Yang–Mills theory is a gauge theory based
on the SU(N) group, or more generally any
compact, reductive Lie algebra. Yang–Mills
theory seeks to describe the behavior of elementary
particles using these non-abelian Lie groups
and is at the core of the unification of the
electromagnetic force and weak forces (i.e.
U(1) × SU(2)) as well as quantum chromodynamics,
the theory of the strong force (based on SU(3)).
Thus it forms the basis of our understanding
of the Standard Model of particle physics.
== History and theoretical description ==
In a private correspondence, Wolfgang Pauli
formulated in 1953 a six-dimensional theory
of Einstein's field equations of general relativity,
extending the five-dimensional theory of Kaluza,
Klein, Fock and others to a higher-dimensional
internal space. However, there is no evidence
that Pauli developed the Lagrangian of a gauge
field or the quantization of it. Because Pauli
found that his theory "leads to some rather
unphysical shadow particles", he refrained
from publishing his results formally. Although
Pauli did not publish his six-dimensional
theory, he gave two talks about it in Zürich.
Recent research shows that an extended Kaluza–Klein
theory is in general not equivalent to Yang–Mills
theory, as the former contains additional
terms.In early 1954, Chen Ning Yang and Robert
Mills extended the concept of gauge theory
for abelian groups, e.g. quantum electrodynamics,
to nonabelian groups to provide an explanation
for strong interactions. The idea by Yang–Mills
was criticized by Pauli, as the quanta of
the Yang–Mills field must be massless in
order to maintain gauge invariance. The idea
was set aside until 1960, when the concept
of particles acquiring mass through symmetry
breaking in massless theories was put forward,
initially by Jeffrey Goldstone, Yoichiro Nambu,
and Giovanni Jona-Lasinio.
This prompted a significant restart of Yang–Mills
theory studies that proved successful in the
formulation of both electroweak unification
and quantum chromodynamics (QCD). The electroweak
interaction is described by SU(2) × U(1)
group while QCD is an SU(3) Yang–Mills theory.
The electroweak theory is obtained by combining
SU(2) with U(1), where quantum electrodynamics
(QED) is described by a U(1) group, and is
replaced in the unified electroweak theory
by a U(1) group representing a weak hypercharge
rather than electric charge. The massless
bosons from the SU(2) × U(1) theory mix after
spontaneous symmetry breaking to produce the
3 massive weak bosons, and the photon field.
The Standard Model combines the strong interaction
with the unified electroweak interaction (unifying
the weak and electromagnetic interaction)
through the symmetry group SU(2) × U(1) × SU(3).
In the current epoch the strong interaction
is not unified with the electroweak interaction,
but from the observed running of the coupling
constants it is believed they all converge
to a single value at very high energies.
Phenomenology at lower energies in quantum
chromodynamics is not completely understood
due to the difficulties of managing such a
theory with a strong coupling. This may be
the reason why confinement has not been theoretically
proven, though it is a consistent experimental
observation. Proof that QCD confines at low
energy is a mathematical problem of great
relevance, and an award has been proposed
by the Clay Mathematics Institute for whoever
is also able to show that the Yang–Mills
theory has a mass gap and its existence.
== Mathematical overview ==
Yang–Mills theories are a special example
of gauge theory with a non-abelian symmetry
group given by the Lagrangian
L
g
f
=
−
1
2
Tr
⁡
(
F
2
)
=
−
1
4
F
a
μ
ν
F
μ
ν
a
{\displaystyle {\mathcal {L}}_{\mathrm {gf}
}=-{\frac {1}{2}}\operatorname {Tr} (F^{2})=-{\frac
{1}{4}}F^{a\mu \nu }F_{\mu \nu }^{a}}
with the generators of the Lie algebra, indexed
by a, corresponding to the F-quantities (the
curvature or field-strength form) satisfying
Tr
⁡
(
T
a
T
b
)
=
1
2
δ
a
b
,
[
T
a
,
T
b
]
=
i
f
a
b
c
T
c
,
{\displaystyle \operatorname {Tr} (T^{a}T^{b})={\frac
{1}{2}}\delta ^{ab},\quad [T^{a},T^{b}]=if^{abc}T^{c},}
where the fabc are structure constants of
the Lie algebra, and the covariant derivative
defined as
D
μ
=
I
∂
μ
−
i
g
T
a
A
μ
a
{\displaystyle D_{\mu }=I\partial _{\mu }-igT^{a}A_{\mu
}^{a}}
where I is the identity matrix (matching the
size of the generators),
A
μ
a
{\displaystyle A_{\mu }^{a}}
is the vector potential, and g is the coupling
constant. In four dimensions, the coupling
constant g is a pure number and for a SU(N)
group one has
a
,
b
,
c
=
1
…
N
2
−
1.
{\displaystyle a,b,c=1\ldots N^{2}-1.}
The relation
F
μ
ν
a
=
∂
μ
A
ν
a
−
∂
ν
A
μ
a
+
g
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}+gf^{abc}A_{\mu
}^{b}A_{\nu }^{c}}
can be derived by the commutator
[
D
μ
,
D
ν
]
=
−
i
g
T
a
F
μ
ν
a
.
{\displaystyle [D_{\mu },D_{\nu }]=-igT^{a}F_{\mu
\nu }^{a}.}
The field has the property of being self-interacting
and equations of motion that one obtains are
said to be semilinear, as nonlinearities are
both with and without derivatives. This means
that one can manage this theory only by perturbation
theory, with small nonlinearities.
Note that the transition between "upper" ("contravariant")
and "lower" ("covariant") vector or tensor
components is trivial for a indices (e.g.
f
a
b
c
=
f
a
b
c
{\displaystyle f^{abc}=f_{abc}}
), whereas for μ and ν it is nontrivial,
corresponding e.g. to the usual Lorentz signature,
η
μ
ν
=
d
i
a
g
(
+
−
−
−
)
{\displaystyle \eta _{\mu \nu }={\rm {diag}}(+---)}
.
From the given Lagrangian one can derive the
equations of motion given by
∂
μ
F
μ
ν
a
+
g
f
a
b
c
A
μ
b
F
μ
ν
c
=
0.
{\displaystyle \partial ^{\mu }F_{\mu \nu
}^{a}+gf^{abc}A^{\mu b}F_{\mu \nu }^{c}=0.}
Putting
F
μ
ν
=
T
a
F
μ
ν
a
{\displaystyle F_{\mu \nu }=T^{a}F_{\mu \nu
}^{a}}
, these can be rewritten as
(
D
μ
F
μ
ν
)
a
=
0.
{\displaystyle (D^{\mu }F_{\mu \nu })^{a}=0.}
A Bianchi identity holds
(
D
μ
F
ν
κ
)
a
+
(
D
κ
F
μ
ν
)
a
+
(
D
ν
F
κ
μ
)
a
=
0
{\displaystyle (D_{\mu }F_{\nu \kappa })^{a}+(D_{\kappa
}F_{\mu \nu })^{a}+(D_{\nu }F_{\kappa \mu
})^{a}=0}
which is equivalent to the Jacobi identity
[
D
μ
,
[
D
ν
,
D
κ
]
]
+
[
D
κ
,
[
D
μ
,
D
ν
]
]
+
[
D
ν
,
[
D
κ
,
D
μ
]
]
=
0
{\displaystyle [D_{\mu },[D_{\nu },D_{\kappa
}]]+[D_{\kappa },[D_{\mu },D_{\nu }]]+[D_{\nu
},[D_{\kappa },D_{\mu }]]=0}
since
[
D
μ
,
F
ν
κ
a
]
=
D
μ
F
ν
κ
a
{\displaystyle [D_{\mu },F_{\nu \kappa }^{a}]=D_{\mu
}F_{\nu \kappa }^{a}}
. Define the dual strength tensor
F
~
μ
ν
=
1
2
ε
μ
ν
ρ
σ
F
ρ
σ
{\displaystyle {\tilde {F}}^{\mu \nu }={\frac
{1}{2}}\varepsilon ^{\mu \nu \rho \sigma }F_{\rho
\sigma }}
, then the Bianchi identity can be rewritten
as
D
μ
F
~
μ
ν
=
0.
{\displaystyle D_{\mu }{\tilde {F}}^{\mu \nu
}=0.}
A source
J
μ
a
{\displaystyle J_{\mu }^{a}}
enters into the equations of motion as
∂
μ
F
μ
ν
a
+
g
f
a
b
c
A
b
μ
F
μ
ν
c
=
−
J
ν
a
.
{\displaystyle \partial ^{\mu }F_{\mu \nu
}^{a}+gf^{abc}A^{b\mu }F_{\mu \nu }^{c}=-J_{\nu
}^{a}.}
Note that the currents must properly change
under gauge group transformations.
We give here some comments about the physical
dimensions of the coupling. In D dimensions,
the field scales as
[
A
]
=
[
L
2
−
D
2
]
{\displaystyle [A]=[L^{\frac {2-D}{2}}]}
and so the coupling must scale as
[
g
2
]
=
[
L
D
−
4
]
{\displaystyle [g^{2}]=[L^{D-4}]}
. This implies that Yang–Mills theory is
not renormalizable for dimensions greater
than four. Furthermore, for D = 4, the coupling
is dimensionless and both the field and the
square of the coupling have the same dimensions
of the field and the coupling of a massless
quartic scalar field theory. So, these theories
share the scale invariance at the classical
level.
== Quantization ==
A method of quantizing the Yang–Mills theory
is by functional methods, i.e. path integrals.
One introduces a generating functional for
n-point functions as
Z
[
j
]
=
∫
[
d
A
]
exp
⁡
[
−
i
2
∫
d
4
x
Tr
⁡
(
F
μ
ν
F
μ
ν
)
+
i
∫
d
4
x
j
μ
a
(
x
)
A
a
μ
(
x
)
]
,
{\displaystyle Z[j]=\int [dA]\exp \left[-{\frac
{i}{2}}\int d^{4}x\operatorname {Tr} (F^{\mu
\nu }F_{\mu \nu })+i\int d^{4}x\,j_{\mu }^{a}(x)A^{a\mu
}(x)\right],}
but this integral has no meaning as it is
because the potential vector can be arbitrarily
chosen due to the gauge freedom. This problem
was already known for quantum electrodynamics
but here becomes more severe due to non-abelian
properties of the gauge group. A way out has
been given by Ludvig Faddeev and Victor Popov
with the introduction of a ghost field (see
Faddeev–Popov ghost) that has the property
of being unphysical since, although it agrees
with Fermi–Dirac statistics, it is a complex
scalar field, which violates the spin–statistics
theorem. So, we can write the generating functional
as
Z
[
j
,
ε
¯
,
ε
]
=
∫
[
d
A
]
[
d
c
¯
]
[
d
c
]
exp
⁡
{
i
S
F
[
∂
A
,
A
]
+
i
S
g
f
[
∂
A
]
+
i
S
g
[
∂
c
,
∂
c
¯
,
c
,
c
¯
,
A
]
}
exp
⁡
{
i
∫
d
4
x
j
μ
a
(
x
)
A
a
μ
(
x
)
+
i
∫
d
4
x
[
c
¯
a
(
x
)
ε
a
(
x
)
+
ε
¯
a
(
x
)
c
a
(
x
)
]
}
{\displaystyle {\begin{aligned}Z[j,{\bar {\varepsilon
}},\varepsilon ]&=\int [dA][d{\bar {c}}][dc]\exp
\left\{iS_{F}[\partial A,A]+iS_{gf}[\partial
A]+iS_{g}[\partial c,\partial {\bar {c}},c,{\bar
{c}},A]\right\}\\&\exp \left\{i\int d^{4}xj_{\mu
}^{a}(x)A^{a\mu }(x)+i\int d^{4}x[{\bar {c}}^{a}(x)\varepsilon
^{a}(x)+{\bar {\varepsilon }}^{a}(x)c^{a}(x)]\right\}\end{aligned}}}
being
S
F
=
−
1
2
∫
d
4
x
Tr
⁡
(
F
μ
ν
F
μ
ν
)
{\displaystyle S_{F}=-{\frac {1}{2}}\int \operatorname
{d} \!^{4}x\operatorname {Tr} (F^{\mu \nu
}F_{\mu \nu })}
for the field,
S
g
f
=
−
1
2
ξ
∫
d
4
x
(
∂
⋅
A
)
2
{\displaystyle S_{gf}=-{\frac {1}{2\xi }}\int
\operatorname {d} \!^{4}x(\partial \cdot A)^{2}}
for the gauge fixing and
S
g
=
−
∫
d
4
x
(
c
¯
a
∂
μ
∂
μ
c
a
+
g
c
¯
a
f
a
b
c
∂
μ
A
b
μ
c
c
)
{\displaystyle S_{g}=-\int \operatorname {d}
\!^{4}x({\bar {c}}^{a}\partial _{\mu }\partial
^{\mu }c^{a}+g{\bar {c}}^{a}f^{abc}\partial
_{\mu }A^{b\mu }c^{c})}
for the ghost. This is the expression commonly
used to derive Feynman's rules (see Feynman
diagram). Here we have ca for the ghost field
while ξ fixes the gauge's choice for the
quantization. Feynman's rules obtained from
this functional are the following
These rules for Feynman diagrams can be obtained
when the generating functional given above
is rewritten as
Z
[
j
,
ε
¯
,
ε
]
=
exp
⁡
(
−
i
g
∫
d
4
x
δ
i
δ
ε
¯
a
(
x
)
f
a
b
c
∂
μ
i
δ
δ
j
μ
b
(
x
)
i
δ
δ
ε
c
(
x
)
)
×
exp
⁡
(
−
i
g
∫
d
4
x
f
a
b
c
∂
μ
i
δ
δ
j
ν
a
(
x
)
i
δ
δ
j
μ
b
(
x
)
i
δ
δ
j
c
ν
(
x
)
)
×
exp
⁡
(
−
i
g
2
4
∫
d
4
x
f
a
b
c
f
a
r
s
i
δ
δ
j
μ
b
(
x
)
i
δ
δ
j
ν
c
(
x
)
i
δ
δ
j
r
μ
(
x
)
i
δ
δ
j
s
ν
(
x
)
)
×
Z
0
[
j
,
ε
¯
,
ε
]
{\displaystyle {\begin{aligned}Z[j,{\bar {\varepsilon
}},\varepsilon ]&=\exp \left(-ig\int d^{4}x\,{\frac
{\delta }{i\delta {\bar {\varepsilon }}^{a}(x)}}f^{abc}\partial
_{\mu }{\frac {i\delta }{\delta j_{\mu }^{b}(x)}}{\frac
{i\delta }{\delta \varepsilon ^{c}(x)}}\right)\\&\qquad
\times \exp \left(-ig\int d^{4}xf^{abc}\partial
_{\mu }{\frac {i\delta }{\delta j_{\nu }^{a}(x)}}{\frac
{i\delta }{\delta j_{\mu }^{b}(x)}}{\frac
{i\delta }{\delta j^{c\nu }(x)}}\right)\\&\qquad
\qquad \times \exp \left(-i{\frac {g^{2}}{4}}\int
d^{4}xf^{abc}f^{ars}{\frac {i\delta }{\delta
j_{\mu }^{b}(x)}}{\frac {i\delta }{\delta
j_{\nu }^{c}(x)}}{\frac {i\delta }{\delta
j^{r\mu }(x)}}{\frac {i\delta }{\delta j^{s\nu
}(x)}}\right)\\&\qquad \qquad \qquad \times
Z_{0}[j,{\bar {\varepsilon }},\varepsilon
]\end{aligned}}}
with
Z
0
[
j
,
ε
¯
,
ε
]
=
exp
⁡
(
−
∫
d
4
x
d
4
y
ε
¯
a
(
x
)
C
a
b
(
x
−
y
)
ε
b
(
y
)
)
exp
⁡
(
1
2
∫
d
4
x
d
4
y
j
μ
a
(
x
)
D
a
b
μ
ν
(
x
−
y
)
j
ν
b
(
y
)
)
{\displaystyle Z_{0}[j,{\bar {\varepsilon
}},\varepsilon ]=\exp \left(-\int d^{4}xd^{4}y{\bar
{\varepsilon }}^{a}(x)C^{ab}(x-y)\varepsilon
^{b}(y)\right)\exp \left({\tfrac {1}{2}}\int
d^{4}xd^{4}yj_{\mu }^{a}(x)D^{ab\mu \nu }(x-y)j_{\nu
}^{b}(y)\right)}
being the generating functional of the free
theory. Expanding in g and computing the functional
derivatives, we are able to obtain all the
n-point functions with perturbation theory.
Using LSZ reduction formula we get from the
n-point functions the corresponding process
amplitudes, cross sections and decay rates.
The theory is renormalizable and corrections
are finite at any order of perturbation theory.
For quantum electrodynamics the ghost field
decouples because the gauge group is abelian.
This can be seen from the coupling between
the gauge field and the ghost field that is
c
¯
a
f
a
b
c
∂
μ
A
b
μ
c
c
{\displaystyle {\bar {c}}^{a}f^{abc}\partial
_{\mu }A^{b\mu }c^{c}}
. For the abelian case, all the structure
constants
f
a
b
c
{\displaystyle f^{abc}}
are zero and so there is no coupling. In the
non-abelian case, the ghost field appears
as a useful way to rewrite the quantum field
theory without physical consequences on the
observables of the theory such as cross sections
or decay rates.
One of the most important results obtained
for Yang–Mills theory is asymptotic freedom.
This result can be obtained by assuming that
the coupling constant g is small (so small
nonlinearities), as for high energies, and
applying perturbation theory. The relevance
of this result is due to the fact that a Yang–Mills
theory that describes strong interaction and
asymptotic freedom permits proper treatment
of experimental results coming from deep inelastic
scattering.
To obtain the behavior of the Yang–Mills
theory at high energies, and so to prove asymptotic
freedom, one applies perturbation theory assuming
a small coupling. This is verified a posteriori
in the ultraviolet limit. In the opposite
limit, the infrared limit, the situation is
the opposite, as the coupling is too large
for perturbation theory to be reliable. Most
of the difficulties that research meets is
just managing the theory at low energies.
That is the interesting case, being inherent
to the description of hadronic matter and,
more generally, to all the observed bound
states of gluons and quarks and their confinement
(see hadrons). The most used method to study
the theory in this limit is to try to solve
it on computers (see lattice gauge theory).
In this case, large computational resources
are needed to be sure the correct limit of
infinite volume (smaller lattice spacing)
is obtained. This is the limit the results
must be compared with. Smaller spacing and
larger coupling are not independent of each
other, and larger computational resources
are needed for each. As of today, the situation
appears somewhat satisfactory for the hadronic
spectrum and the computation of the gluon
and ghost propagators, but the glueball and
hybrids spectra are yet a questioned matter
in view of the experimental observation of
such exotic states. Indeed, the σ resonance
is not seen in any of such lattice computations
and contrasting interpretations have been
put forward. This is a hotly debated issue.
== Open problems ==
Yang–Mills theories met with general acceptance
in the physics community after Gerard 't Hooft,
in 1972, worked out their renormalization,
relying on a formulation of the problem worked
out by his advisor Martinus Veltman. (Their
work was recognized by the 1999 Nobel prize
in physics.) Renormalizability is obtained
even if the gauge bosons described by this
theory are massive, as in the electroweak
theory, provided the mass is only an "acquired"
one, generated by the Higgs mechanism.
The mathematics of the Yang–Mills theory
is a very active field of research, yielding
e.g. invariants of differentiable structures
on four-dimensional manifolds via work of
Simon Donaldson. Furthermore, the field of
Yang–Mills theories was included in the
Clay Mathematics Institute's list of "Millennium
Prize Problems". Here the prize-problem consists,
especially, in a proof of the conjecture that
the lowest excitations of a pure Yang–Mills
theory (i.e. without matter fields) have a
finite mass-gap with regard to the vacuum
state. Another open problem, connected with
this conjecture, is a proof of the confinement
property in the presence of additional Fermion
particles.
In physics the survey of Yang–Mills theories
does not usually start from perturbation analysis
or analytical methods, but more recently from
systematic application of numerical methods
to lattice gauge theories.
== See also
