Gauge theory
For a generally accessible and less technical introduction to the topic, see Introduction to gauge theory.
In physics, a
gauge theory is a type of
field theory in which the
Lagrangian is
invariant under a
continuous group of local transformations.
The term
gauge refers to redundant degrees of freedom in the Lagrangian. The transformations between possible gauges, called
gauge transformations, form a
Lie group which is referred to as the
symmetry group or the
gauge group of the theory. Associated with any Lie group is the
Lie algebra of
group generators. For each group generator there necessarily arises a corresponding
vector field called the
gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called
gauge invariance). When such a theory is quantized, the
quanta of the gauge fields are called
gauge bosons. If the symmetry group is
non-commutative, the gauge theory is referred to as
non-abelian, the usual example being the
Yang–Mills theory.
Gauge theories are important as the successful field theories explaining the dynamics of
elementary particles.
Quantum electrodynamics is an
abelian gauge theory with the symmetry group
U(1) and has one gauge field, the
electromagnetic field, with the
photon being the gauge boson. The
Standard Model is a non-abelian gauge theory with the symmetry group
U(1)×
SU(2)×
SU(3) and has a total of twelve gauge bosons: the
photon, three
weak bosons and eight
gluons.
Many powerful theories in physics are described by
Lagrangians which are
invariant under some 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. The requirement of
local symmetry, the cornerstone of gauge theories, is a stricter constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in space-time. Gauge symmetries can be viewed as analogues of the
equivalence principle of
general relativity in which each point in spacetime is allowed a choice of local reference (coordinate) frame. Both symmetries reflect a redundancy in the description of a system.
Historically, these ideas were first stated in the context of
classical electromagnetism and later in
general relativity. However, the modern importance of gauge symmetries appeared first in the
relativistic quantum mechanics of
electrons —
quantum electrodynamics, elaborated on below. Today, gauge theories are useful in
condensed matter,
nuclear and
high energy physics among other subfields.
History and importance
The earliest field theory having a gauge symmetry was
Maxwell's formulation of
electrodynamics in 1864. The importance of this symmetry remained unnoticed in the earliest formulations. Similarly unnoticed,
Hilbert had derived the
Einstein field equations by postulating the invariance of the
action under a general coordinate transformation. Later
Hermann Weyl, in an attempt to unify
general relativity and
electromagnetism, conjectured (incorrectly, as it turned out) that
Eichinvarianz or invariance under the change of
scale (or "gauge") might also be a local symmetry of general relativity. After the development of
quantum mechanics, Weyl,
Vladimir Fock and
Fritz London modified gauge by replacing the scale factor with a
complex quantity and turned the scale transformation into a change of
phase — a
U(1) gauge symmetry. This explained the
electromagnetic field effect on the
wave function of a
charged quantum mechanical
particle. This was the first widely recognised gauge theory, popularised by
Pauli in the 1940s.
[1]
In 1954, 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. This is 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. Asymptotic freedom was believed to be an important characteristic of strong interactions. This motivated searching for a strong force gauge theory. 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 1970s, Sir
Michael Atiyah began 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.
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.
Description
Global and local symmetries
In physics, the mathematical description of any physical situation usually contains excess degrees of freedom; the same physical situation is equally well described by many equivalent mathematical configurations. For instance, in Newtonian dynamics, if two configurations are related by a Galilean transformation—an inertial change of reference frame—they represent the same physical situation. These transformations form a group of "symmetries" of the theory, and a physical situation corresponds not to an individual mathematical configuration but to a class of configurations related to one another by this symmetry group. This idea can be generalized to include local as well as global symmetries, analogous to much more abstract "changes of coordinates" in a situation where there is no preferred "inertial" coordinate system that covers the entire physical system. A gauge theory is a mathematical model that has symmetries of this kind, together with a set of techniques for making physical predictions consistent with the symmetries of the model.
Example of global symmetry
When a quantity occurring in the mathematical configuration is not just a number but has some geometrical significance, such as a velocity or an axis of rotation, its representation as numbers arranged in a vector or matrix is also changed by a coordinate transformation. For instance, if one description of a pattern of fluid flow states that the fluid velocity in the neighborhood of (x=1, y=0) is 1 m/s in the positive x direction, then a description of the same situation in which the coordinate system has been rotated clockwise by 90 degrees will state that the fluid velocity in the neighborhood of (x=0, y=1) is 1 m/s in the positive y direction. The coordinate transformation has affected both the coordinate system used to identify the location of the measurement and the basis in which its value is expressed. As long as this transformation is performed globally (affecting the coordinate basis in the same way at every point), the effect on values that represent the rate of change of some quantity along some path in space and time as it passes through point P is the same as the effect on values that are truly local to P.
Use of fiber bundles to describe local symmetries
In order to adequately describe physical situations in more complex theories, it is often necessary to introduce a "coordinate basis" for some of the objects of the theory that do not have this simple relationship to the coordinates used to label points in space and time. (In mathematical terms, the theory involves a
fiber bundle in which the fiber at each point of the base space consists of possible coordinate bases for use when describing the values of objects at that point.) In order to spell out a mathematical configuration, one must choose a particular coordinate basis at each point (a
local section of the fiber bundle) and express the values of the objects of the theory (usually "
fields" in the physicist's sense) using this basis. Two such mathematical configurations are equivalent (describe the same physical situation) if they are related by a transformation of this abstract coordinate basis (a change of local section, or
gauge transformation).
In most gauge theories, the set of possible transformations of the abstract gauge basis at an individual point in space and time is a finite-dimensional
Lie group. The simplest such group is
U(1), which appears in the modern formulation of
quantum electrodynamics (QED) via its use of
complex numbers. QED is generally regarded as the first, and simplest, physical gauge theory. The set of possible gauge transformations of the entire configuration of a given gauge theory also forms a group, the
gauge group of the theory. An element of the gauge group can be parameterized by a smoothly varying function from the points of spacetime to the (finite-dimensional) Lie group, whose value at each point represents the action of the gauge transformation on the fiber over that point.
A gauge transformation with constant parameter at every point in space and time is analogous to a rigid rotation of the geometric coordinate system; it represents a
global symmetry of the gauge representation. As in the case of a rigid rotation, this gauge transformation affects expressions that represent the rate of change along a path of some gauge-dependent quantity in the same way as those that represent a truly local quantity. A gauge transformation whose parameter is
not a constant function is referred to as a
local symmetry; its effect on expressions that involve a
derivative is qualitatively different from that on expressions that don't. (This is analogous to a non-inertial change of reference frame, which can produce a
Coriolis effect.)
Gauge fields
The "gauge covariant" version of a gauge theory accounts for this effect by introducing a
gauge field (in mathematical language, an
Ehresmann connection) and formulating all rates of change in terms of the
covariant derivative with respect to this connection. The gauge field becomes an essential part of the description of a mathematical configuration. A configuration in which the gauge field can be eliminated by a gauge transformation has the property that its
field strength (in mathematical language, its
curvature) is zero everywhere; a gauge theory is
not limited to these configurations. In other words, the distinguishing characteristic of a gauge theory is that the gauge field does not merely compensate for a poor choice of coordinate system; there is generally no gauge transformation that makes the gauge field vanish.
When analyzing the
dynamics of a gauge theory, the gauge field must be treated as a dynamical variable, similarly to other objects in the description of a physical situation. In addition to its
interaction with other objects via the covariant derivative, the gauge field typically contributes
energy in the form of a "self-energy" term. One can obtain the equations for the gauge theory by:
- starting from a naïve ansatz without the gauge field (in which the derivatives appear in a "bare" form);
- listing those global symmetries of the theory that can be characterized by a continuous parameter (generally an abstract equivalent of a rotation angle);
- computing the correction terms that result from allowing the symmetry parameter to vary from place to place; and
- reinterpreting these correction terms as couplings to one or more gauge fields, and giving these fields appropriate self-energy terms and dynamical behavior.
This is the sense in which a gauge theory "extends" a global symmetry to a local symmetry, and closely resembles the historical development of the gauge theory of gravity known as
general relativity.
Physical experiments
Gauge theories are used to model the results of physical experiments, essentially by:
- limiting the universe of possible configurations to those consistent with the information used to set up the experiment, and then
- computing the probability distribution of the possible outcomes that the experiment is designed to measure.
The mathematical descriptions of the "setup information" and the "possible measurement outcomes" (loosely speaking, the "boundary conditions" of the experiment) are generally not expressible without reference to a particular coordinate system, including a choice of gauge. (If nothing else, one assumes that the experiment has been adequately isolated from "external" influence, which is itself a gauge-dependent statement.) Mishandling gauge dependence in boundary conditions is a frequent source of
anomalies in gauge theory calculations, and gauge theories can be broadly classified by their approaches to anomaly avoidance.
Continuum theories
The two gauge theories mentioned above (continuum electrodynamics and general relativity) are examples of continuum field theories. The techniques of calculation in a continuum theory implicitly assume that:
- given a completely fixed choice of gauge, the boundary conditions of an individual configuration can in principle be completely described;
- given a completely fixed gauge and a complete set of boundary conditions, the principle of least action determines a unique mathematical configuration (and therefore a unique physical situation) consistent with these bounds;
- the likelihood of possible measurement outcomes can be determined by:
- establishing a probability distribution over all physical situations determined by boundary conditions that are consistent with the setup information,
- establishing a probability distribution of measurement outcomes for each possible physical situation, and
- convolving these two probability distributions to get a distribution of possible measurement outcomes consistent with the setup information; and
- fixing the gauge introduces no anomalies in the calculation, due either to gauge dependence in describing partial information about boundary conditions or to incompleteness of the theory.
These assumptions are close enough to valid, across a wide range of energy scales and experimental conditions, to allow these theories to make accurate predictions about almost all of the phenomena encountered in daily life, from light, heat, and electricity to eclipses and spaceflight. They fail only at the smallest and largest scales (due to omissions in the theories themselves) and when the mathematical techniques themselves break down (most notably in the case of
turbulence and other
chaotic phenomena).
Quantum field theories
Other than these "classical" continuum field theories, the most widely known gauge theories are
quantum field theories, including
quantum electrodynamics and the
Standard Model of elementary particle physics. The starting point of a quantum field theory is much like that of its continuum analog: a gauge-covariant
action integral which characterizes "allowable" physical situations according to the
principle of least action. However, continuum and quantum theories differ significantly in how they handle the excess degrees of freedom represented by gauge transformations. Continuum theories, and most pedagogical treatments of the simplest quantum field theories, use a
gauge fixing prescription to reduce the orbit of mathematical configurations that represent a given physical situation to a smaller orbit related by a smaller gauge group (the global symmetry group, or perhaps even the trivial group).
More sophisticated quantum field theories, in particular those which involve a
non-abelian gauge group, break the gauge symmetry within the techniques of
perturbation theory by introducing additional fields (the
Faddeev–Popov ghosts) and counterterms motivated by
anomaly cancellation, in an approach known as
BRST quantization. While these concerns are in one sense highly technical, they are also closely related to the nature of measurement, the limits on knowledge of a physical situation, and the interactions between incompletely specified experimental conditions and incompletely understood physical theory. The mathematical techniques that have been developed in order to make gauge theories tractable have found many other applications, from
solid-state physics and
crystallography to
low-dimensional topology.
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