Introduction to gauge theory

In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally.

Explanation

Many powerful theories in physics are described by Lagrangians which are invariant under certain symmetry transformation groups. When they are invariant under a transformation identically performed at every point in the space in which the physical processes occur; they are said to have a global symmetry. In a gauge theory the requirement of global transformations is relaxed such that the Lagrangian has local symmetries as well. This can be viewed as a generalization of the equivalence principle of general relativity in which each point in spacetime is allowed a choice of local reference (coordinate) frame. As in that situation, gauge "symmetries" reflect a redundancy in the description of a system. Historically, these ideas were first noticed in the context of classical electromagnetism and later in general relativity. However, the modern importance of gauge symmetries appeared in relativistic quantum mechanics of electrons (see discussions below). Today, gauge theories are useful in condensed matter, nuclear and high energy physics among other subfields.

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.

Yang-Mills theories are a particular example of gauge theories with non-abelian symmetry groups specified by the Yang-Mills action (Other gauge theories with a non-abelian gauge symmetry also exist, e.g., the Chern-Simons model).

There is a certain inaccuracy in the way the term symmetry is used in part of the physics literature, especially in the more elementary books about elementary particles and field theory. In (quantum) physics, symmetry is a transformation between physical states that preserves the expectation values of all observables O (in particular the Hamiltonian). S: |φ> → |ψ> = S|φ>; |<ψ|O|ψ>|²=|<φ|O|φ>|² . The usual formulation of the physics theories uses fields, which sometimes are not physical quantities. Such are the gauge fields (fiber bundle connections for the mathematicians), which provide a redundant but convenient description of the physical degrees of freedom. The gauge (local) "symmetries" are a reflection of this redundancy. The physical quantities are certain equivalence classes of gauge fields. An analogy can be made with the construction of the real numbers. We can use sequences of rational numbers that have the same limit. Of course, each real number is represented by infinitely many such sequences. We can choose a particular well defined sequence to be a representative of the real number. This corresponds to the procedure of gauge fixing in gauge theories. The fact that gauge fields are not physical degrees of freedom becomes very clear when we try to quantize them. Then we are forced to work in one way or another with the physical quantities by removing the redundancy (the gauge symmetry). Another important illustration of the problem with the gauge “symmetries” is when we have anomalies. By definition these are symmetries which exist in the classical system, but not in its quantum counterpart. Anomalies are something quite usual and also an experimental fact - e.g. the axial anomaly in the strong interactions (broken symmetries). However, because gauge symmetries are not symmetries, gauge anomalies are not something that just complicates the quantum theory but something that kills it, i.e. there are no gauge "anomalies", because such theories don't exist. This is why having the exact relation between the number of flavours and quark colours in the Standard model is so important - otherwise there is a gauge anomaly and the theory does not exist. For the same reason, string theories are defined in 10 dimensions. Only then do the anomalies cancel.

Importance

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.

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.

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 exotic R⁴s, 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.

A simple gauge symmetry example from electrodynamics

The definition of electrical ground in an electric circuit is an example of a gauge symmetry; when the electric potentials across all points in a circuit are raised by the same amount, the circuit would still operate identically; as the potential differences (voltages) in the circuit are unchanged. A common illustration of this fact is the sight of a small bird perched on a high voltage power line without electrocution, because the bird is insulated from the ground (as long as it doesn't complete the circuit by accidentally touching another wire or some grounded structure).

This is called a global gauge symmetry^[1]. The absolute value of the potential is immaterial; what matters to circuit operation is the potential differences across the components of the circuit. The definition of the ground point is arbitrary, but once that point is set, then that definition must be followed globally.

In contrast, if some symmetry could be defined arbitrarily from one position to the next, that would be a local gauge symmetry.

• ^ James S. Trefil 1983, The moment of creation. Scribner, ISBN 0-684-17963-6 pages 92-93.

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 "motivated" 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 φ_i

${\mathcal {S}}=\int \,\mathrm {d} ^{4}x\sum _{i=1}^{n}\left[{\frac {1}{2}}\partial _{\mu }\varphi _{i}\partial ^{\mu }\varphi _{i}-{\frac {1}{2}}m^{2}\varphi _{i}^{2}\right].$

The Lagrangian (density) can be compactly written as

$\ {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\Phi )^{T}\partial ^{\mu }\Phi -{\frac {1}{2}}m^{2}\Phi ^{T}\Phi$

by introducing a vector of fields

$\ \Phi =(\varphi _{1},\varphi _{2},\ldots ,\varphi _{n})^{T}.$

The term $\partial _{\mu }$ is sometimes confusing to those who have not seen it before. It is simply the use of Einstein notation to describe the partial derivative of $\Phi$ in each of the four dimensions. It is now transparent that the Lagrangian is invariant under the transformation

$\Phi \mapsto G\Phi$

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

$\ J_{\mu }^{a}=i\partial _{\mu }\Phi ^{T}T^{a}\Phi$

where the T^a 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),

$\ \partial _{\mu }(G\Phi )^{T}\partial ^{\mu }(G\Phi )\neq \partial _{\mu }\Phi ^{T}\partial ^{\mu }\Phi .$

This suggests defining a "derivative" D with the property

$\ D_{\mu }(G(x)\Phi (x))=G(x)D_{\mu }\Phi .$

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

$\ D_{\mu }=\partial _{\mu }+gA_{\mu }(x)$

where the gauge field A(x) is defined to have the transformation law

$\ A_{\mu }(x)\mapsto G(x)A_{\mu }(x)G^{-1}(x)-{\frac {1}{g}}\partial _{\mu }G(x)G^{-1}(x)$

and g is the coupling constant - a quantity defining the strength of an interaction.

The gauge field is an element of the Lie algebra, and can therefore be expanded as

$\ A_{\mu }(x)=\sum _{a}A_{\mu }^{a}(x)T^{a}$

There are therefore as many gauge fields as there are generators of the Lie algebra.

Finally, we now have a locally gauge invariant Lagrangian

$\ {\mathcal {L}}_{\mathrm {loc} }={\frac {1}{2}}(D_{\mu }\Phi )^{T}D^{\mu }\Phi -{\frac {1}{2}}m^{2}\Phi ^{T}\Phi .$

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

Feynman diagram of scalar bosons interacting via a gauge boson

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

$\ {\mathcal {L}}_{\mathrm {int} }={\frac {g}{2}}\Phi ^{T}A_{\mu }^{T}\partial ^{\mu }\Phi +{\frac {g}{2}}(\partial _{\mu }\Phi )^{T}A^{\mu }\Phi +{\frac {g^{2}}{2}}(A_{\mu }\Phi )^{T}A^{\mu }\Phi .$

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 Yang-Mills 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, one possible form for the gauge field Lagrangian is (conventionally) written as

$\ {\mathcal {L}}_{\mathrm {gf} }=-{\frac {1}{2}}\operatorname {Tr} (F^{\mu \nu }F_{\mu \nu })$

with

$\ F_{\mu \nu }={\frac {1}{ig}}[D_{\mu },D_{\nu }]$

and the trace being taken over the vector space of the fields. This is called the Yang-Mills action. Other gauge invariant actions also exist (e.g. nonlinear electrodynamics, Born-Infeld action, Chern-Simons model, theta term etc.).

Note that in this Lagrangian there is not a field $\Phi$ whose transformation counterweights the one of $A$. 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 quantization, 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

$\ {\mathcal {L}}={\mathcal {L}}_{\mathrm {loc} }+{\mathcal {L}}_{\mathrm {gf} }={\mathcal {L}}_{\mathrm {global} }+{\mathcal {L}}_{\mathrm {int} }+{\mathcal {L}}_{\mathrm {gf} }$

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

${\mathcal {S}}=\int {\bar {\psi }}(i\hbar c\,\gamma ^{\mu }\partial _{\mu }-mc^{2})\psi \,\mathrm {d} ^{4}x.$

The global symmetry for this system is

$\ \psi \mapsto e^{i\theta }\psi .$

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

$\ D_{\mu }=\partial _{\mu }-i{\frac {e}{\hbar }}A_{\mu }.$

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

$\ {\mathcal {L}}_{\mathrm {int} }={\frac {e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)A_{\mu }(x)=J^{\mu }(x)A_{\mu }(x).$

where $J^{\mu }(x)$ 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 $A_{\mu }(x)$ 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.

$\ {\mathcal {L}}_{\mathrm {QED} }={\bar {\psi }}(i\hbar c\,\gamma ^{\mu }D_{\mu }-mc^{2})\psi -{\frac {1}{4\mu _{0}}}F_{\mu \nu }F^{\mu \nu }.$

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

Mathematical formalism

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.

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 is extremely complicated. But still, nonlinear sigma models transform nonlinearly, so there are applications.

If we have 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.

We can define a connection (gauge connection) on this principal bundle, yielding a covariant derivative ∇ in each associated vector bundle. If we choose a local frame (a local basis of sections) then we can represent this covariant derivative by the connection form A, a Lie algebra-valued 1-form which is called the gauge potential in physics and which is evidently not an intrinsic but a frame-dependent quantity. From this connection form we can construct the curvature form F, a Lie algebra-valued 2-form which is an intrinsic quantity, by

${\mathbf {F}}=\mathrm {d} {\mathbf {A}}+{\mathbf {A}}\wedge {\mathbf {A}}$

where d stands for the exterior derivative and $\wedge$ stands for the wedge product. (Keep in mind that ${\mathbf {A}}$ is an element of the vector space spanned by the generators $T^{a}$, and so the components of ${\mathbf {A}}$ do not commute with one another. Hence the wedge product ${\mathbf {A}}\wedge {\mathbf {A}}$ does not vanish.)

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

$\delta _{\varepsilon }{\mathbf {A}}=[\varepsilon ,{\mathbf {A}}]-\mathrm {d} \epsilon$

where $[\cdot ,\cdot ]$ is the Lie bracket.

One nice thing is that if $\delta _{\varepsilon }X=\varepsilon X$, then $\delta _{\varepsilon }DX=\varepsilon DX$ where D is the covariant derivative

$DX\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {d} X+{\mathbf {A}}X.$

Also, $\delta _{\varepsilon }{\mathbf {F}}=\varepsilon {\mathbf {F}}$, 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

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

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:

$\chi ^{(\rho )}\left({\mathcal {P}}\left\{e^{\int _{\gamma }A}\right\}\right)$

where χ is the character of a complex representation ρ and ${\mathcal {P}}$ 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:

• The scale anomaly, which gives rise to a running coupling constant. In QED this gives rise to the phenomenon of the Landau pole. In Quantum Chromodynamics (QCD) this leads to asymptotic freedom.
• The chiral anomaly in either chiral or vector field theories with fermions. This has close connection with topology through the notion of instantons. In QCD this anomaly causes the decay of a pion to two photons.
• The gauge anomaly, which must cancel in any consistent physical theory. In the electroweak theory this cancellation requires an equal number of quarks and leptons.

References

• George Svetlichny, Preparation for Gauge Theory, an introduction to the mathematical aspects

• C. Becchi, Introduction to Gauge Theories, an elementary introduction to quantum gauge fields.

• 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) [ISBN 0-19-851961-3]

• D.A. Bromley (2000). Gauge Theory of Weak Interactions. Springer. ISBN 3-540-67672-4.

• Gordon L. Kane (1987). Modern Elementary Particle Physics. Perseus Books. ISBN 0-201-11749-5.

• Paul H. Frampton , Gauge Field Theories, Second Edition, Wiley (2000).

See also

• Aharonov-Bohm effect
• Bell's Theorem
• Coulomb gauge
• Electroweak theory
• Field theoretical formulation of the standard model
• Gauge covariant derivative
• Kaluza-Klein theory
• Lorenz gauge
• Quantum chromodynamics
• Quantum gauge theory
• Symmetry in physics
• Weyl gauge

External Links

Yang-Mills equations on Dispersive Wiki