# Gauge Field Theories That is, Maxwell's equations have a gauge symmetry.

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 n non-interacting real scalar fields , with 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 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 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. 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. This new "derivative" is called a gauge covariant derivative and takes the form.

Where g is called the coupling constant; a quantity defining the strength of an interaction.

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After a simple calculation we can see that the gauge field A x must transform as follows. The difference between this Lagrangian and the original globally gauge-invariant Lagrangian is seen to be the interaction Lagrangian. This term introduces interactions 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. In the quantized version of the obtained 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. 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.

This formulation of the Lagrangian is called a Yang—Mills action. Other gauge invariant actions also exist e. 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. 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. An appropriate covariant derivative is then. Identifying the "charge" e not to be confused with the 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 A x with the four- vector potential of electromagnetic field results in an interaction Lagrangian.

The gauge principle is therefore seen to naturally introduce the so-called minimal coupling of the electromagnetic field to the electron field. 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.

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 representations realizations , but these are extremely complicated. Still, nonlinear sigma models transform nonlinearly, so there are applications.

If there is 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. 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 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 2-form that is an intrinsic quantity, is constructed from a connection form by. Under such an infinitesimal gauge transformation,. 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. A quantity which is gauge-invariant i. 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.

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. 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. 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.

## Quantum Field Theory

Precise computations in such schemes often require supercomputing , 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 an anomaly. Among the most well known are:.

A pure gauge is the set of field configurations obtained by a gauge transformation on the null-field configuration, i. So it is a particular "gauge orbit" in the field configuration's space. From Wikipedia, the free encyclopedia. Physical theory with fields invariant under the action of local "gauge" Lie groups. For a more accessible and less technical introduction to this topic, see Introduction to gauge theory.

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