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Gauge theory

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Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. Gauge theories with non-Abelian symmetry groups are also sometimes known as Yang-Mills theories. 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 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) × SU(2) × U(1). Modern theories like string theory, as well as some formulations of general relativity, are in one way or another, gauge theories.

Sometimes, the term gauge symmetry is used in a more general sense to include any local symmetry, like for example, diffeomorphisms. This sense of the term will not be used in this article.

Contents

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 charged quantum mechanical particle. This was the first gauge theory. It was popularised by Pauli in the 1940s, e.g. R.M. P.13, 203 (http://prola.aps.org/abstract/RMP/v13/i3/p203_1).

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. (Ronald Shaw, working under Abdus Salam, independently introduced the same notion in his doctoral thesis.) 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 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 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 the seventies, Sir Michael Atiyah began a program of studying the mathematics of solutions to the classical Yang-Mills equations. In 1983, Atiyah'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. Michael Freedman used Donaldson's work to exhibit "fake R4"s, that is, 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


An example : Scalar O(n) gauge theory

The following illustrates how local gauge invariance can be "derived" heuristically 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 φi

<math> \mathcal{S} = \int \, d^4 x \sum_{i=1}^n \frac{1}{2} \partial_\mu \varphi_i \partial^\mu \varphi_i - \frac{1}{2}m^2 \varphi_i^2. <math>

The Lagrangian can be compactly written as

<math>\ L = \frac{1}{2} (\partial_\mu \Phi)^T \partial^\mu \Phi - \frac{1}{2}m^2 \Phi^T \Phi <math>

by introducing a vector of fields

<math>\ \Phi = ( \varphi_1, \varphi_2,\ldots, \varphi_n)^T. <math>

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

<math> \Phi \mapsto G \Phi <math>

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

<math>\ J^{a}_{\mu} = i\partial_\mu \Phi^T T^{a} \Phi<math>

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

<math>\ \partial_\mu (G \Phi)^T \partial^\mu G \Phi \neq \partial_\mu \Phi^T \partial^\mu \Phi. <math>

This suggests defining a "derivative" D with the property

<math>\ D_\mu (G(x) \Phi(x)) = G(x) D_\mu \Phi. <math>

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

<math>\ D_\mu = \partial_\mu - (\partial_\mu G(x)) G^{-1}(x) = \partial_\mu + g A_\mu(x) <math>

where the gauge field A(x) is defined as

<math>\ A_{\mu}(x) = \frac{1}{g} G(x) \partial_\mu G^{-1}(x) = \sum_a A_{\mu}^a (x) T^a <math>

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

Finally, we now have a locally gauge invariant Lagrangian

<math>\ L_\mathrm{loc} = \frac{1}{2} (D_\mu \Phi)^T D^\mu \Phi -\frac{1}{2}m^2 \Phi^T \Phi. <math>

Pauli calls gauge transformation of the first type to the one applied to fields as <math>\Phi<math>, while the compensating transformation in <math>A<math> is said to be a gauge transformation of the second type.

Feynman diagram of scalar bosons interacting via a gauge boson
Enlarge
Feynman diagram of scalar bosons interacting via a gauge boson

The difference between this Lagrangian and the original globally gauge-invariant Langrangian is seen to be the interaction Lagrangian

<math>\ L_\mathrm{int} = \frac{g}{2} \Phi^T A_{\mu}^T \partial^\mu \Phi + \frac{g}{2} (\partial_\mu \Phi)^T A^{\mu} \Phi + \frac{g^2}{2} (A_\mu \Phi)^T A^\mu \Phi. <math>

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

<math>\ L_\mathrm{gf} = - \frac{1}{4} \operatorname{Tr}(F^{\mu \nu} F_{\mu \nu}) <math>

with

<math>\ F_{\mu \nu} = [D_\mu, D_\nu] <math>

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

Note that in this Lagrangian there is not a field <math>\Phi<math> whose transformation counterweights the one of <math>A<math>. Invariance of this term under gauge transformations is a particular case of a prior classical (or geometrical, if you prefer) symmetry. This symmetry must be restricted in order to perform quantisation, the procedure being denominated gauge fixing, but even after restriction, gauge transformations are possible (see Sakurai, Advanced Quantum Mechanics, sect 1-4).

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

<math>\ L = L_\mathrm{loc} + L_\mathrm{gf} = L_\mathrm{global} + L_\mathrm{int} + L_\mathrm{gf} <math>

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)

<math> \mathcal{S} = \int \bar\psi(i \gamma_\mu \partial^\mu - m) \psi \, d^4x.

<math>

The global symmetry for this system is

<math>\ \psi \mapsto e^{i \theta} \psi.<math>

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

<math>\ D_\mu = \partial_\mu + i e A_\mu. <math>

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

<math>\ L_\mathrm{int} = \bar\psi(x) \gamma_\mu \psi(x) A^{\mu}(x) = J_{\mu}(x)

A^{\mu}(x). <math>

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.

<math>\ L = \bar\psi(i\gamma_\mu D^\mu - m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}. <math>

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.

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, such as 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 models 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 group of G-bundle automorphisms is called the group of gauge transformations. The space of smooth (although in physics, we often don't deal with smooth functions) sections of this bundle forms a principal homogeneous space of the group of gauge transformations. A choice of a section of the principal bundle is called a gauge fixing.

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 curvature form, by

<math>\bold{F}=d\bold{A}+\bold{A}\wedge\bold{A}<math>

where d stands for the exterior derivative and <math>\wedge<math> 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,

<math>\delta_\varepsilon \bold{A}=[\varepsilon,\bold{A}]-d\epsilon<math>

where <math>[\bullet,\bullet]<math> is the Lie bracket.

One thing nice is if <math>\delta_\varepsilon X=\varepsilon X<math>, then <math>\delta_\varepsilon DX=\varepsilon DX<math> where D is the covariant derivative

<math>DX\equiv dX+\bold{A}X.<math>

Also, <math>\delta_\varepsilon \bold{F}=\varepsilon \bold{F}<math>, 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

<math>\frac{1}{4g^2}\int \operatorname{Tr}[*F\wedge F]<math>

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:

<math>\chi^{(\rho)}\left(\mathcal{P}\left\{e^{\int_\gamma A}\right\}\right)<math>

where χ is the character of a complex representation ρ and <math>\mathcal{P}<math> represents the path-ordered operator.

Quantization of gauge theories

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 allow simplification of some computations: for example Ward identities connect different renormalization constants.

Methods and aims

The first gauge theory to be 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 nowadays 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 that are geared 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 which are geared to these problems (such as lattice gauge theory) may be called non-perturbative quantization schemes. Precise computations in such schemes often require supercomputing, and are therefore less well developed currently than other schemes.

Anomalies

Some of the classical symmetries of the theory are then seen not to hold in the quantum theory — a phenomenon called an anomaly. Among the most well known are:

See also

References

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