Definitions

# Gauge theory

Gauge theory is a peculiar quantum field theory where the Lagrangian is invariant under certain transformations.

The transformations (called local gauge transformations) form a Lie group which is referred to as the symmetry group or the gauge group of the theory. For each group parameter there is a corresponding vector field called gauge field which helps to make the Lagrangian invariant. The quanta of the gauge field are called gauge bosons.

If the symmetry group is non-commutative, the gauge theory is referred to as non-abelian or Yang-Mills theory.

Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1) and one gauge field, the electromagnetic field, with the photon being the gauge boson.

The standard model is a non-abelian gauge theory with the symmetry group U(1)×SU(2)×SU(3) and twelve gauge bosons: the photon, three weak bosons $Z^0$ and $W^pm$; and eight gluons.

## Description

In physics, the mathematical description of any physical situation usually contains excess 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 of "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.

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 (x=1, y=0) is 1 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 will state that the fluid velocity in the neighborhood of (x=0, y=1) is 1 m/s in the positive 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.

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 does 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 "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 continuum electrodynamics using complex numbers; this 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, whose value 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 don't. (This is analogous to a non-inertial change of reference frame, which can produce a Coriolis effect.)

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 of a gauge theory, the gauge field must be treated as a dynamical variable, similarly to other objects in the description of a physical situation. In addition to its 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.

Gauge theories are used to model the results of physical experiments, essentially by:

• 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.

The mathematical descriptions of the "setup information" and the "possible measurement outcomes" (loosely speaking, the "boundary conditions" of the experiment) are generally not expressible without reference to a particular coordinate system, including a choice of gauge. (If nothing else, one assumes that the experiment has been adequately isolated from "external" influence, which is itself a gauge-dependent statement.) Mishandling gauge dependence in boundary conditions is a frequent source of anomalies in gauge theory calculations, and gauge theories can be broadly classified by their approaches to anomaly avoidance.

The two gauge theories mentioned above (continuum electrodynamics and general relativity) are examples of continuum field theories. The techniques of calculation in a continuum theory implicitly assume that:

• given a completely fixed choice of gauge, the boundary conditions of an individual configuration can in principle be completely described;
• given a completely fixed gauge and a complete set of boundary conditions, the principle of least action determines a unique mathematical configuration (and therefore a unique physical situation) consistent with these bounds;
• the likelihood of possible measurement outcomes can be determined by:
• establishing a probability distribution over all physical situations determined by boundary conditions that are consistent with the setup information,
• establishing a probability distribution of measurement outcomes for each possible physical situation, and
• convolving these two probability distributions to get a distribution of possible measurement outcomes consistent with the setup information; and
• 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.

These assumptions are close enough to valid, 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, from light, heat, and electricity to eclipses and spaceflight. 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 chaotic phenomena).

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 which 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 which involve a non-abelian gauge group, break the gauge symmetry within the techniques of perturbation theory by introducing additional fields (the Faddeev-Popov ghosts) 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. 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.

## History

The earliest field theory having a gauge symmetry was Maxwell's formulation of electrodynamics in 1864. The importance of this symmetry remained unnoticed in the earliest formulations. Similarly unnoticed, Hilbert had derived the Einstein field equations by postulating the invariance of the action under a general coordinate transformation. Later Hermann Weyl, in an attempt to unify general relativity and electromagnetism, conjectured (incorrectly, as it turned out) that Eichinvarianz or invariance under the change of scale (or "gauge") might also be a local symmetry of general relativity. After the development of quantum mechanics, Weyl, Vladimir Fock and Fritz London modified gauge by replacing the scale factor with a complex quantity and turned the scale transformation into a change of phase—a U(1) gauge symmetry). This explained the electromagnetic field effect on the wave function of a charged quantum mechanical particle. This was the first widely recognised gauge theory. It was popularised by Pauli in the 1940s, e.g. R.M. P.13, 203

In 1954, 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. This is 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.

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 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, Sir 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 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.

An extensive historical discussion can be found in Woit.

## Explanation

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. The requirement of local symmetry is much more strict than the requirement of global symmetry. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in space-time. 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.

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.

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).

There is a certain inaccuracy in the way the term symmetry is used in some physics literature, especially in 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 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 represent 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 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.

## 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 will 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, or by dropping something).

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.

In contrast, if some symmetry could be defined arbitrarily from one position to the next, that would be a local gauge symmetry. In fact, the example above (of electromagnetism) is actually a local gauge symmetry where the particular transformation of the potential of the electric field is just the constant one (a local symmetry is a global one but not vice versa); there are an infinite number of transformations of the electromagnetic field's potential A that are not constant but can be made consistent as explained below. (Although the number of transformations is infinite, the number of classes of transformation is finite--just 1 class, the addition of the gradient of a scalar function.)

## Classical gauge theory

This section requires some familiarity with classical or quantum field theory, and the use of Lagrangians.

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

$mathcal\left\{S\right\} = int , mathrm\left\{d\right\}^4 x sum_\left\{i=1\right\}^n left\left[frac\left\{1\right\}\left\{2\right\} partial_mu varphi_i partial^mu varphi_i - frac\left\{1\right\}\left\{2\right\}m^2 varphi_i^2 right\right].$

The Lagrangian (density) can be compactly written as

$mathcal\left\{L\right\} = frac\left\{1\right\}\left\{2\right\} \left(partial_mu Phi\right)^T partial^mu Phi - frac\left\{1\right\}\left\{2\right\}m^2 Phi^T Phi$

by introducing a vector of fields

$Phi = \left(varphi_1, varphi_2,ldots, varphi_n\right)^T .$

The term $partial_mu$ is Einstein notation for the partial derivative of $Phi$ in each of the four dimensions. It is now transparent that the Lagrangian is invariant under the transformation

$Phi mapsto G Phi$

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

$J^\left\{a\right\}_\left\{mu\right\} = ipartial_mu Phi^T T^\left\{a\right\} Phi$

where the Ta matrices are generators 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 space-time coordinates x.

Unfortunately, the G matrices do not "pass through" the derivatives. When G = G(x),

$partial_mu \left(G Phi\right)^T partial^mu \left(G Phi\right) neq partial_mu Phi^T partial^mu Phi.$

This suggests defining a "derivative" D with the property

$D_mu \left(G\left(x\right) Phi\left(x\right)\right) = G\left(x\right) D_mu Phi.$

It can be checked that such a "derivative" (called a covariant derivative) is

$D_mu = partial_mu + g A_mu\left(x\right)$

where the gauge field A(x) is defined to have the transformation law

$A_\left\{mu\right\}\left(x\right) mapsto G\left(x\right)A_\left\{mu\right\}\left(x\right)G^\left\{-1\right\}\left(x\right) - frac\left\{1\right\}\left\{g\right\} \left(partial_mu G\left(x\right)\right) G^\left\{-1\right\}\left(x\right)$

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

$A_\left\{mu\right\}\left(x\right)= sum_a A_\left\{mu\right\}^a \left(x\right) T^a$

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

Finally, we now have a locally gauge invariant Lagrangian

$mathcal\left\{L\right\}_mathrm\left\{loc\right\} = frac\left\{1\right\}\left\{2\right\} \left(D_mu Phi\right)^T D^mu Phi -frac\left\{1\right\}\left\{2\right\}m^2 Phi^T Phi.$

Pauli calls gauge transformation of the first type to the one applied to fields as $Phi$, while the compensating transformation in $A$ is said to be a gauge transformation of the second type.

The difference between this Lagrangian and the original globally gauge-invariant Lagrangian is seen to be the interaction Lagrangian

$mathcal\left\{L\right\}_mathrm\left\{int\right\} = frac\left\{g\right\}\left\{2\right\} Phi^T A_\left\{mu\right\}^T partial^mu Phi + frac\left\{g\right\}\left\{2\right\} \left(partial_mu Phi\right)^T A^\left\{mu\right\} Phi + frac\left\{g^2\right\}\left\{2\right\} \left(A_mu Phi\right)^T A^mu Phi.$

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.

### 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 $A\left(x\right)$ 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

$mathcal\left\{L\right\}_mathrm\left\{gf\right\} = - frac\left\{1\right\}\left\{2\right\} operatorname\left\{Tr\right\}\left(F^\left\{mu nu\right\} F_\left\{mu nu\right\}\right)$

with

$F_\left\{mu nu\right\} = frac\left\{1\right\}\left\{ig\right\}\left[D_mu, D_nu\right]$

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.).

Note that in this Lagrangian there is not a field $Phi$ 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).

The complete Lagrangian for the O(n) gauge theory is now

$mathcal\left\{L\right\} = mathcal\left\{L\right\}_mathrm\left\{loc\right\} + mathcal\left\{L\right\}_mathrm\left\{gf\right\} = mathcal\left\{L\right\}_mathrm\left\{global\right\} + mathcal\left\{L\right\}_mathrm\left\{int\right\} + mathcal\left\{L\right\}_mathrm\left\{gf\right\}$

### 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

$mathcal\left\{S\right\} = int barpsi\left(i hbar c , gamma^mu partial_mu - m c^2 \right) psi , mathrm\left\{d\right\}^4x.$

The global symmetry for this system is

$psi mapsto e^\left\{i theta\right\} psi.$

The gauge group here is U(1), just the phase angle of the field, with a constant θ.

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

An appropriate covariant derivative is then

$D_mu = partial_mu - i frac\left\{e\right\}\left\{hbar\right\} A_mu.$

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

$mathcal\left\{L\right\}_mathrm\left\{int\right\} = frac\left\{e\right\}\left\{hbar\right\}barpsi\left(x\right) gamma^mu psi\left(x\right) A_\left\{mu\right\}\left(x\right) = J^\left\{mu\right\}\left(x\right)$
A_{mu}(x).

where $J^\left\{mu\right\}\left(x\right)$ is the usual four vector electric current density. 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 $A_\left\{mu\right\}\left(x\right)$ 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.

$mathcal\left\{L\right\}_\left\{mathrm\left\{QED\right\}\right\} = barpsi\left(ihbar c , gamma^mu D_mu - m c^2 \right)psi - frac\left\{1\right\}\left\{4 mu_0\right\}F_\left\{munu\right\}F^\left\{munu\right\}.$

## 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.

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 representations which transform covariantly pointwise (called by physicists gauge transformations of the first kind), representations which transform as a connection form (called by physicists gauge transformations of the second kind) (this is an affine representation) and other more general representations, such as the B field in BF theory. And of course, we can consider more general nonlinear representations (realizations), but that is extremely complicated. But still, nonlinear sigma models transform nonlinearly, so there are applications.

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.

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

$bold\left\{F\right\}=mathrm\left\{d\right\}bold\left\{A\right\}+bold\left\{A\right\}wedgebold\left\{A\right\}$

where d stands for the exterior derivative and $wedge$ stands for the wedge product. ($bold\left\{A\right\}$ is an element of the vector space spanned by the generators $T^\left\{a\right\}$, and so the components of $bold\left\{A\right\}$ do not commute with one another. Hence the wedge product $bold\left\{A\right\}wedgebold\left\{A\right\}$ does not vanish.)

Infinitesimal gauge transformations form a Lie algebra, which is characterized by a smooth Lie algebra valued scalar, ε. Under such an infinitesimal gauge transformation,

$delta_varepsilon bold\left\{A\right\}=\left[varepsilon,bold\left\{A\right\}\right]-mathrm\left\{d\right\}varepsilon$

where $\left[cdot,cdot\right]$ is the Lie bracket.

One nice thing is that if $delta_varepsilon X=varepsilon X$, then $delta_varepsilon DX=varepsilon DX$ where D is the covariant derivative

$DX stackrel\left\{mathrm\left\{def\right\}\right\}\left\{=\right\} mathrm\left\{d\right\}X+bold\left\{A\right\}X.$

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

Not all gauge transformations can be generated by infinitesimal gauge transformations in general. An example is 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.

The Yang-Mills action is now given by

$frac\left\{1\right\}\left\{4g^2\right\}int operatorname\left\{Tr\right\}\left[*Fwedge F\right]$

where * stands for the Hodge dual and the integral is defined as in differential geometry.

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:

$chi^\left\{\left(rho\right)\right\}left\left(mathcal\left\{P\right\}left\left\{e^\left\{int_gamma A\right\}right\right\}right\right)$

where χ is the character of a complex representation ρ and $mathcal\left\{P\right\}$ represents the path-ordered operator.

## 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.

### 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 now handled by a variety of means. Methods for quantization are covered in the article on quantization.

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.

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 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. Among the most well known are: