory]], also postulate the existence of a gauge boson known as the [[graviton]]. Gauge symmetries can be viewed as analogues of the [[principle of general covariance]] of general relativity in which the coordinate system can be chosen freely under arbitrary [[diffeomorphism]]s of spacetime. Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, [[gauge theory gravity]], replaces the principle of general covariance with a true gauge principle with new gauge fields.

Historically, these ideas were first stated in the context of [[classical electromagnetism]] and later in [[general relativity]]. However, the modern importance of gauge symmetries appeared first in the [[relativistic quantum mechanics]] of [[electron]]s{{spaced ndash}}[[quantum electrodynamics]], elaborated on below. Today, gauge theories are useful in [[condensed matter physics|condensed matter]], [[nuclear physics|nuclear]] and [[high energy physics]] among other subfields.

== History ==
<!--'History of gauge theory' redirects here-->
The concept and the name of gauge theory derives from the work of [[Hermann Weyl]] in 1918.<ref name=Brading>{{cite journal | last1 = Brading | first1 = Katherine|author1-link=Katherine Brading | date=March 2002 | title = Which Symmetry? Noether, Weyl, and Conservation of Electric Charge | url = https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=1c6e78e69b177018a8b3013b69d8fe5fb109c1b7 | journal = Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics | volume = 33 | issue = 1 | pages = 3–22 | doi=10.1016/S1355-2198(01)00033-8| bibcode = 2002SHPMP..33....3B }}</ref> Weyl, in an attempt to generalize the geometrical ideas of [[general relativity]] to include [[electromagnetism]], conjectured that ''Eichinvarianz'' or invariance under the change of [[scale (measurement)|scale]] (or "gauge") might also be a local symmetry of general relativity. After the development of [[quantum mechanics]], Weyl, [[Vladimir Fock]]<ref>{{Cite journal |last1=Jackson |first1=J. D. |last2=Okun |first2=L. B. |date=2001-09-14 |title=Historical roots of gauge invariance |url=https://link.aps.org/doi/10.1103/RevModPhys.73.663 |journal=Reviews of Modern Physics |language=en |volume=73 |issue=3 |pages=663–680 |doi=10.1103/RevModPhys.73.663 |issn=0034-6861 |quote=The discovery of the symmetry under gauge transformations (1 a,b,c) of the quantum mechanical system of a charged particle interacting with electromagnetic fields is due to Fock (1926b)|arxiv=hep-ph/0012061 |bibcode=2001RvMP...73..663J }}</ref> and [[Fritz London]] replaced the simple scale factor with a [[complex number|complex]] quantity and turned the scale transformation into a change of [[phase (waves)|phase]], which is a [[U(1)]] gauge symmetry. This explained the [[electromagnetic field]] effect on the [[wave function]] of a [[electric charge|charge]]d quantum mechanical [[Elementary particle|particle]]. Weyl's 1929 paper introduced the modern concept of gauge invariance<ref name="O’Raifeartaigh" >{{Cite journal |last1=O’Raifeartaigh |first1=Lochlainn |last2=Straumann |first2=Norbert |date=2000-01-01 |title=Gauge theory: Historical origins and some modern developments |url=https://link.aps.org/doi/10.1103/RevModPhys.72.1 |journal=Reviews of Modern Physics |language=en |volume=72 |issue=1 |pages=1–23 |doi=10.1103/RevModPhys.72.1 |bibcode=2000RvMP...72....1O |issn=0034-6861}}</ref> subsequently popularized by [[Wolfgang Pauli]] in his 1941 review.<ref>{{cite journal | last1 = Pauli | first1 = Wolfgang | author-link = Wolfgang Pauli | year = 1941 | title = Relativistic Field Theories of Elementary Particles | url = http://prola.aps.org/abstract/RMP/v13/i3/p203_1 | journal = Rev. Mod. Phys. | volume = 13 | issue = 3 | pages = 203–32 | doi=10.1103/revmodphys.13.203| bibcode = 1941RvMP...13..203P }}</ref> In retrospect, [[James Clerk Maxwell]]'s formulation, in 1864–65, of [[classical electrodynamics|electrodynamics]] in "[[A Dynamical Theory of the Electromagnetic Field]]" suggested the possibility of invariance, when he stated that any vector field whose curl vanishes—and can therefore normally be written as a [[gradient]] of a function—could be added to the vector potential without affecting the [[magnetic field]]. Similarly unnoticed, [[David Hilbert]] had derived the [[Einstein field equations]] by postulating the invariance of the [[Action (physics)|action]] under a general coordinate transformation. The importance of these symmetry invariances remained unnoticed until Weyl's work.

Inspired by Pauli's descriptions of connection between charge conservation and field theory driven by invariance, [[Chen Ning Yang]] sought a field theory for [[atomic nucleus|atomic nuclei]] binding based on conservation of nuclear [[isospin]].<ref name=Baggott40>{{Cite book |last=Baggott |first=J. E. |title=The quantum story: a history in 40 moments |date=2013 |publisher=Oxford Univ. Press |isbn=978-0-19-956684-6 |edition=Impression: 3 |location=Oxford}}</ref>{{rp|202}} In 1954, Yang and [[Robert Mills (physicist)|Robert Mills]] generalized the gauge invariance of electromagnetism, constructing a theory based on the action of the (non-abelian) SU(2) symmetry [[group (mathematics)|group]] on the [[isospin]] doublet of [[proton]]s and [[neutron]]s.<ref>{{cite journal |author=Yang C. N., Mills R. L. |title=Conservation of Isotopic Spin and Isotopic Gauge Invariance |journal=[[Phys. Rev.]] |volume=96 |year=1954 |issue=1 |pages=191–195 |doi=10.1103/PhysRev.96.191|bibcode = 1954PhRv...96..191Y |doi-access=free }}</ref> This is similar to the action of the [[U(1)]] group on the [[spinor]] [[field (physics)|field]]s of [[quantum electrodynamics]].

The [[Yang–Mills theory]] became the prototype theory to resolve some of the confusion in [[elementary particle physics]].
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]]. Asymptotic freedom was believed to be an important characteristic of strong interactions. This motivated searching for a strong force gauge theory. This theory, now known as [[quantum chromodynamics]], is a gauge theory with the action of the SU(3) group on the [[color charge|color]] triplet of [[quarks]]. The [[Standard Model]] unifies the description of electromagnetism, weak interactions and strong interactions in the language of gauge theory.

In the 1970s, [[Michael Atiyah]] began 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 manifold|differentiable]] classification of [[smooth function|smooth]] 4-[[manifold]]s is very different from their classification [[up to]] [[homeomorphism]].<ref>{{Cite journal |last=Donaldson |first=Simon K. |title=Self-dual connections and the topology of smooth 4-manifolds |journal=[[Bull. Amer. Math. Soc.]] |volume=8 |issue=1 |year=1983 |pages=81–83 |doi=10.1090/S0273-0979-1983-15090-5 | mr=0682827 |doi-access=free }}</ref> [[Michael Freedman]] used Donaldson's work to exhibit [[exotic R4|exotic '''R'''<sup>4</sup>]]s, that is, exotic [[differentiable structure]]s on [[Euclidean space|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]] that enabled the calculation of certain [[topology|topological]] invariants<ref>{{citation|last1=Seiberg|first1= N.|author-link1=Nathan Seiberg|last2= Witten|first2= E. |author-link2=Edward Witten|title=Electric-magnetic duality, monopole condensation, and confinement in ''N''=2 supersymmetric Yang–Mills theory|journal= Nuclear Physics B |volume= 426  |year=1994a|issue= 1|pages=19–52|doi=10.1016/0550-3213(94)90124-4|mr=1293681|arxiv=hep-th/9407087|bibcode=1994NuPhB.426...19S|s2cid= 14361074}}; {{citation|title=Erratum|journal= Nuclear Physics B |volume= 430  |year=1994|issue= 2|pages=485–486|doi=10.1016/0550-3213(94)00449-8|mr=1303306|bibcode=1994NuPhB.430..485.}}</ref><ref>{{citation|last1=Seiberg|first1= N.|author-link1=Nathan Seiberg|last2= Witten|first2= E. |author-link2=Edward Witten|title=Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD|journal= Nuclear Physics B |volume= 431  |year=1994b|issue= 3|pages=484–550|doi=10.1016/0550-3213(94)90214-3|mr=1306869|arxiv=hep-th/9408099|bibcode=1994NuPhB.431..484S|s2cid= 17584951}}</ref> (the [[Seiberg–Witten invariant]]s). These contributions to mathematics from gauge theory have led to a renewed interest in this area.

The importance of gauge theories in physics is exemplified in the success of the mathematical formalism in providing a unified framework to describe the [[quantum field theory|quantum field theories]] of [[electromagnetism]], the [[weak force]] and the [[strong force]]. This theory, known as the [[Standard model|Standard Model]], accurately describes experimental predictions regarding three of the four [[fundamental force]]s 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 [[general relativity]], are, in one way or another, gauge theories.

: ''See Jackson and Okun''<ref name=JacksonOkun>
{{cite journal
 |last1=Jackson |first1=JD
 |last2=Okun | first2=LB
 |title=Historical roots of gauge invariance
 |journal=Reviews of Modern Physics
 |volume=73
 |issue=3
 |pages=663
 |year=2001
 |doi=10.1103/RevModPhys.73.663
 |arxiv=hep-ph/0012061
 |bibcode=2001RvMP...73..663J
 |s2cid=8285663
 }}</ref>'' for early history of gauge and Pickering''<ref name=Pickering>
{{cite book
 |last=Pickering |first=A.
 |title=Constructing Quarks
 |publisher=[[University of Chicago Press]]
 |year=1984
 |isbn=0-226-66799-5
}}</ref>'' for more about the history of gauge and quantum field theories.''

== Description ==

=== Global and local symmetries ===

==== Global symmetry ====

In [[physics]], the mathematical description of any physical situation usually contains excess [[Degrees of freedom (physics and chemistry)|degrees of freedom]]; the same physical situation is equally well described by many equivalent mathematical configurations. For instance, in [[Newtonian dynamics]], if two configurations are related by a [[Galilean transformation]] (an [[inertial]] change of reference frame) they represent the same physical situation. These transformations form a [[group (mathematics)|group]] of "[[symmetry in physics|symmetries]]" of the theory, and a physical situation corresponds not to an individual mathematical configuration but to a class of configurations related to one another by this symmetry group.

This idea can be generalized to include local as well as global symmetries, analogous to much more abstract "changes of coordinates" in a situation where there is no preferred "[[inertial]]" coordinate system that covers the entire physical system. A gauge theory is a mathematical model that has symmetries of this kind, together with a set of techniques for making physical predictions consistent with the symmetries of the model.

==== Example of global symmetry ====
When a quantity occurring in the mathematical configuration is not just a number but has some geometrical significance, such as a velocity or an axis of rotation, its representation as numbers arranged in a vector or matrix is also changed by a coordinate transformation. For instance, if one description of a pattern of fluid flow states that the fluid velocity in the neighborhood of ({{nowrap|1=''x'' = 1}}, {{nowrap|1=''y'' = 0}}) is 1&nbsp;m/s in the positive ''x'' direction, then a description of the same situation in which the coordinate system has been rotated clockwise by 90 degrees states that the fluid velocity in the neighborhood of ({{nowrap|1=''x'' = 0}}, {{nowrap|1=''y''= −1}}) is 1&nbsp;m/s in the negative ''y'' direction. The coordinate transformation has affected both the coordinate system used to identify the ''location'' of the measurement and the basis in which its ''value'' is expressed. As long as this transformation is performed globally (affecting the coordinate basis in the same way at every point), the effect on values that represent the ''rate of change'' of some quantity along some path in space and time as it passes through point ''P'' is the same as the effect on values that are truly local to ''P''.

==== Local symmetry ====

===== Use of fiber bundles to describe local symmetries =====
In order to adequately describe physical situations in more complex theories, it is often necessary to introduce a "coordinate basis" for some of the objects of the theory that do not have this simple relationship to the coordinates used to label points in space and time. (In mathematical terms, the theory involves a [[fiber bundle]] in which the fiber at each point of the base space consists of possible coordinate bases for use when describing the values of objects at that point.) In order to spell out a mathematical configuration, one must choose a particular coordinate basis at each point (a ''local section'' of the fiber bundle) and express the values of the objects of the theory (usually "[[field theory (physics)|fields]]" in the physicist's sense) using this basis. Two such mathematical configurations are equivalent (describe the same physical situation) if they are related by a transformation of this abstract coordinate basis (a change of local section, or ''gauge transformation'').

In most gauge theories, the set of possible transformations of the abstract gauge basis at an individual point in space and time is a finite-dimensional Lie group. The simplest such group is [[U(1)]], which appears in the modern formulation of [[quantum electrodynamics#Mathematics|quantum electrodynamics (QED)]] via its use of [[complex number]]s. QED is generally regarded as the first, and simplest, physical gauge theory. The set of possible gauge transformations of the entire configuration of a given gauge theory also forms a group, the ''gauge group'' of the theory. An element of the gauge group can be parameterized by a smoothly varying function from the points of spacetime to the (finite-dimensional) Lie group, such that the value of the function and its derivatives at each point represents the action of the gauge transformation on the fiber over that point.

A gauge transformation with constant parameter at every point in space and time is analogous to a rigid rotation of the geometric coordinate system; it represents a [[global symmetry]] of the gauge representation. As in the case of a rigid rotation, this gauge transformation affects expressions that represent the rate of change along a path of some gauge-dependent quantity in the same way as those that represent a truly local quantity. A gauge transformation whose parameter is ''not'' a constant function is referred to as a [[local symmetry]]; its effect on expressions that involve a [[derivative]] is qualitatively different from that on expressions that do not. (This is analogous to a non-inertial change of reference frame, which can produce a [[Coriolis effect]].)

=== Gauge fields ===
The "gauge covariant" version of a gauge theory accounts for this effect by introducing a ''gauge field'' (in mathematical language, an [[Ehresmann connection]]) and formulating all rates of change in terms of the [[covariant derivative]] with respect to this connection. The gauge field becomes an essential part of the description of a mathematical configuration. A configuration in which the gauge field can be eliminated by a gauge transformation has the property that its [[field strength]] (in mathematical language, its [[curvature]]) is zero everywhere; a gauge theory is ''not'' limited to these configurations. In other words, the distinguishing characteristic of a gauge theory is that the gauge field does not merely compensate for a poor choice of coordinate system; there is generally no gauge transformation that makes the gauge field vanish.

When analyzing the [[Dynamics (physics)|dynamics]] of a gauge theory, the gauge field must be treated as a dynamical variable, similar to other objects in the description of a physical situation. In addition to its [[Fundamental interaction|interaction]] with other objects via the covariant derivative, the gauge field typically contributes [[energy]] in the form of a "self-energy" term. One can obtain the equations for the gauge theory by:
* starting from a naïve [[ansatz]] without the gauge field (in which the derivatives appear in a "bare" form);
* listing those global symmetries of the theory that can be characterized by a continuous parameter (generally an abstract equivalent of a rotation angle);
* computing the correction terms that result from allowing the symmetry parameter to vary from place to place; and
* reinterpreting these correction terms as couplings to one or more gauge fields, and giving these fields appropriate self-energy terms and dynamical behavior.
This is the sense in which a gauge theory "extends" a global symmetry to a local symmetry, and closely resembles the historical development of the gauge theory of gravity known as [[general relativity]].

=== Physical experiments ===
Gauge theories used to model the results of physical experiments engage in:
* limiting the universe of possible configurations to those consistent with the information used to set up the experiment, and then
* computing the probability distribution of the possible outcomes that the experiment is designed to measure.
We cannot express the mathematical descriptions of the "setup information" and the "possible measurement outcomes", or the "boundary conditions" of the experiment, without reference to a particular coordinate system, including a choice of gauge. One assumes an adequate experiment isolated from "external" influence that is itself a gauge-dependent statement. Mishandling gauge dependence calculations in boundary conditions is a frequent source of [[anomaly (physics)|anomalies]], and approaches to anomaly avoidance classifies gauge theories{{clarify|date=February 2014}}.

=== Continuum theories ===
The two gauge theories mentioned above, continuum electrodynamics and general relativity, are continuum field theories. The techniques of calculation in a [[Continuum mechanics|continuum theory]] implicitly assume that:
* given a completely fixed choice of gauge, the boundary conditions of an individual configuration are completely described
* given a completely fixed gauge and a complete set of boundary conditions, the least action determines a unique mathematical configuration and therefore a unique physical situation consistent with these bounds
* fixing the gauge introduces no anomalies in the calculation, due either to gauge dependence in describing partial information about boundary conditions or to incompleteness of the theory.
Determination of the likelihood of possible measurement outcomes proceed by:
* establishing a probability distribution over all physical situations determined by boundary conditions consistent with the setup information
* establishing a probability distribution of measurement outcomes for each possible physical situation
* [[convolution|convolving]] these two probability distributions to get a distribution of possible measurement outcomes consistent with the setup information
These assumptions have enough validity across a wide range of energy scales and experimental conditions to allow these theories to make accurate predictions about almost all of the phenomena encountered in daily life: light, heat, and electricity, eclipses, spaceflight, etc. They fail only at the smallest and largest scales due to omissions in the theories themselves, and when the mathematical techniques themselves break down, most notably in the case of [[turbulence]] and other [[chaos (physics)|chaotic]] phenomena.

=== Quantum field theories ===
{{main|Quantum field theory}}
Other than these classical continuum field theories, the most widely known gauge theories are [[quantum field theories]], including [[quantum electrodynamics]] and the [[Standard Model]] of elementary particle physics. The starting point of a quantum field theory is much like that of its continuum analog: a gauge-covariant [[action integral]] that characterizes "allowable" physical situations according to the [[principle of least action]]. However, continuum and quantum theories differ significantly in how they handle the excess degrees of freedom represented by gauge transformations. Continuum theories, and most pedagogical treatments of the simplest quantum field theories, use a [[gauge fixing]] prescription to reduce the orbit of mathematical configurations that represent a given physical situation to a smaller orbit related by a smaller gauge group (the global symmetry group, or perhaps even the trivial group).

More sophisticated quantum field theories, in particular those that involve a non-abelian gauge group, break the gauge symmetry within the techniques of [[Perturbation theory (quantum mechanics)|perturbation theory]] by introducing additional fields (the [[Faddeev–Popov ghost]]s) and counterterms motivated by [[anomaly cancellation]], in an approach known as [[BRST quantization]]. While these concerns are in one sense highly technical, they are also closely related to the nature of measurement, the limits on knowledge of a physical situation, and the interactions between incompletely specified experimental conditions and incompletely understood physical theory.{{Citation needed|date=November 2014}} The mathematical techniques that have been developed in order to make gauge theories tractable have found many other applications, from [[solid-state physics]] and [[crystallography]] to [[low-dimensional topology]].

== Classical gauge theory ==

=== Classical electromagnetism ===

In [[electrostatics]], one can either discuss the electric field, '''E''', or its corresponding [[electric potential]], ''V''. Knowledge of one makes it possible to find the other, except that potentials differing by a constant, <math>V \mapsto V+C</math>, correspond to the same electric field. This is because the electric field relates to ''changes'' in the potential from one point in space to another, and the constant ''C'' would cancel out when subtracting to find the change in potential. In terms of [[vector calculus]], the electric field is the [[gradient]] of the potential, <math>\mathbf{E} = -\nabla V</math>. Generalizing from static electricity to electromagnetism, we have a second potential, the [[magnetic vector potential|vector potential]] '''A''', with
: <math>\begin{align}
  \mathbf{E} &= -\nabla V - \frac{\partial \mathbf{A}}{\partial t}\\
  \mathbf{B} &=  \nabla \times \mathbf{A}
\end{align}</math>

The general gauge transformations now become not just <math>V \mapsto V+C</math> but
: <math>\begin{align}
  \mathbf{A} &\mapsto \mathbf{A} + \nabla f\\
           V &\mapsto V - \frac{\partial f}{\partial t}
\end{align}</math>
where ''f'' is any twice continuously differentiable function that depends on position and time. The electromagnetic fields remain the same under the gauge transformation.

=== Example: scalar O(''n'') gauge theory ===
: ''The remainder of this section requires some familiarity with [[classical field theory|classical]] or [[quantum field theory]], and the use of [[Lagrangian (field theory)|Lagrangian]]s.''
: ''Definitions in this section: [[gauge group]], [[gauge field]], [[interaction Lagrangian]], [[gauge boson]].''

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

Consider a set of <math>n</math> non-interacting real [[field (physics)|scalar field]]s, with equal masses ''m''. This system is described by an [[action (physics)|action]] that is the sum of the (usual) action for each scalar field <math>\varphi_i</math>
: <math> \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]</math>

The Lagrangian (density) can be compactly written as
: <math>\ \mathcal{L} = \frac{1}{2} (\partial_\mu \Phi)^\mathsf{T} \partial^\mu \Phi - \frac{1}{2}m^2 \Phi^\mathsf{T} \Phi </math>
by introducing a [[vector (geometry)|vector]] of fields
: <math>\ \Phi^\mathsf{T} = ( \varphi_1, \varphi_2,\ldots, \varphi_n) </math>

The term <math>\partial_\mu\Phi</math> is the [[partial derivative]] of <math>\Phi</math> along dimension <math>\mu</math>.

It is now transparent that the Lagrangian is invariant under the transformation
: <math>\ \Phi \mapsto \Phi' = G \Phi </math>
whenever ''G'' is a ''constant'' [[matrix (mathematics)|matrix]] belonging to the ''n''-by-''n'' [[orthogonal group]] O(''n''). This is seen to preserve the Lagrangian, since the derivative of <math>\Phi'</math> transforms identically to <math>\Phi</math> and both quantities appear inside [[dot product]]s in the Lagrangian (orthogonal transformations preserve the dot product).
: <math>\ (\partial_\mu \Phi) \mapsto (\partial_\mu \Phi)' = G \partial_\mu \Phi </math>

This characterizes 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-structure]]s. Incidentally, [[Noether's theorem]] implies that invariance under this group of transformations leads to the conservation of the ''currents''
: <math>\ J^{a}_{\mu} = i\partial_\mu \Phi^\mathsf{T} T^{a} \Phi</math>
where the ''T<sup>a</sup>'' matrices are [[generating set of a group|generator]]s of the SO(''n'') group. There is one conserved current for every generator.

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 [[spacetime]] [[coordinate]]s ''x''.

In this case, the ''G'' matrices do not "pass through" the derivatives, when ''G'' = ''G''(''x''),
: <math>\ \partial_\mu (G \Phi) \neq G (\partial_\mu \Phi) </math>

The failure of the derivative to commute with "G" introduces an additional term (in keeping with the product rule), which spoils the invariance of the Lagrangian. In order to rectify this we define a new derivative operator such that the derivative of <math>\Phi'</math> again transforms identically with <math>\Phi</math>
: <math>\ (D_\mu \Phi)' = G D_\mu \Phi</math>

This new "derivative" is called a [[Gauge covariant derivative|(gauge) covariant derivative]] and takes the form
: <math>\ D_\mu = \partial_\mu - i g A_\mu </math>
where ''g'' is called the coupling constant; a quantity defining the strength of an interaction.
After a simple calculation we can see that the '''gauge field''' ''A''(''x'') must transform as follows
: <math>\ A'_\mu = G A_\mu G^{-1} - \frac{i}{g} (\partial_\mu G)G^{-1}</math>

The gauge field is an element of the Lie algebra, and can therefore be expanded as
: <math>\ A_{\mu} =  \sum_a A_{\mu}^a T^a </math>

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

Finally, we now have a ''locally gauge invariant'' Lagrangian
: <math>\ \mathcal{L}_\mathrm{loc} = \frac{1}{2} (D_\mu \Phi)^\mathsf{T} D^\mu \Phi -\frac{1}{2}m^2 \Phi^\mathsf{T} \Phi</math>

Pauli uses the term ''gauge transformation of the first type'' to mean the transformation of <math>\Phi</math>, while the compensating transformation in <math>A</math> is called a ''gauge transformation of the second type''.

[[Image:Feynman-Diagram.svg|thumb|right|200px|[[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'''
: <math>\ \mathcal{L}_\mathrm{int} = i\frac{g}{2} \Phi^\mathsf{T} A_{\mu}^\mathsf{T} \partial^\mu \Phi + i\frac{g}{2}  (\partial_\mu \Phi)^\mathsf{T} A^{\mu} \Phi - \frac{g^2}{2} (A_\mu \Phi)^\mathsf{T} A^\mu \Phi</math>

This term introduces [[Fundamental interaction|interaction]]s between the ''n'' scalar fields just as a consequence of the demand for local gauge invariance. However, to make this interaction physical and not completely arbitrary, the mediator ''A''(''x'') needs to propagate in space. That is dealt with in the next section by adding yet another term, <math>\mathcal{L}_{\mathrm{gf}}</math>, to the Lagrangian. In the [[Quantization (physics)|quantized]] version of the obtained [[classical field theory]], the [[quantum|quanta]] of the gauge field ''A''(''x'') are called [[gauge boson]]s. The interpretation of the interaction Lagrangian in quantum field theory is of [[scalar (physics)|scalar]] [[boson]]s interacting by the exchange of these gauge bosons.

=== Yang–Mills Lagrangian for the gauge field ===
{{Main|Yang–Mills theory}}
The picture of a classical gauge theory developed in the previous section is almost complete, except for the fact that to define the covariant derivatives ''D'', one needs to know the value of the gauge field <math>A(x)</math> at all spacetime 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 that generates this field equation is locally gauge invariant as well, one possible form for the gauge field Lagrangian is
: <math>\mathcal{L}_\text{gf} = -\frac{1}{2} \operatorname{tr}\left(F^{\mu \nu} F_{\mu \nu}\right) = -\frac{1}{4} F^{a \mu \nu} F^a_{\mu \nu}</math>
where the <math>F^a_{\mu \nu}</math> are obtained from potentials <math>A^a_\mu</math>, being the components of <math>A(x)</math>, by
: <math>F_{\mu \nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g\sum_{b,c} f^{abc} A_\mu^b A_\nu^c</math>
and the <math>f^{abc}</math> are the [[structure constants]] of the Lie algebra of the generators of the gauge group. This formulation of the Lagrangian is called a '''Yang–Mills action'''. Other gauge invariant actions also exist (e.g., [[nonlinear electrodynamics]], [[Born–Infeld action]], [[Chern–Simons model]], [[strong CP problem|theta term]], etc.).

In this Lagrangian term there is no field whose transformation counterweighs the one of <math>A</math>. Invariance of this term under gauge transformations is a particular case of ''a priori'' classical (geometrical) symmetry. This symmetry must be restricted in order to perform quantization, the procedure being denominated [[gauge fixing]], but even after restriction, gauge transformations may be possible.<ref>J. J. Sakurai, ''Advanced Quantum Mechanics'', Addison-Wesley, 1967, sect. 1–4.</ref>

The complete Lagrangian for the gauge theory is now
: <math>\mathcal{L} = \mathcal{L}_\text{loc} + \mathcal{L}_\text{gf} = \mathcal{L}_\text{global} + \mathcal{L}_\text{int} + \mathcal{L}_\text{gf}</math>

=== 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 that generates the electron field's [[Dirac equation]] is
: <math>\mathcal{S} = \int \bar{\psi}\left(i \hbar c \, \gamma^\mu \partial_\mu - mc^2\right) \psi \, \mathrm{d}^4 x</math>

The global symmetry for this system is
: <math>\psi \mapsto e^{i \theta} \psi</math>

The gauge group here is [[U(1)]], just rotations of the [[complex number|phase angle]] of the field, with the particular rotation determined by the constant {{math|''θ''}}.

"Localising" this symmetry implies the replacement of {{math|''θ''}} by {{math|''θ''(''x'')}}. An appropriate covariant derivative is then
: <math>D_\mu = \partial_\mu - i \frac{e}{\hbar} A_\mu</math>

Identifying the "charge" {{math|''e''}} (not to be confused with the mathematical constant [[E (mathematical constant)|e]] in the symmetry description) with the usual [[electric charge]] (this is the origin of the usage of the term in gauge theories), and the gauge field {{math|''A''(''x'')}} with the four-[[vector potential]] of the [[electromagnetic field]] results in an interaction Lagrangian
: <math>\mathcal{L}_\text{int} = \frac{e}{\hbar}\bar{\psi}(x) \gamma^\mu \psi(x) A_\mu(x) = J^\mu(x) A_\mu(x)</math>
where <math>J^\mu(x) = \frac{e}{\hbar}\bar{\psi}(x) \gamma^\mu \psi(x)</math> is the electric current [[four vector]] in the [[Dirac equation|Dirac field]]. The [[gauge principle]] is therefore seen to naturally introduce the so-called [[minimal coupling]] of the electromagnetic field to the electron field.

Adding a Lagrangian for the gauge field <math>A_\mu(x)</math> in terms of the [[electromagnetic tensor|field strength tensor]] exactly as in electrodynamics, one obtains the Lagrangian used as the starting point in [[quantum electrodynamics]].
: <math>\mathcal{L}_\text{QED} = \bar{\psi}\left(i\hbar c \, \gamma^\mu D_\mu - mc^2\right)\psi - \frac{1}{4 \mu_0}F_{\mu\nu}F^{\mu\nu}</math>

{{see also|Dirac equation|Maxwell's equations|Quantum electrodynamics}}

== Mathematical formalism ==
{{See also|Gauge theory (mathematics)}}
Gauge theories are usually discussed in the language of [[differential geometry]]. Mathematically, a ''gauge'' is just a choice of a (local) [[section (fiber bundle)|section]] of some [[principal bundle]]. A '''gauge transformation''' is just a transformation between two such sections.

Although gauge theory is dominated by the study of [[connection form|connections]] (primarily because it's mainly studied by [[high-energy physics|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 representation]]s (i.e., affine [[module (mathematics)|modules]]) of the gauge transformations can be classified as sections of a [[jet bundle]] satisfying certain properties. There are representations that transform covariantly pointwise (called by physicists gauge transformations of the first kind), representations that transform as a [[connection form]] (called by physicists gauge transformations of the second kind, an affine representation)—and other more general representations, such as the B field in [[BF theory]]. There are more general [[nonlinear realization|nonlinear representations]] (realizations), but these are extremely complicated. Still, [[nonlinear sigma model]]s transform nonlinearly, so there are applications.

If there is a [[principal bundle]] ''P'' whose [[Fibre bundle|base space]] is [[space]] or [[spacetime]] and [[Fibre bundle|structure group]] is a Lie group, then the sections of ''P'' form a [[principal homogeneous space]] of the group of gauge transformations.

[[Connection form|Connection]]s (gauge connection) define this principal bundle, yielding a [[covariant derivative]] ∇ in each [[associated vector bundle]]. If a local frame is chosen (a local basis of sections), then this covariant derivative is represented by the [[connection form]] ''A'', a Lie algebra-valued [[Differential form|1-form]], which is called the '''gauge potential''' in [[physics]]. This is evidently not an intrinsic but a frame-dependent quantity. The [[curvature form]] ''F'', a Lie algebra-valued [[Differential form|2-form]] that is an intrinsic quantity, is constructed from a connection form by
: <math>\mathbf{F} = \mathrm{d}\mathbf{A} + \mathbf{A}\wedge\mathbf{A}</math>
where d stands for the [[exterior derivative]] and <math>\wedge</math> stands for the [[wedge product]]. (<math>\mathbf{A}</math> is an element of the vector space spanned by the generators <math>T^{a}</math>, and so the components of <math>\mathbf{A}</math> do not commute with one another. Hence the wedge product <math>\mathbf{A}\wedge\mathbf{A}</math> does not vanish.)

Infinitesimal gauge transformations form a Lie algebra, which is characterized by a smooth Lie-algebra-valued [[scalar (mathematics)|scalar]], ε. Under such an [[infinitesimal]] gauge transformation,
: <math>\delta_\varepsilon \mathbf{A} = [\varepsilon,\mathbf{A}] - \mathrm{d}\varepsilon</math>
where <math>[\cdot,\cdot]</math> is the Lie bracket.

One nice thing is that if <math>\delta_\varepsilon X = \varepsilon X</math>, then <math>\delta_\varepsilon DX = \varepsilon DX</math> where D is the covariant derivative
: <math>DX\ \stackrel{\mathrm{def}}{=}\ \mathrm{d}X + \mathbf{A}X</math>

Also, <math>\delta_\varepsilon \mathbf{F} = [\varepsilon, \mathbf{F}]</math>, which means <math>\mathbf{F}</math> transforms covariantly.

Not all gauge transformations can be generated by [[infinitesimal]] gauge transformations in general. An example is when the [[Fibre bundle|base manifold]] is a [[Compact space|compact]] [[manifold]] without [[Boundary (topology)|boundary]] such that the [[homotopy]] class of mappings from that [[manifold]] to the Lie group is nontrivial. See [[instanton]] for an example.

The ''Yang–Mills action'' is now given by
: <math>\frac{1}{4g^2}\int \operatorname{Tr}[{\star}F \wedge F]</math>
where <math>{\star}</math> is the [[Hodge star operator]] and the integral is defined as in [[differential form|differential geometry]].

A quantity which is '''gauge-invariant''' (i.e., [[Invariant (physics)|invariant]] under gauge transformations) is the [[Wilson loop]], which is defined over any closed path, γ, as follows:
: <math>\chi^{(\rho)}\left(\mathcal{P}\left\{e^{\int_\gamma A}\right\}\right)</math>
where ''χ'' is the [[group representation|character]] of a complex [[group representation|representation]] ρ and <math>\mathcal{P}</math> represents the path-ordered operator.

The formalism of gauge theory carries over to a general setting. For example, it is sufficient to ask that a [[vector bundle]] have a [[metric connection]]; when one does so, one finds that the metric connection satisfies the Yang–Mills equations of motion.

== 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 allows simplification of some computations: for example [[Ward identities]] connect different [[renormalization]] constants.

=== Methods and aims ===
The first gauge theory 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 now handled by a variety of means. Methods for quantization are covered in the article on [[quantization (physics)|quantization]].

The main point to quantization is to be able to compute [[probability amplitude|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.

When the [[running coupling]] of the theory is small enough, then all required quantities may be computed in [[perturbation theory]]. Quantization schemes intended to simplify such computations (such as [[quantization (physics)#Canonical quantization|canonical quantization]]) may be called '''perturbative quantization schemes'''. At present some of these methods lead to the most precise experimental tests of gauge theories.

However, in most gauge theories, there are many interesting questions which are non-perturbative. Quantization schemes suited 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.

=== Anomalies ===
Some of the symmetries of the classical theory are then seen not to hold in the quantum theory; a phenomenon called an '''[[anomaly (physics)|anomaly]]'''. Among the most well known are:
* The [[scale anomaly]], which gives rise to a ''running coupling constant''. In QED this gives rise to the phenomenon of the [[Landau pole]]. In [[quantum chromodynamics]] (QCD) this leads to [[asymptotic freedom]].
* The [[chiral anomaly]] in either chiral or vector field theories with fermions. This has close connection with [[topology]] through the notion of [[instanton]]s. In QCD this anomaly causes the decay of a [[pion]] to two [[photon]]s.
* The [[gauge anomaly]], which must cancel in any consistent physical theory. In the [[electroweak theory]] this cancellation requires an equal number of [[quark]]s and [[lepton]]s.

== Pure gauge ==
A pure gauge is the set of field configurations obtained by a [[gauge transformation]] on the null-field configuration, i.e., a gauge transform of zero. So it is a particular "gauge orbit" in the field configuration's space.

Thus, in the abelian case, where <math>A_\mu (x) \rightarrow A'_\mu(x) = A_\mu(x)+ \partial_\mu f(x)</math>, the pure gauge is just the set of field configurations <math>A'_\mu(x) = \partial_\mu f(x)</math> for all {{math|''f''(''x'')}}.

== See also ==
{{colbegin}}
* [[Gauge principle]]
* [[Aharonov–Bohm effect]]
* [[Coulomb gauge]]
* [[Electroweak theory]]
* [[Gauge covariant derivative]]
* [[Gauge fixing]]
* [[Gauge gravitation theory]]
* [[Gauge group (mathematics)]]
* [[Kaluza–Klein theory]]
* [[Lorenz gauge]]
* [[Quantum chromodynamics]]
* [[Gluon field]]
* [[Gluon field strength tensor]]
* [[Quantum electrodynamics]]
* [[Electromagnetic four-potential]]
* [[Electromagnetic tensor]]
* [[Quantum field theory]]
* [[Standard Model]]
* [[Standard Model (mathematical formulation)]]
* [[Symmetry breaking]]
* [[Symmetry in physics]]
* [[Charge (physics)]]
* [[Symmetry in quantum mechanics]]
* [[Fock symmetry in theory of hydrogen|Fock symmetry]]
* [[Ward identities]]
* [[Yang–Mills theory]]
* [[Yang–Mills existence and mass gap]]
* [[1964 PRL symmetry breaking papers]]
* [[Gauge theory (mathematics)]]
{{colend}}

== References ==
{{reflist}}

== Bibliography ==

; General readers :
* Schumm, Bruce (2004) ''[https://archive.org/details/deepdownthingsbr00schu <!-- quote="gauge theory". --> Deep Down Things]''. Johns Hopkins University Press. Esp. chpt. 8. A serious attempt by a physicist to explain gauge theory and the [[Standard Model]] with little formal mathematics.

; Texts :
* {{cite book|last1=Greiner|first1=Walter|last2=Müller|first2=Berndt|title=Gauge Theory of Weak Interactions|publisher=[[Springer Science+Business Media|Springer]]|year=2000|isbn=3-540-67672-4}}
* {{cite book |first1=T.-P. |last1=Cheng|author2-link=Ling-Fong Li |first2=L.-F. |last2=Li |title=Gauge Theory of Elementary Particle Physics |publisher=[[Oxford University Press]] |year=1983 |isbn=0-19-851961-3 }}
* {{cite book |last=Frampton |first=P. |author-link=Paul Frampton |title=Gauge Field Theories |edition=3rd |publisher=[[Wiley-VCH]] |year=2008 }}
* {{cite book |last=Kane |first=G.L. |title=Modern Elementary Particle Physics |publisher=Perseus Books |year=1987 |isbn=0-201-11749-5}}

; Articles :
* {{cite arXiv |year=1997 |eprint=hep-ph/9705211 |title= Introduction to Gauge Theories |last1=Becchi | first1=C.}}
* {{cite web |last=Gross |first=D. |author-link=David Gross |year=1992 |url=http://psroc.phys.ntu.edu.tw/cjp/download.php?type=paper&vol=30&num=7&page=955 |title=Gauge theory – Past, Present and Future|access-date=2009-04-23}}
* {{cite journal |doi=10.1119/1.1491265 |arxiv=physics/0204034 |title=From Lorenz to Coulomb and other explicit gauge transformations |last1=Jackson | first1=J.D. | author-link=John David Jackson (physicist) |year=2002 |journal=Am. J. Phys. |volume=70 |pages= 917–928 |bibcode = 2002AmJPh..70..917J |issue=9 |s2cid=119652556 }}
* {{cite arXiv |year=1999 |eprint=math-ph/9902027 |title=Preparation for Gauge Theory |last1=Svetlichny |first1=George }}

== External links ==
{{wikiquote}}
* {{springer|title=Gauge transformation|id=p/g043400}}
* [https://web.archive.org/web/20131109161828/http://wiki.math.toronto.edu/DispersiveWiki/index.php/Yang-Mills Yang–Mills equations on DispersiveWiki]
* [http://www.scholarpedia.org/article/Gauge_theories Gauge theories on Scholarpedia]

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