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Gauge theory
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Gauge theory

Gauge theories are a class of physical theories which are based on the idea that symmetry transformations can only be performed locally. Most physical theories are described by Lagrangians which are invariant under certain transformations, when the transformations are identically performed at every space-time point - they have global symmetries. Gauge theory extends this idea by requiring that the Lagrangians must possess local symmetries as well - it should be possible to perform these symmetry transformations in a particular region of space-time without affecting what happens in another region. This requirement is sometimes philosophically seen as a generalized version of the equivalence principle of general relativity.

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)XSU(2)XU(1). Modern theories like string theory, as well as some formulations of general relativity, are in one way or another, gauge theories.

Table of contents
1 A brief history
2 Classical gauge theory
3 Mathematical formalism
4 References

A brief history

The earliest physical theory which had a gauge symmetry was Maxwell's electrodynamics. However, the importance of this symmetry remained unnoticed in the earliest formulations. After Einstein's development of general relativity, Hermann Weyl, in an attempt to unify general relativity and electromagnetism, conjectured that Eichinvarianz or invariance under the change of scale (or "gauge") might also be a local symmetry of the theory of general relativity. This conjecture was found to lead to some unphysical results. However after the development of quantum mechanics, Weyl, Vladimir Fock and Fritz London realized that the idea, with some modifications (replacing the scale factor with a complex quantity, and turning the scale transformation into a change of phase - a U(1) gauge symmetry) provided a neat explanation for the effect of an electromagnetic field on the wave function of a chargedd quantum mechanical particle. This was the first gauge theory.

In the 1950s, 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. 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, similar to the action of the U(1) group on the spinor fieldss of quantum electrodynamics. 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, that was believed to be an important characteristic of strong interactions - thereby motivating the search for a gauge theory of the strong force. 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 1983, Simon Donaldson used techniques developed in gauge theory (instantons) to show that the differentiable classification of smooth 4-manifolds is very different from their classification upto homeomorphism, and exhibited 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.

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


A generic example : Scalar O(n) gauge theory

The following illustrates how local gauge invariance is postulated 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 fieldss, with equal masses m. This system is described by an action which is the sum of the (usual) action for each scalar field φi

The Lagrangian can be compactly written as

by introducing a vector of fields

It is now transparent that the Lagrangian is invariant under the transformation

whenever G is a constant matrix, belonging to the n-dimensional orthogonal group O(n). This is the global symmetry of this particular Lagrangian, and the symmetry group is often called the gauge group. Incidentally, Noether's theorem implies that invariance under this group of transformations leads to the conservation of the current

where the Ta matrices are generators of the O(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),

This suggests defining a "derivative" D with the property

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

where the gauge field A(x) is defined as

and g is known as the "charge" - a quantity defining the strength of an interaction.

Finally, we now have a locally gauge invariant Lagrangian

The difference between this and the original globally gauge invariant Langrangian 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. 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 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(x) 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, the most general form of the gauge field Lagrangian is (conventionally) written as

with

and the
trace being taken over the vector space of the fields.

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

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 (conventionally)

The global symmetry for this system is

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

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

where J(x) is the usual four vector electric current density. The gauge principle is therefore seen to introduce the so-called minimal coupling of the electromagnetic field to the electron field in a natural fashion.

Adding a Lagrangian for the gauge field A(x) 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.

See also: Dirac equation, Maxwell's equations, Quantum electrodynamics

Mathematical formalism

Mathematically, a gauge is some degree of freedom within a theory that has no observable effect. A gauge transformation is thus a transformation of this degree of freedom which does not modify any physical observable properties. Gauge theories are usually discussed in the language of differential geometry.

If we have a principal bundle whose base space is space or spacetime and structure group is a Lie group, then, the space of smooth (although in physics, we often don't deal with smooth functions) sections of this bundle forms a group, called the group of gauge transformations. We can define a connection (gauge connection) on this principal bundle, yielding a Lie algebra-valued 1-form, A, which is called the gauge potential in physics. From this 1-form, we can construct a Lie algebra-valued 2-form, F, called the field strength, by

where d stands for the exterior derivative and stands for the wedge product.

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

where is the Lie bracket.

One thing nice is if , then where D is the covariant derivative

Also, , which means F transforms covariantly.

One thing to note is that not all gauge transformations can be generated by infinitesimal gauge transformations in general; for example, 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

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:

where χ is the character of a complex representation ρ and represents the path ordered operator.

References