In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations. The term gauge refers to. quatrième section, j’aborderai le rôle de la symétrie de jauge dans la procédure entités de la théorie) sur l’espace-temps4, l’invariance de jauge implique. “Optique Géométrique et invariance de jauge: Solutions oscillantes d’amplitude critique pour les équations de Yang-Mills.” Séminaire Équations aux dérivées.
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In physicsa gauge theory invariancs a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations. The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian. The transformations between possible gauges, called gauge transformationsform a Lie group—referred to as the symmetry group ve the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators.
For each group generator there necessarily arises a corresponding field usually a vector field called the gauge field.
Gauge theory – Wikipedia
Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations called gauge invariance. When such a theory is quantized [ clarification needed ]the quanta of the gauge fields are called gauge bosons.
If the symmetry group is non-commutative, then the gauge theory is referred to as non-abelian gauge theorythe 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 in the spacetime in which the physical processes occur, they are said to have a global symmetry.
Local symmetrythe 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 gauge theory with the symmetry group U 1 and has one gauge field, the electromagnetic four-potentialwith the photon being the gauge boson.
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 gravitybeginning with gauge gravitation theoryalso 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 diffeomorphisms of spacetime.
Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, gauge theory gravityreplaces 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. Today, gauge theories are useful in condensed matternuclear and high energy physics among other subfields. The earliest field theory having a gauge symmetry was Maxwell ‘s formulation, in —65, of electrodynamics ” A Dynamical Theory of the Electromagnetic Field ” which stated that any function whose curl vanishes—and can therefore normally be written as a gradient —could be added to the vector potential without affecting the magnetic field.
Similarly unnoticed, Hilbert had derived the Einstein field equations by postulating the invariance of the action under a general coordinate transformation. Later Hermann Weylin an attempt to unify general relativity and electromagnetismconjectured 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 mechanicsWeyl, 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 phasewhich is 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, popularised by Pauli in Inattempting to resolve some of the great confusion in dee particle physicsChen Ning Yang and Robert Mills introduced non-abelian gauge theories as models to understand the strong interaction holding together nucleons in atomic nuclei. Generalizing the gauge invariance of electromagnetism, they attempted jwuge 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 forceand 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 chromodynamicsis 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 s, Michael Atiyah began studying the mathematics of solutions to the classical Yang—Mills equations.
InAtiyah’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.
This led to an increasing interest in gauge theory for its own sake, independent of its successes in fundamental physics. InEdward Witten and Nathan Seiberg invented gauge-theoretic techniques based on supersymmetry that dee the calculation of certain topological invariants  jaute the Seiberg—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 tremendous success of the mathematical formalism in providing a unified framework to describe the quantum field theories of electromagnetismthe weak force and the strong force.
Modern theories like string theoryas well as general relativityare, in one way or another, gauge theories. In physicsthe mathematical description of any physical situation usually contains excess degrees of freedom ; the same physical situation unvariance equally well described by many equivalent mathematical configurations.
For instance, in Newtonian dynamicsif 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 invariwnce also changed by a coordinate transformation.
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 pointthe 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.
Jaute 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.
Introduction to gauge theory
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 1which appears in the modern formulation of quantum electrodynamics QED via its use of complex numbers. QED is generally regarded as the first, and simplest, physical gauge theory. The set unvariance 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 jquge 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 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, similar 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:. 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.
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 anomaliesand 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 a continuum theory implicitly assume that:.
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: 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 theoriesincluding 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: 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 by introducing additional fields the Faddeev—Popov ghosts and counterterms motivated by anomaly cancellationin 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. Historically, the first example of gauge symmetry discovered was classical electromagnetism. In electrostaticsone can either discuss the electric field, Eor its corresponding electric potentialV.
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. Generalizing from static electricity to electromagnetism, we have a second potential, the vector potential Awith.
The fields remain the same under the gauge transformation, and therefore Maxwell’s equations are still satisfied. That is, Maxwell’s equations have a gauge symmetry. The following illustrates how local gauge invarianve can be “motivated” heuristically jxuge from global symmetry properties, and how it leads to an interaction between originally non-interacting fields.
Consider a set of n non-interacting real scalar fieldswith equal masses m. This characterizes the global symmetry of this particular Lagrangian, and the symmetry group is often called the gauge group ; the mathematical term is structure groupespecially in the theory of G-structures.
Incidentally, Noether’s theorem implies that invariance under this group of transformations leads to the conservation of the currents. 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.