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Inphysics, agauge theory is a type offield theory in which theLagrangian, and hence the dynamics of the system itself, does not change underlocal transformations according to certain smooth families of operations (Lie groups). Formally, the Lagrangian isinvariant under these transformations.
The term "gauge" refers to any specific mathematical formalism to regulate redundantdegrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, calledgauge transformations, form a Lie group—referred to as thesymmetry group or thegauge group of the theory. Associated with any Lie group is theLie algebra ofgroup generators. For each group generator there necessarily arises a correspondingfield (usually avector field) called thegauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (calledgauge invariance). When such a theory isquantized, thequanta of the gauge fields are calledgauge bosons. If the symmetry group is non-commutative, then the gauge theory is referred to asnon-abelian gauge theory, the usual example being theYang–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 ateverypoint in thespacetime in which the physical processes occur, they are said to have aglobal 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 ofelementary particles.Quantum electrodynamics is anabelian gauge theory with the symmetry groupU(1) and has one gauge field, theelectromagnetic four-potential, with thephoton being the gauge boson. TheStandard Model is a non-abelian gauge theory with the symmetry group U(1) ×SU(2) ×SU(3) and has a total of twelve gauge bosons: thephoton, threeweak bosons and eightgluons.
Gauge theories are also important in explaininggravitation in the theory ofgeneral relativity. Its case is somewhat unusual in that the gauge field is atensor, theLanczos tensor. Theories ofquantum gravity, beginning withgauge gravitation theory, also postulate the existence of a gauge boson known as thegraviton. Gauge symmetries can be viewed as analogues of theprinciple of general covariance of general relativity in which the coordinate system can be chosen freely under arbitrarydiffeomorphisms 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 ofclassical electromagnetism and later ingeneral relativity. However, the modern importance of gauge symmetries appeared first in therelativistic quantum mechanics ofelectrons – quantum electrodynamics, elaborated on below. Today, gauge theories are useful incondensed matter,nuclear andhigh energy physics among other subfields.
The concept and the name of gauge theory derives from the work ofHermann Weyl in 1918.[1] Weyl, in an attempt to generalize the geometrical ideas ofgeneral relativity to includeelectromagnetism, conjectured thatEichinvarianz or invariance under the change ofscale (or "gauge") might also be a local symmetry of general relativity. After the development ofquantum mechanics, Weyl,Vladimir Fock[2] andFritz London replaced the simple scale factor with acomplex quantity and turned the scale transformation into a change ofphase, which is aU(1) gauge symmetry. This explained theelectromagnetic field effect on thewave function of acharged quantum mechanicalparticle. Weyl's 1929 paper introduced the modern concept of gauge invariance[3] subsequently popularized byWolfgang Pauli in his 1941 review.[4] In retrospect,James Clerk Maxwell's formulation, in 1864–65, ofelectrodynamics 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 agradient of a function—could be added to the vector potential without affecting themagnetic field. Similarly unnoticed,David Hilbert had derived theEinstein field equations by postulating the invariance of theaction 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 foratomic nuclei binding based on conservation of nuclearisospin.[5]: 202 In 1954, Yang andRobert Mills generalized the gauge invariance of electromagnetism, constructing a theory based on the action of the (non-abelian) SU(2) symmetrygroup on theisospin doublet ofprotons andneutrons.[6] This is similar to the action of theU(1) group on thespinorfields ofquantum electrodynamics.
TheYang–Mills theory became the prototype theory to resolve some of the confusion inelementary particle physics.This idea later found application in thequantum field theory of theweak force, and its unification with electromagnetism in theelectroweak theory. Gauge theories became even more attractive when it was realized that non-abelian gauge theories reproduced a feature calledasymptotic 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 asquantum chromodynamics, is a gauge theory with the action of the SU(3) group on thecolor triplet ofquarks. TheStandard 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 classicalYang–Mills equations. In 1983, Atiyah's studentSimon Donaldson built on this work to show that thedifferentiable classification ofsmooth 4-manifolds is very different from their classificationup tohomeomorphism.[7]Michael Freedman used Donaldson's work to exhibitexoticR4s, that is, exoticdifferentiable structures onEuclidean 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 andNathan Seiberg invented gauge-theoretic techniques based onsupersymmetry that enabled the calculation of certaintopological invariants[8][9] (theSeiberg–Witten invariants). 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 thequantum field theories ofelectromagnetism, theweak force and thestrong force. This theory, known as theStandard Model, accurately describes experimental predictions regarding three of the fourfundamental forces of nature, and is a gauge theory with the gauge groupSU(3) × SU(2) × U(1). Modern theories likestring theory, as well asgeneral relativity, are, in one way or another, gauge theories.
Inphysics, the mathematical description of any physical situation usually contains excessdegrees of freedom; the same physical situation is equally well described by many equivalent mathematical configurations. For instance, inNewtonian dynamics, if two configurations are related by aGalilean transformation (aninertial change of reference frame) they represent the same physical situation. These transformations form agroup 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 positivex 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 (x = 0,y= −1) is 1 m/s in the negativey direction. The coordinate transformation has affected both the coordinate system used to identify thelocation of the measurement and the basis in which itsvalue 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 therate of change of some quantity along some path in space and time as it passes through pointP is the same as the effect on values that are truly local toP.
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 afiber 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 (alocal 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, orgauge 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 isU(1), which appears in the modern formulation ofquantum electrodynamics (QED) via its use ofcomplex numbers. 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, thegauge 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 aglobal 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 isnot a constant function is referred to as alocal symmetry; its effect on expressions that involve aderivative is qualitatively different from that on expressions that do not. (This is analogous to a non-inertial change of reference frame, which can produce aCoriolis effect.)
The "gauge covariant" version of a gauge theory accounts for this effect by introducing agauge field (in mathematical language, anEhresmann connection) and formulating all rates of change in terms of thecovariant 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 itsfield strength (in mathematical language, itscurvature) is zero everywhere; a gauge theory isnot 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 thedynamics 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 itsinteraction with other objects via the covariant derivative, the gauge field typically contributesenergy in the form of a "self-energy" term. One can obtain the equations for the gauge theory by:
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 asgeneral relativity.
Gauge theories used to model the results of physical experiments engage in:
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 ofanomalies, and approaches to anomaly avoidance classifies gauge theories[clarification needed].
The two gauge theories mentioned above, continuum electrodynamics and general relativity, are continuum field theories. The techniques of calculation in acontinuum theory implicitly assume that:
Determination of the likelihood of possible measurement outcomes proceed by:
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 ofturbulence and otherchaotic phenomena.
Other than these classical continuum field theories, the most widely known gauge theories arequantum field theories, includingquantum electrodynamics and theStandard Model of elementary particle physics. The starting point of a quantum field theory is much like that of its continuum analog: a gauge-covariantaction integral that characterizes "allowable" physical situations according to theprinciple 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 agauge 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 ofperturbation theory by introducing additional fields (theFaddeev–Popov ghosts) and counterterms motivated byanomaly cancellation, in an approach known asBRST 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] The mathematical techniques that have been developed in order to make gauge theories tractable have found many other applications, fromsolid-state physics andcrystallography tolow-dimensional topology.
Inelectrostatics, one can either discuss the electric field,E, or its correspondingelectric potential,V. Knowledge of one makes it possible to find the other, except that potentials differing by a constant,, correspond to the same electric field. This is because the electric field relates tochanges in the potential from one point in space to another, and the constantC would cancel out when subtracting to find the change in potential. In terms ofvector calculus, the electric field is thegradient of the potential,. Generalizing from static electricity to electromagnetism, we have a second potential, thevector potentialA, with
The general gauge transformations now become not just but
wheref is any twice continuously differentiable function that depends on position and time. The electromagnetic fields remain the same under the gauge transformation.
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 non-interacting realscalar fields, with equal massesm. This system is described by anaction that is the sum of the (usual) action for each scalar field
The Lagrangian (density) can be compactly written as
by introducing avector of fields
The term is thepartial derivative of along dimension.
It is now transparent that the Lagrangian is invariant under the transformation
wheneverG is aconstantmatrix belonging to then-by-northogonal group O(n). This is seen to preserve the Lagrangian, since the derivative of transforms identically to and both quantities appear insidedot products in the Lagrangian (orthogonal transformations preserve the dot product).
This characterizes theglobal symmetry of this particular Lagrangian, and the symmetry group is often called thegauge group; the mathematical term isstructure group, especially in the theory ofG-structures. Incidentally,Noether's theorem implies that invariance under this group of transformations leads to the conservation of thecurrents
where theTa matrices aregenerators of the SO(n) group. There is one conserved current for every generator.
Now, demanding that this Lagrangian should havelocal O(n)-invariance requires that theG matrices (which were earlier constant) should be allowed to become functions of thespacetimecoordinatesx.
In this case, theG matrices do not "pass through" the derivatives, whenG =G(x),
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 again transforms identically with
This new "derivative" is called a(gauge) covariant derivative and takes the form
whereg is called the coupling constant; a quantity defining the strength of an interaction.After a simple calculation we can see that thegauge fieldA(x) must transform as follows
The gauge field is an element of the Lie algebra, and can therefore be expanded as
There are therefore as many gauge fields as there are generators of the Lie algebra.
Finally, we now have alocally gauge invariant Lagrangian
Pauli uses the termgauge transformation of the first type to mean the transformation of, while the compensating transformation in is called agauge transformation of the second type.
The difference between this Lagrangian and the originalglobally gauge-invariant Lagrangian is seen to be theinteraction Lagrangian
This term introducesinteractions between then scalar fields just as a consequence of the demand for local gauge invariance. However, to make this interaction physical and not completely arbitrary, the mediatorA(x) needs to propagate in space. That is dealt with in the next section by adding yet another term,, to the Lagrangian. In thequantized version of the obtainedclassical field theory, thequanta of the gauge fieldA(x) are calledgauge bosons. The interpretation of the interaction Lagrangian in quantum field theory is ofscalarbosons interacting by the exchange of these gauge bosons.
The picture of a classical gauge theory developed in the previous section is almost complete, except for the fact that to define the covariant derivativesD, one needs to know the value of the gauge field 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
where the are obtained from potentials, being the components of, by
and the are thestructure constants of the Lie algebra of the generators of the gauge group. This formulation of the Lagrangian is called aYang–Mills action. Other gauge invariant actions also exist (e.g.,nonlinear electrodynamics,Born–Infeld action,Chern–Simons model,theta term, etc.).
In this Lagrangian term there is no field whose transformation counterweighs the one of. Invariance of this term under gauge transformations is a particular case ofa priori classical (geometrical) symmetry. This symmetry must be restricted in order to perform quantization, the procedure being denominatedgauge fixing, but even after restriction, gauge transformations may be possible.[12]
The complete Lagrangian for the gauge theory is now
As a simple application of the formalism developed in the previous sections, consider the case ofelectrodynamics, with only theelectron field. The bare-bones action that generates the electron field'sDirac equation is
The global symmetry for this system is
The gauge group here isU(1), just rotations of thephase angle of the field, with the particular rotation determined by the constantθ.
"Localising" this symmetry implies the replacement ofθ byθ(x). An appropriate covariant derivative is then
Identifying the "charge"e (not to be confused with the mathematical constante in the symmetry description) with the usualelectric charge (this is the origin of the usage of the term in gauge theories), and the gauge fieldA(x) with the four-vector potential of theelectromagnetic field results in an interaction Lagrangian
where is the electric currentfour vector in theDirac field. Thegauge principle is therefore seen to naturally introduce the so-calledminimal coupling of the electromagnetic field to the electron field.
Adding a Lagrangian for the gauge field in terms of thefield strength tensor exactly as in electrodynamics, one obtains the Lagrangian used as the starting point inquantum electrodynamics.
Gauge theories are usually discussed in the language ofdifferential geometry. Mathematically, agauge is just a choice of a (local)section of someprincipal bundle. Agauge transformation is just a transformation between two such sections.
Although gauge theory is dominated by the study ofconnections (primarily because it's mainly studied byhigh-energy physicists), the idea of a connection is not central to gauge theory in general. In fact, a result in general gauge theory shows thataffine representations (i.e., affinemodules) of the gauge transformations can be classified as sections of ajet bundle satisfying certain properties. There are representations that transform covariantly pointwise (called by physicists gauge transformations of the first kind), representations that transform as aconnection form (called by physicists gauge transformations of the second kind, an affine representation)—and other more general representations, such as the B field inBF theory. There are more generalnonlinear representations (realizations), but these are extremely complicated. Still,nonlinear sigma models transform nonlinearly, so there are applications.
If there is aprincipal bundleP whosebase space isspace orspacetime andstructure group is a Lie group, then the sections ofP form aprincipal homogeneous space of the group of gauge transformations.
Connections (gauge connection) define this principal bundle, yielding acovariant derivative ∇ in eachassociated vector bundle. If a local frame is chosen (a local basis of sections), then this covariant derivative is represented by theconnection formA, a Lie algebra-valued1-form, which is called thegauge potential inphysics. This is evidently not an intrinsic but a frame-dependent quantity. Thecurvature formF, a Lie algebra-valued2-form that is an intrinsic quantity, is constructed from a connection form by
where d stands for theexterior derivative and stands for thewedge product. ( is an element of the vector space spanned by the generators, and so the components of do not commute with one another. Hence the wedge product does not vanish.)
Infinitesimal gauge transformations form a Lie algebra, which is characterized by a smooth Lie-algebra-valuedscalar, ε. Under such aninfinitesimal gauge transformation,
where is the Lie bracket.
One nice thing is that if, then where D is the covariant derivative
Also,, which means transforms covariantly.
Not all gauge transformations can be generated byinfinitesimal gauge transformations in general. An example is when thebase manifold is acompactmanifold withoutboundary such that thehomotopy class of mappings from thatmanifold to the Lie group is nontrivial. Seeinstanton for an example.
TheYang–Mills action is now given by
where is theHodge star operator and the integral is defined as indifferential geometry.
A quantity which isgauge-invariant (i.e.,invariant under gauge transformations) is theWilson loop, which is defined over any closed path, γ, as follows:
whereχ is thecharacter of a complexrepresentation ρ and represents the path-ordered operator.
The formalism of gauge theory carries over to a general setting. For example, it is sufficient to ask that avector bundle have ametric connection; when one does so, one finds that the metric connection satisfies the Yang–Mills equations of motion.
Gauge theories may be quantized by specialization of methods which are applicable to anyquantum 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 exampleWard identities connect differentrenormalization constants.
The first gauge theory quantized wasquantum electrodynamics (QED). The first methods developed for this involved gauge fixing and then applyingcanonical quantization. TheGupta–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 onquantization.
The main point to quantization is to be able to computequantum amplitudes for various processes allowed by the theory. Technically, they reduce to the computations of certaincorrelation functions in thevacuum state. This involves arenormalization of the theory.
When therunning coupling of the theory is small enough, then all required quantities may be computed inperturbation theory. Quantization schemes intended to simplify such computations (such ascanonical quantization) may be calledperturbative 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 aslattice gauge theory) may be callednon-perturbative quantization schemes. Precise computations in such schemes often requiresupercomputing, and are therefore less well-developed currently than other schemes.
Some of the symmetries of the classical theory are then seen not to hold in the quantum theory; a phenomenon called ananomaly. Among the most well known are:
A pure gauge is the set of field configurations obtained by agauge 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, the pure gauge is just the set of field configurations for allf(x).
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)
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