2 Classical gauge theory

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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

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2.1 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 fields, with equal masses m. This system is described by an action which is the sum of the (usual) action for each scalar field φ_i

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-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. Incidentally, Noether's theorem implies that invariance under this group of transformations leads to the conservation of the current

where the T^a 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.

2.2 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

2.3 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

3 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.

Note that 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 essential or 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 reps which transform covariantly pointwise (called by physicists gauge transformations of the first kind), reps which transform as a connection form (called by physicists gauge transformations of the second kind) (note that this is an affine rep) and other more general reps, like for example the B field in BF theory . And of course, we can consider more general nonlinear reps (realizations), but that's really complicated. But still, nonlinear sigma model s transform nonlinearly, so there are applications.

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.

4 References

• George Svetlichny, Preparation for Gauge Theory, an introduction to the mathematical aspects
• David Gross, Gauge theory - Past, Present and Future, notes from a talk
• Ta-Pei Cheng, Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Oxford University Press, 1983) [BooksEnthsiast.com]

Differential geometry Differential topology Theoretical physics

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