In physics, a gauge theory is a type of field
theory in which the Lagrangian is invariant
under certain Lie groups of local transformations.
The term gauge refers to any specific mathematical
formalism to regulate redundant degrees of
freedom in the Lagrangian. The transformations
between possible gauges, called gauge transformations,
form a Lie group—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 field (usually
a 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, then the gauge theory
is referred to as non-abelian gauge theory,
the usual example being the Yang–Mills theory.
Many powerful theories in physics are described
by Lagrangians that are invariant under some
symmetry transformation groups. When they
are invariant under a transformation identically
performed at every point in the spacetime
in which the physical processes occur, they
are said to have a global symmetry. Local
symmetry, the cornerstone of gauge theories,
is a stronger constraint. In fact, a global
symmetry is just a local symmetry whose group's
parameters are fixed in spacetime (the same
way a constant value can be understood as
a function of a certain parameter, the output
of which is always the same).
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
four-potential, 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.
Gauge theories are also important in explaining
gravitation in the theory of general relativity.
Its case is somewhat unusual in that the gauge
field is a tensor, the Lanczos tensor. Theories
of quantum gravity, beginning with gauge gravitation
theory, also postulate the existence of a
gauge boson known as the graviton. Gauge symmetries
can be viewed as analogues of the principle
of general covariance of general relativity
in which the coordinate system can be chosen
freely under arbitrary diffeomorphisms of
spacetime. Both gauge invariance and diffeomorphism
invariance reflect a redundancy in the description
of the system. An alternative theory of gravitation,
gauge theory gravity, replaces the principle
of general covariance with a true gauge principle
with new gauge fields.
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 ==
The earliest field theory having a gauge symmetry
was Maxwell's formulation, in 1864–65, of
electrodynamics ("A Dynamical Theory of the
Electromagnetic Field") which stated that
any function whose curl vanishes—and can
therefore normally be written as a gradient—could
be added to the vector potential without affecting
the magnetic field. 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
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, which is 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 1941.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, 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
that enabled the calculation of certain topological
invariants (the Seiberg–Witten invariants).
These contributions to mathematics from gauge
theory have led to a renewed interest in this
area.
The importance of gauge theories in physics
is exemplified in 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 general relativity, are,
in one way or another, gauge theories.
See Pickering for more about the history of
gauge and quantum field theories.
== Description ==
=== Global and local symmetries ===
==== Global symmetry ====
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
states 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.
==== Local symmetry ====
===== 
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, such that
the value of the function and its derivatives
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, similar 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 used to model the results of
physical experiments engage in:
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.We cannot express the
mathematical descriptions of the "setup information"
and the "possible measurement outcomes", or
the "boundary conditions" of the experiment,
without reference to a particular coordinate
system, including a choice of gauge. One assumes
an adequate experiment isolated from "external"
influence that is itself a gauge-dependent
statement. Mishandling gauge dependence calculations
in boundary conditions is a frequent source
of anomalies, and approaches to anomaly avoidance
classifies gauge theories.
=== Continuum theories ===
The two gauge theories mentioned above, continuum
electrodynamics and general relativity, are
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
are completely described
given a completely fixed gauge and a complete
set of boundary conditions, the least action
determines a unique mathematical configuration
and therefore a unique physical situation
consistent with these bounds
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.Determination
of the likelihood of possible measurement
outcomes proceed by:
establishing a probability distribution over
all physical situations determined by boundary
conditions consistent with the setup information
establishing a probability distribution of
measurement outcomes for each possible physical
situation
convolving these two probability distributions
to get a distribution of possible measurement
outcomes consistent with the setup informationThese
assumptions have enough validity 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: light,
heat, and electricity, eclipses, spaceflight,
etc. 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 that 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 that 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.
== Classical gauge theory ==
=== 
Classical electromagnetism ===
Historically, the first example of gauge symmetry
discovered was classical electromagnetism.
In electrostatics, one can either discuss
the electric field, E, or its corresponding
electric potential, V. Knowledge of one makes
it possible to find the other, except that
potentials differing by a constant,
V
→
V
+
C
{\displaystyle V\rightarrow V+C}
, correspond to the same electric field. This
is because the electric field relates to changes
in the potential from one point in space to
another, and the constant C would cancel out
when subtracting to find the change in potential.
In terms of vector calculus, the electric
field is the gradient of the potential,
E
=
−
∇
V
{\displaystyle \mathbf {E} =-\nabla V}
. Generalizing from static electricity to
electromagnetism, we have a second potential,
the vector potential A, with
E
=
−
∇
V
−
∂
A
∂
t
B
=
∇
×
A
{\displaystyle {\begin{aligned}\mathbf {E}
&=-\nabla V-{\frac {\partial \mathbf {A} }{\partial
t}}\\\mathbf {B} &=\nabla \times \mathbf {A}
\end{aligned}}}
The general gauge transformations now become
not just
V
→
V
+
C
{\displaystyle V\rightarrow V+C}
but
A
→
A
+
∇
f
V
→
V
−
∂
f
∂
t
{\displaystyle {\begin{aligned}\mathbf {A}
&\rightarrow \mathbf {A} +\nabla f\\V&\rightarrow
V-{\frac {\partial f}{\partial t}}\end{aligned}}}
where f is any twice differentiable function
that depends on position and time. The fields
remain the same under the gauge transformation,
and therefore Maxwell's equations are still
satisfied. That is, Maxwell's equations have
a gauge symmetry.
=== An example: Scalar O(n) gauge theory ===
The remainder of 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.The following
illustrates how local gauge invariance can
be "motivated" heuristically starting from
global symmetry properties, and how it leads
to an interaction between originally non-interacting
fields.
Consider a set of n non-interacting real scalar
fields, with equal masses m. This system is
described by an action that is the sum of
the (usual) action for each scalar field
φ
i
{\displaystyle \varphi _{i}}
S
=
∫
d
4
x
∑
i
=
1
n
[
1
2
∂
μ
φ
i
∂
μ
φ
i
−
1
2
m
2
φ
i
2
]
{\displaystyle {\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
L
=
1
2
(
∂
μ
Φ
)
T
∂
μ
Φ
−
1
2
m
2
Φ
T
Φ
{\displaystyle \ {\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
Φ
=
(
φ
1
,
φ
2
,
…
,
φ
n
)
T
{\displaystyle \ \Phi =(\varphi _{1},\varphi
_{2},\ldots ,\varphi _{n})^{T}}
The term
∂
μ
{\displaystyle \partial _{\mu }}
is Einstein notation for the partial derivative
of
Φ
{\displaystyle \Phi }
in each of the four dimensions.
It is now transparent that the Lagrangian
is invariant under the transformation
Φ
↦
Φ
′
=
G
Φ
{\displaystyle \ \Phi \mapsto \Phi '=G\Phi
}
whenever G is a constant matrix belonging
to the n-by-n orthogonal group O(n). This
is seen to preserve the Lagrangian, since
the derivative of
Φ
{\displaystyle \Phi }
transforms identically to
Φ
{\displaystyle \Phi }
and both quantities appear inside dot products
in the Lagrangian (orthogonal transformations
preserve the dot product).
(
∂
μ
Φ
)
↦
(
∂
μ
Φ
)
′
=
G
∂
μ
Φ
{\displaystyle \ (\partial _{\mu }\Phi )\mapsto
(\partial _{\mu }\Phi )'=G\partial _{\mu }\Phi
}
This characterizes the global symmetry of
this particular Lagrangian, and the symmetry
group is often called the gauge group; the
mathematical term is structure group, especially
in the theory of G-structures. Incidentally,
Noether's theorem implies that invariance
under this group of transformations leads
to the conservation of the currents
J
μ
a
=
i
∂
μ
Φ
T
T
a
Φ
{\displaystyle \ J_{\mu }^{a}=i\partial _{\mu
}\Phi ^{T}T^{a}\Phi }
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.
In this case, the G matrices do not "pass
through" the derivatives, when G = G(x),
∂
μ
(
G
Φ
)
≠
G
(
∂
μ
Φ
)
{\displaystyle \ \partial _{\mu }(G\Phi )\neq
G(\partial _{\mu }\Phi )}
The failure of the derivative to commute with
"G" introduces an additional term (in keeping
with the product rule), which spoils the invariance
of the Lagrangian. In order to rectify this
we define a new derivative operator such that
the derivative of
Φ
{\displaystyle \Phi }
again transforms identically with
Φ
{\displaystyle \Phi }
(
D
μ
Φ
)
′
=
G
D
μ
Φ
{\displaystyle \ (D_{\mu }\Phi )'=GD_{\mu
}\Phi }
This new "derivative" is called a (gauge)
covariant derivative and takes the form
D
μ
=
∂
μ
−
i
g
A
μ
{\displaystyle \ D_{\mu }=\partial _{\mu }-igA_{\mu
}}
Where g is called the coupling constant; a
quantity defining the strength of an interaction.
After a simple calculation we can see that
the gauge field A(x) must transform as follows
A
μ
′
=
G
A
μ
G
−
1
+
i
g
(
∂
μ
G
)
G
−
1
{\displaystyle \ A'_{\mu }=GA_{\mu }G^{-1}+{\frac
{i}{g}}(\partial _{\mu }G)G^{-1}}
The gauge field is an element of the Lie algebra,
and can therefore be expanded as
A
μ
=
∑
a
A
μ
a
T
a
{\displaystyle \ A_{\mu }=\sum _{a}A_{\mu
}^{a}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
L
l
o
c
=
1
2
(
D
μ
Φ
)
T
D
μ
Φ
−
1
2
m
2
Φ
T
Φ
{\displaystyle \ {\mathcal {L}}_{\mathrm {loc}
}={\frac {1}{2}}(D_{\mu }\Phi )^{T}D^{\mu
}\Phi -{\frac {1}{2}}m^{2}\Phi ^{T}\Phi }
Pauli uses the term gauge transformation of
the first type to mean the transformation
of
Φ
{\displaystyle \Phi }
, while the compensating transformation in
A
{\displaystyle A}
is called a gauge transformation of the second
type.
The difference between this Lagrangian and
the original globally gauge-invariant Lagrangian
is seen to be the interaction Lagrangian
L
i
n
t
=
i
g
2
Φ
T
A
μ
T
∂
μ
Φ
+
i
g
2
(
∂
μ
Φ
)
T
A
μ
Φ
−
g
2
2
(
A
μ
Φ
)
T
A
μ
Φ
{\displaystyle \ {\mathcal {L}}_{\mathrm {int}
}=i{\frac {g}{2}}\Phi ^{T}A_{\mu }^{T}\partial
^{\mu }\Phi +i{\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.
However, to make this interaction physical
and not completely arbitrary, the mediator
A(x) needs to propagate in space. That is
dealt with in the next section by adding yet
another term,
L
g
f
{\displaystyle {\mathcal {L}}_{\mathrm {gf}
}}
, to the Lagrangian. In the quantized version
of the obtained 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 ===
The picture of a classical gauge theory developed
in the previous section 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
)
{\displaystyle 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 that
generates this field equation is locally gauge
invariant as well, one possible form for the
gauge field Lagrangian is
L
gf
=
−
1
2
Tr
⁡
(
F
μ
ν
F
μ
ν
)
=
−
1
4
F
a
μ
ν
F
μ
ν
a
{\displaystyle {\mathcal {L}}_{\text{gf}}=-{\frac
{1}{2}}\operatorname {Tr} \left(F^{\mu \nu
}F_{\mu \nu }\right)=-{\frac {1}{4}}F^{a\mu
\nu }F_{\mu \nu }^{a}}
where the
F
μ
ν
a
{\displaystyle F_{\mu \nu }^{a}}
are obtained from potentials
A
μ
a
{\displaystyle A_{\mu }^{a}}
, being the components of
A
(
x
)
{\displaystyle A(x)}
, by
F
μ
ν
a
=
∂
μ
A
ν
a
−
∂
ν
A
μ
a
+
g
∑
b
,
c
f
a
b
c
A
μ
b
A
ν
c
{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu
}A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+g\sum
_{b,c}f^{abc}A_{\mu }^{b}A_{\nu }^{c}}
and the
f
a
b
c
{\displaystyle f^{abc}}
are the structure constants of the Lie algebra
of the generators of the gauge group. This
formulation of the Lagrangian is called a
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 term there is
no field whose transformation counterweighs
the one of
A
{\displaystyle A}
. Invariance of this term under gauge transformations
is a particular case of a priori classical
(geometrical) symmetry. This symmetry must
be restricted in order to perform quantization,
the procedure being denominated gauge fixing,
but even after restriction, gauge transformations
may be possible.The complete Lagrangian for
the gauge theory is now
L
=
L
loc
+
L
gf
=
L
global
+
L
int
+
L
gf
{\displaystyle {\mathcal {L}}={\mathcal {L}}_{\text{loc}}+{\mathcal
{L}}_{\text{gf}}={\mathcal {L}}_{\text{global}}+{\mathcal
{L}}_{\text{int}}+{\mathcal {L}}_{\text{gf}}}
=== An 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 that generates
the electron field's Dirac equation is
S
=
∫
ψ
¯
(
i
ℏ
c
γ
μ
∂
μ
−
m
c
2
)
ψ
d
4
x
{\displaystyle {\mathcal {S}}=\int {\bar {\psi
}}\left(i\hbar c\,\gamma ^{\mu }\partial _{\mu
}-mc^{2}\right)\psi \,\mathrm {d} ^{4}x}
The global symmetry for this system is
ψ
↦
e
i
θ
ψ
{\displaystyle \psi \mapsto e^{i\theta }\psi
}
The gauge group here is U(1), just rotations
of the phase angle of the field, with the
particular rotation determined by the constant
θ.
"Localising" this symmetry implies the replacement
of θ by θ(x). An appropriate covariant derivative
is then
D
μ
=
∂
μ
−
i
e
ℏ
A
μ
{\displaystyle D_{\mu }=\partial _{\mu }-i{\frac
{e}{\hbar }}A_{\mu }}
Identifying the "charge" e (not to be confused
with the mathematical constant e in the symmetry
description) 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
L
int
=
e
ℏ
ψ
¯
(
x
)
γ
μ
ψ
(
x
)
A
μ
(
x
)
=
J
μ
(
x
)
A
μ
(
x
)
{\displaystyle {\mathcal {L}}_{\text{int}}={\frac
{e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu
}\psi (x)A_{\mu }(x)=J^{\mu }(x)A_{\mu }(x)}
where
J
μ
(
x
)
=
e
ℏ
ψ
¯
(
x
)
γ
μ
ψ
(
x
)
{\displaystyle J^{\mu }(x)={\frac {e}{\hbar
}}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)}
is the electric current four vector in the
Dirac field. The gauge principle is therefore
seen to naturally introduce the so-called
minimal coupling of the electromagnetic field
to the electron field.
Adding a Lagrangian for the gauge field
A
μ
(
x
)
{\displaystyle A_{\mu }(x)}
in terms of the field strength tensor exactly
as in electrodynamics, one obtains the Lagrangian
used as the starting point in quantum electrodynamics.
L
QED
=
ψ
¯
(
i
ℏ
c
γ
μ
D
μ
−
m
c
2
)
ψ
−
1
4
μ
0
F
μ
ν
F
μ
ν
{\displaystyle {\mathcal {L}}_{\text{QED}}={\bar
{\psi }}\left(i\hbar c\,\gamma ^{\mu }D_{\mu
}-mc^{2}\right)\psi -{\frac {1}{4\mu _{0}}}F_{\mu
\nu }F^{\mu \nu }}
== 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.
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 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 representations that
transform covariantly pointwise (called by
physicists gauge transformations of the first
kind), representations that transform as a
connection form (called by physicists gauge
transformations of the second kind, an affine
representation)—and other more general representations,
such as the B field in BF theory. There are
more general nonlinear representations (realizations),
but these are extremely complicated. Still,
nonlinear sigma models transform nonlinearly,
so there are applications.
If there is 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.
Connections (gauge connection) define this
principal bundle, yielding a covariant derivative
∇ in each associated vector bundle. If a
local frame is chosen (a local basis of sections),
then this covariant derivative is represented
by the connection form A, a Lie algebra-valued
1-form, which is called the gauge potential
in physics. This is evidently not an intrinsic
but a frame-dependent quantity. The curvature
form F, a Lie algebra-valued 2-form that is
an intrinsic quantity, is constructed from
a connection form by
F
=
d
A
+
A
∧
A
{\displaystyle \mathbf {F} =\mathrm {d} \mathbf
{A} +\mathbf {A} \wedge \mathbf {A} }
where d stands for the exterior derivative
and
∧
{\displaystyle \wedge }
stands for the wedge product. (
A
{\displaystyle \mathbf {A} }
is an element of the vector space spanned
by the generators
T
a
{\displaystyle T^{a}}
, and so the components of
A
{\displaystyle \mathbf {A} }
do not commute with one another. Hence the
wedge product
A
∧
A
{\displaystyle \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,
δ
ε
A
=
[
ε
,
A
]
−
d
ε
{\displaystyle \delta _{\varepsilon }\mathbf
{A} =[\varepsilon ,\mathbf {A} ]-\mathrm {d}
\varepsilon }
where
[
⋅
,
⋅
]
{\displaystyle [\cdot ,\cdot ]}
is the Lie bracket.
One nice thing is that if
δ
ε
X
=
ε
X
{\displaystyle \delta _{\varepsilon }X=\varepsilon
X}
, then
δ
ε
D
X
=
ε
D
X
{\displaystyle \delta _{\varepsilon }DX=\varepsilon
DX}
where D is the covariant derivative
D
X
=
d
e
f
d
X
+
A
X
{\displaystyle DX\ {\stackrel {\mathrm {def}
}{=}}\ \mathrm {d} X+\mathbf {A} X}
Also,
δ
ε
F
=
ε
F
{\displaystyle \delta _{\varepsilon }\mathbf
{F} =\varepsilon \mathbf {F} }
, which means
F
{\displaystyle \mathbf {F} }
transforms covariantly.
Not all gauge transformations can be generated
by infinitesimal gauge transformations in
general. An example is 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
1
4
g
2
∫
Tr
⁡
[
∗
F
∧
F
]
{\displaystyle {\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:
χ
(
ρ
)
(
P
{
e
∫
γ
A
}
)
{\displaystyle \chi ^{(\rho )}\left({\mathcal
{P}}\left\{e^{\int _{\gamma }A}\right\}\right)}
where χ is the character of a complex representation
ρ and
P
{\displaystyle {\mathcal {P}}}
represents the path-ordered operator.
The formalism of gauge theory carries over
to a general setting. For example, it is sufficient
to ask that a vector bundle have a metric
connection; when one does so, one finds that
the metric connection satisfies the Yang-Mills
equations of motion.
== 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
allows simplification of some computations:
for example Ward identities connect different
renormalization constants.
=== Methods and aims ===
The first gauge theory 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 now 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 intended 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 suited 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 symmetries of the classical 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.
== Pure gauge ==
A pure gauge is the set of field configurations
obtained by a gauge transformation on the
null-field configuration, i.e., a gauge-transform
of zero. So it is a particular "gauge orbit"
in the field configuration's space.
Thus, in the abelian case, where
A
μ
(
x
)
→
A
μ
′
(
x
)
=
A
μ
(
x
)
+
∂
μ
f
(
x
)
{\displaystyle A_{\mu }(x)\rightarrow A'_{\mu
}(x)=A_{\mu }(x)+\partial _{\mu }f(x)}
, the pure gauge is just the set of field
configurations
A
μ
′
(
x
)
=
∂
μ
f
(
x
)
{\displaystyle A'_{\mu }(x)=\partial _{\mu
}f(x)}
for all f(x).
== See also
