An instanton (or pseudoparticle) is a notion
appearing in theoretical and mathematical
physics. An instanton is a classical solution
to equations of motion with a finite, non-zero
action, either in quantum mechanics or in
quantum field theory. More precisely, it is
a solution to the equations of motion of the
classical field theory on a Euclidean spacetime.
In such quantum theories, solutions to the
equations of motion may be thought of as critical
points of the action. The critical points
of the action may be local maxima of the action,
local minima, or saddle points. Instantons
are important in quantum field theory because:
they appear in the path integral as the leading
quantum corrections to the classical behavior
of a system, and
they can be used to study the tunneling behavior
in various systems such as a Yang–Mills
theory.Relevant to dynamics, families of instantons
allow to mutually relate the instantons, i.e.
different critical points of the equation
of motion. In physics instantons are particularly
important because the condensation of instantons
(and noise-induced anti-instantons) is believed
to be the explanation of the noise-induced
chaotic phase known as self-organized criticality.
== Mathematics ==
Mathematically, a Yang–Mills instanton is
a self-dual or anti-self-dual connection in
a principal bundle over a four-dimensional
Riemannian manifold that plays the role of
physical space-time in non-abelian gauge theory.
Instantons are topologically nontrivial solutions
of Yang–Mills equations that absolutely
minimize the energy functional within their
topological type. The first such solutions
were discovered in the case of four-dimensional
Euclidean space compactified to the four-dimensional
sphere, and turned out to be localized in
space-time, prompting the names pseudoparticle
and instanton.
Yang–Mills instantons have been explicitly
constructed in many cases by means of twistor
theory, which relates them to algebraic vector
bundles on algebraic surfaces, and via the
ADHM construction, or hyperkähler reduction
(see hyperkähler manifold), a sophisticated
linear algebra procedure. The groundbreaking
work of Simon Donaldson, for which he was
later awarded the Fields medal, used the moduli
space of instantons over a given four-dimensional
differentiable manifold as a new invariant
of the manifold that depends on its differentiable
structure and applied it to the construction
of homeomorphic but not diffeomorphic four-manifolds.
Many methods developed in studying instantons
have also been applied to monopoles. This
is because magnetic monopoles arise as solutions
of a dimensional reduction of the Yang–Mills
equations.
== Quantum mechanics ==
An instanton can be used to calculate the
transition probability for a quantum mechanical
particle tunneling through a potential barrier.
One example of a system with an instanton
effect is a particle in a double-well potential.
In contrast to a classical particle, there
is non-vanishing probability that it crosses
a region of potential energy higher than its
own energy.
=== Motivation of considering instantons ===
Let us consider the quantum mechanics of a
single particle motion inside the double-well
potential
V
(
x
)
=
1
4
(
x
2
−
1
)
2
.
{\displaystyle V(x)={1 \over 4}(x^{2}-1)^{2}.}
The potential energy takes its minimal value
at
x
=
±
1
{\displaystyle x=\pm 1}
, and these are called classical minima because
the particle tends to lie in one of them in
the classical mechanics. There are two lowest
energy states in the classical mechanics.
In quantum mechanics, we solve the Schrödinger
equation
−
ℏ
2
2
m
∂
2
∂
x
2
ψ
+
V
(
x
)
ψ
(
x
)
=
E
ψ
(
x
)
,
{\displaystyle -{\hbar ^{2} \over 2m}{\partial
^{2} \over \partial x^{2}}\psi +V(x)\psi (x)=E\psi
(x),}
to identify the energy eigenstates. If we
do this, we will find only the unique lowest-energy
state instead of two states. The ground-state
wave function localizes at both of the classical
minima
x
=
±
1
{\displaystyle x=\pm 1}
instead of only one of them because of the
quantum interference or quantum tunneling.
Instantons are the tool to understand why
this happens within the semi-classical approximation
of the path-integral formulation in Euclidean
time. We will first see this by using the
WKB approximation that approximately computes
the wave function itself, and will move on
to introduce instantons by using the path
integral formulation.
=== WKB approximation ===
One way to calculate this probability is by
means of the semi-classical WKB approximation,
which requires the value of
ℏ
{\displaystyle \hbar }
to be small. The time independent Schrödinger
equation for the particle reads
d
2
ψ
d
x
2
=
2
m
(
V
(
x
)
−
E
)
ℏ
2
ψ
.
{\displaystyle {\frac {d^{2}\psi }{dx^{2}}}={\frac
{2m(V(x)-E)}{\hbar ^{2}}}\psi .}
If the potential were constant, the solution
would be a plane wave, up to a proportionality
factor,
ψ
=
exp
⁡
(
−
i
k
x
)
{\displaystyle \psi =\exp(-\mathrm {i} kx)\,}
with
k
=
2
m
(
E
−
V
)
ℏ
.
{\displaystyle k={\frac {\sqrt {2m(E-V)}}{\hbar
}}.}
This means that if the energy of the particle
is smaller than the potential energy, one
obtains an exponentially decreasing function.
The associated tunneling amplitude is proportional
to
e
−
1
ℏ
∫
a
b
2
m
(
V
(
x
)
−
E
)
d
x
,
{\displaystyle e^{-{\frac {1}{\hbar }}\int
_{a}^{b}{\sqrt {2m(V(x)-E)}}\,dx},}
where a and b are the beginning and endpoint
of the tunneling trajectory.
=== Path integral interpretation via instantons
===
Alternatively, the use of path integrals allows
an instanton interpretation and the same result
can be obtained with this approach. In path
integral formulation, the transition amplitude
can be expressed as
K
(
a
,
b
;
t
)
=
⟨
x
=
a
|
e
−
i
H
t
ℏ
|
x
=
b
⟩
=
∫
d
[
x
(
t
)
]
e
i
S
[
x
(
t
)
]
ℏ
.
{\displaystyle K(a,b;t)=\langle x=a|e^{-{\frac
{i\mathbb {H} t}{\hbar }}}|x=b\rangle =\int
d[x(t)]e^{\frac {iS[x(t)]}{\hbar }}.}
Following the process of Wick rotation (analytic
continuation) to Euclidean spacetime (
i
t
→
τ
{\displaystyle it\rightarrow \tau }
), one gets
K
E
(
a
,
b
;
τ
)
=
⟨
x
=
a
|
e
−
H
τ
ℏ
|
x
=
b
⟩
=
∫
d
[
x
(
τ
)
]
e
−
S
E
[
x
(
τ
)
]
ℏ
,
{\displaystyle K_{E}(a,b;\tau )=\langle x=a|e^{-{\frac
{\mathbb {H} \tau }{\hbar }}}|x=b\rangle =\int
d[x(\tau )]e^{-{\frac {S_{E}[x(\tau )]}{\hbar
}}},}
with the Euclidean action
S
E
=
∫
τ
a
τ
b
(
1
2
m
(
d
x
d
τ
)
2
+
V
(
x
)
)
d
τ
.
{\displaystyle S_{E}=\int _{\tau _{a}}^{\tau
_{b}}\left({\frac {1}{2}}m\left({\frac {dx}{d\tau
}}\right)^{2}+V(x)\right)d\tau .}
The potential energy changes sign
V
(
x
)
→
−
V
(
x
)
{\displaystyle V(x)\rightarrow -V(x)}
under the Wick rotation and the minima transform
into maxima, thereby
V
(
x
)
{\displaystyle V(x)}
exhibits two "hills" of maximal energy.
Let us now consider the local minimum of the
Euclidean action
S
E
{\displaystyle S_{E}}
with the double-well potential
V
(
x
)
=
1
4
(
x
2
−
1
)
2
{\displaystyle V(x)={1 \over 4}(x^{2}-1)^{2}}
, and we set
m
=
1
{\displaystyle m=1}
just for simplicity of computation. Since
we want to know how the two classically lowest
energy states
x
=
±
1
{\displaystyle x=\pm 1}
are connected, let us set
a
=
−
1
{\displaystyle a=-1}
and
b
=
1
{\displaystyle b=1}
.
For
a
=
−
1
{\displaystyle a=-1}
and
b
=
1
{\displaystyle b=1}
, we can rewrite the Euclidean action as
S
E
=
∫
τ
a
τ
b
d
τ
1
2
(
d
x
d
τ
−
2
V
(
x
)
)
2
+
2
∫
τ
a
τ
b
d
τ
d
x
d
τ
V
(
x
)
{\displaystyle S_{E}=\int _{\tau _{a}}^{\tau
_{b}}d\tau {1 \over 2}\left({dx \over d\tau
}-{\sqrt {2V(x)}}\right)^{2}+{\sqrt {2}}\int
_{\tau _{a}}^{\tau _{b}}d\tau {dx \over d\tau
}{\sqrt {V(x)}}}
=
∫
τ
a
τ
b
d
τ
1
2
(
d
x
d
τ
−
2
V
(
x
)
)
2
+
∫
−
1
1
d
x
1
2
(
1
−
x
2
)
.
{\displaystyle \quad =\int _{\tau _{a}}^{\tau
_{b}}d\tau {1 \over 2}\left({dx \over d\tau
}-{\sqrt {2V(x)}}\right)^{2}+\int _{-1}^{1}dx{1
\over {\sqrt {2}}}(1-x^{2}).}
≥
2
2
3
.
{\displaystyle \quad \geq {2{\sqrt {2}} \over
3}.}
The above inequality is saturated by the solution
of
d
x
d
τ
=
2
V
(
x
)
{\displaystyle {dx \over d\tau }={\sqrt {2V(x)}}}
with the condition
x
(
τ
a
)
=
−
1
{\displaystyle x(\tau _{a})=-1}
and
x
(
τ
b
)
=
1
{\displaystyle x(\tau _{b})=1}
. Such solutions exist, and the solution takes
the simple form when
τ
a
=
−
∞
{\displaystyle \tau _{a}=-\infty }
and
τ
b
=
∞
{\displaystyle \tau _{b}=\infty }
. The explicit formula for the instanton solution
is given by
x
(
τ
)
=
tanh
⁡
(
1
2
(
τ
−
τ
0
)
)
.
{\displaystyle x(\tau )=\tanh \left({1 \over
{\sqrt {2}}}(\tau -\tau _{0})\right).}
Here
τ
0
{\displaystyle \tau _{0}}
is an arbitrary constant. Since this solution
jumps from one classical vacuum
x
=
−
1
{\displaystyle x=-1}
to another classical vacuum
x
=
1
{\displaystyle x=1}
instantaneously around
τ
=
τ
0
{\displaystyle \tau =\tau _{0}}
, it is called an instanton.
=== Explicit formula for double-well potential
===
The explicit formula for the eigenenergies
of the Schrödinger equation with double-well
potential has been given by Müller-Kirsten
with derivation by both a perturbation method
(plus boundary conditions) applied to the
Schrödinger equation, and explicit derivation
from the path integral (and WKB). The result
is the following. Defining parameters of the
Schrödinger equation and the potential by
the equations
d
2
y
(
z
)
d
z
2
+
[
E
−
V
(
z
)
]
y
(
z
)
=
0
,
{\displaystyle {\frac {d^{2}y(z)}{dz^{2}}}+[E-V(z)]y(z)=0,}
and
V
(
z
)
=
−
1
4
z
2
h
4
+
1
2
c
2
z
4
,
c
2
>
0
,
h
4
>
0
,
{\displaystyle V(z)=-{\frac {1}{4}}z^{2}h^{4}+{\frac
{1}{2}}c^{2}z^{4},\;\;\;c^{2}>0,\;h^{4}>0,}
the eigenvalues for
q
0
=
1
,
3
,
5
,
.
.
.
{\displaystyle q_{0}=1,3,5,...}
are found to be:
E
±
(
q
0
,
h
2
)
=
−
h
8
2
5
c
2
+
1
2
q
0
h
2
−
c
2
(
3
q
0
2
+
1
)
2
h
4
−
2
c
4
q
0
8
h
10
(
17
q
0
2
+
19
)
+
O
(
1
h
16
)
{\displaystyle E_{\pm }(q_{0},h^{2})=-{\frac
{h^{8}}{2^{5}c^{2}}}+{\frac {1}{\sqrt {2}}}q_{0}h^{2}-{\frac
{c^{2}(3q_{0}^{2}+1)}{2h^{4}}}-{\frac {{\sqrt
{2}}c^{4}q_{0}}{8h^{10}}}(17q_{0}^{2}+19)+O({\frac
{1}{h^{16}}})}
∓
2
q
0
+
1
h
2
(
h
6
/
2
c
2
)
q
0
/
2
π
2
q
0
/
4
[
(
q
0
−
1
)
/
2
]
!
e
−
h
6
/
6
2
c
2
.
{\displaystyle \mp {\frac {2^{q_{0}+1}h^{2}(h^{6}/2c^{2})^{q_{0}/2}}{{\sqrt
{\pi }}2^{q_{0}/4}[(q_{0}-1)/2]!}}e^{-h^{6}/6{\sqrt
{2}}c^{2}}.}
Clearly these eigenvalues are asymptotically
(
h
2
→
∞
{\displaystyle h^{2}\rightarrow \infty }
) degenerate as expected as a consequence
of the harmonic part of the potential.
=== Results ===
Results obtained from the mathematically well-defined
Euclidean path integral may be Wick-rotated
back and give the same physical results as
would be obtained by appropriate treatment
of the (potentially divergent) Minkowskian
path integral. As can be seen from this example,
calculating the transition probability for
the particle to tunnel through a classically
forbidden region (
V
(
x
)
{\displaystyle V(x)}
) with the Minkowskian path integral corresponds
to calculating the transition probability
to tunnel through a classically allowed region
(with potential −V(X)) in the Euclidean
path integral (pictorially speaking – in
the Euclidean picture – this transition
corresponds to a particle rolling from one
hill of a double-well potential standing on
its head to the other hill). This classical
solution of the Euclidean equations of motion
is often named "kink solution" and is an example
of an instanton. In this example, the two
"vacua" (i.e. ground states) of the double-well
potential, turn into hills in the Euclideanized
version of the problem.
Thus, the instanton field solution of the
(Euclidean, i. e., with imaginary time) (1
+ 1)-dimensional field theory – first quantized
quantum mechanical description – allows
to be interpreted as a tunneling effect between
the two vacua (ground states – higher states
require periodic instantons) of the physical
(1-dimensional space + real time) Minkowskian
system. In the case of the double-well potential
written
V
(
ϕ
)
=
m
4
2
g
2
(
1
−
g
2
ϕ
2
m
2
)
2
{\displaystyle V(\phi )={\frac {m^{4}}{2g^{2}}}\left(1-{\frac
{g^{2}\phi ^{2}}{m^{2}}}\right)^{2}}
the instanton, i.e. solution of
d
2
ϕ
d
τ
2
=
V
′
(
ϕ
)
,
{\displaystyle {\frac {d^{2}\phi }{d\tau ^{2}}}=V'(\phi
),}
(i.e. with energy
E
c
l
=
0
{\displaystyle E_{cl}=0}
), is
ϕ
c
(
τ
)
=
m
g
tanh
⁡
[
m
(
τ
−
τ
0
)
]
,
{\displaystyle \phi _{c}(\tau )={\frac {m}{g}}\tanh
\left[m(\tau -\tau _{0})\right],}
where
τ
=
i
t
{\displaystyle \tau =it}
is the Euclidean time.
Note that a naïve perturbation theory around
one of those two vacua alone (of the Minkowskian
description) would never show this non-perturbative
tunneling effect, dramatically changing the
picture of the vacuum structure of this quantum
mechanical system. In fact the naive perturbation
theory has to be supplemented by boundary
conditions, and these supply the nonperturbative
effect, as is evident from the above explicit
formula and analogous calculations for other
potentials such as a cosine potential (cf.
Mathieu function) or other periodic potentials
(cf. e.g. Lamé function and spheroidal wave
function) and irrespective of whether one
uses the Schrödinger equation or the path
integral.Therefore, the perturbative approach
may not completely describe the vacuum structure
of a physical system. This may have important
consequences, for example, in the theory of
"axions" where the non-trivial QCD vacuum
effects (like the instantons) spoil the Peccei–Quinn
symmetry explicitly and transform massless
Nambu–Goldstone bosons into massive pseudo-Nambu–Goldstone
ones.
=== Periodic instantons ===
In one-dimensional field theory or quantum
mechanics one defines as ``instanton´´ a
field configuration which is a solution of
the classical (Newton-like) equation of motion
with Euclidean time and finite Euclidean action.
In the context of soliton theory the corresponding
solution is known as a kink. In view of their
analogy with the behaviour of classical particles
such configurations or solutions, as well
as others, are collectively known as pseudoparticles
or pseudoclassical configurations. The ``instanton´´
(kink) solution is accompanied by another
solution known as ``anti-instanton´´ (anti-kink)
, and instanton and anti-instanton are distinguished
by ``topological charges´´ +1 and −1 respectively,
but have the same Euclidean action.
``Periodic instantons´´ are a generalization
of instantons. In explicit form they are expressible
in terms of Jacobian elliptic functions which
are periodic functions (effectively generalisations
of trigonometrical functions). In the limit
of infinite period these periodic instantons
– frequently known as ``bounces´´, ``bubbles´´
or the like – reduce to instantons.
The stability of these pseudoclassical configurations
can be investigated by expanding the Lagrangian
defining the theory around the pseudoparticle
configuration and then investigating the equation
of small fluctuations around it. For all versions
of quartic potentials (double-well, inverted
double-well) and periodic (Mathieu) potentials
these equations were discovered to be Lamé
equations, see Lamé functions. The eigenvalues
of these equations are known and permit in
the case of instability the calculation of
decay rates by evaluation of the path integral.
=== Instantons in reaction rate theory ===
In the context of reaction rate theory periodic
instantons are used to calculate the rate
of tunneling of atoms in chemical reactions.
The progress of a chemical reaction can be
described as the movement of pseudoparticle
on a high dimensional potential energy surface
(PES). The thermal rate constant
k
{\displaystyle k}
can then be related to the imaginary part
of the free energy
F
{\displaystyle F}
by
k
(
β
)
=
−
2
ℏ
Im
F
=
2
β
ℏ
Im
ln
(
Z
k
)
≈
2
ℏ
β
Im
Z
k
Re
Z
k
,
Re
Z
k
≫
Im
Z
k
{\displaystyle k(\beta )=-{\frac {2}{\hbar
}}{\text{Im}}\mathrm {F} ={\frac {2}{\beta
\hbar }}{\text{Im}}\ {\text{ln}}(Z_{k})\approx
{\frac {2}{\hbar \beta }}{\frac {{\text{Im}}Z_{k}}{{\text{Re}}Z_{k}}},\
\ {\text{Re}}Z_{k}\gg {\text{Im}}Z_{k}}
whereby
Z
k
{\displaystyle Z_{k}}
is the canonical partition which is calculated
by taking the trace of the Boltzmann operator
in the position representation.
Z
k
=
Tr
(
e
−
β
H
^
)
=
∫
d
x
⟨
x
|
e
−
β
H
^
|
x
⟩
{\displaystyle Z_{k}={\text{Tr}}(e^{-\beta
{\hat {H}}})=\int d\mathbf {x} \left\langle
\mathbf {x} \left|e^{-\beta {\hat {H}}}\right|\mathbf
{x} \right\rangle }
Using a wick rotation and identifying the
Euclidean time with
ℏ
β
=
1
/
(
k
b
T
)
{\displaystyle \hbar \beta =1/(k_{b}T)}
one obtains a path integral representation
for the partition function in mass weighted
coordinates
Z
k
=
∮
⁡
D
x
(
τ
)
e
−
S
E
[
x
(
τ
)
]
/
ℏ
,
S
E
=
∫
0
β
ℏ
(
x
˙
2
2
+
V
(
x
(
τ
)
)
)
d
τ
{\displaystyle Z_{k}=\oint {\mathcal {D}}\mathbf
{x} (\tau )e^{-S_{E}[\mathbf {x} (\tau )]/\hbar
},\ \ \ S_{E}=\int _{0}^{\beta \hbar }\left({\frac
{\dot {\mathbf {x} }}{2}}^{2}+V(\mathbf {x}
(\tau ))\right)d\tau }
The path integral is then approximated via
a steepest descent integration which takes
only into account the contributions from the
classical solutions and quadratic fluctuations
around them. This yields for the rate constant
expression in mass weighted coordinates
k
(
β
)
=
2
β
ℏ
(
det
[
−
∂
2
∂
τ
2
+
V
″
(
x
RS
(
τ
)
)
]
det
[
−
∂
2
∂
τ
2
+
V
″
(
x
Inst
(
τ
)
)
]
)
1
2
exp
⁡
(
−
S
E
[
x
inst
(
τ
)
+
S
E
[
x
RS
(
τ
)
]
ℏ
)
{\displaystyle k(\beta )={\frac {2}{\beta
\hbar }}\left({\frac {{\text{det}}\left[-{\frac
{\partial ^{2}}{\partial \tau ^{2}}}+\mathbf
{V} ''(x_{\text{RS}}(\tau ))\right]}{{\text{det}}\left[-{\frac
{\partial ^{2}}{\partial \tau ^{2}}}+\mathbf
{V} ''(x_{\text{Inst}}(\tau ))\right]}}\right)^{\frac
{1}{2}}{\exp \left({\frac {-S_{E}[x_{\text{inst}}(\tau
)+S_{E}[x_{\text{RS}}(\tau )]}{\hbar }}\right)}}
where
x
Inst
{\displaystyle \mathbf {x} _{\text{Inst}}}
is a periodic instanton and
x
RS
{\displaystyle \mathbf {x} _{\text{RS}}}
is the trivial solution of the pseudoparticle
at rest which represents the reactant state
configuration.
=== Inverted double-well formula ===
As for the double-well potential one can derive
the eigenvalues for the inverted double-well
potential. In this case, however, the eigenvalues
are complex. Defining parameters by the equations
d
2
y
d
z
2
+
[
E
−
V
(
z
)
]
y
(
z
)
=
0
,
V
(
z
)
=
1
4
h
4
z
2
−
1
2
c
2
z
4
,
{\displaystyle {\frac {d^{2}y}{dz^{2}}}+[E-V(z)]y(z)=0,\;\;\;V(z)={\frac
{1}{4}}h^{4}z^{2}-{\frac {1}{2}}c^{2}z^{4},}
the eigenvalues as given by Müller-Kirsten
are, for
q
0
=
1
,
3
,
5
,
.
.
.
,
{\displaystyle q_{0}=1,3,5,...,}
E
=
1
2
q
0
h
2
−
3
c
2
4
h
4
(
q
0
2
+
1
)
−
q
0
c
4
h
10
(
4
q
0
2
+
29
)
+
O
(
1
h
16
)
±
i
2
q
0
h
2
(
h
6
/
2
c
2
)
q
0
/
2
(
2
π
)
1
/
2
[
(
q
0
−
1
)
/
2
]
!
e
−
h
6
/
6
c
2
.
{\displaystyle E={\frac {1}{2}}q_{0}h^{2}-{\frac
{3c^{2}}{4h^{4}}}(q_{0}^{2}+1)-{\frac {q_{0}c^{4}}{h^{10}}}(4q_{0}^{2}+29)+O({\frac
{1}{h^{16}}})\pm i{\frac {2^{q_{0}}h^{2}(h^{6}/2c^{2})^{q_{0}/2}}{(2\pi
)^{1/2}[(q_{0}-1)/2]!}}e^{-h^{6}/6c^{2}}.}
The imaginary part of this expression agrees
with the well known result of Bender and Wu.
In their notation
ℏ
=
1
,
q
0
=
2
K
+
1
,
h
6
/
2
c
2
=
ϵ
.
{\displaystyle \hbar =1,q_{0}=2K+1,h^{6}/2c^{2}=\epsilon
.}
== Quantum field theory ==
In studying Quantum Field Theory (QFT), the
vacuum structure of a theory may draw attention
to instantons. Just as a double-well quantum
mechanical system illustrates, a naïve vacuum
may not be the true vacuum of a field theory.
Moreover, the true vacuum of a field theory
may be an "overlap" of several topologically
inequivalent sectors, so called "topological
vacua".
A well understood and illustrative example
of an instanton and its interpretation can
be found in the context of a QFT with a non-abelian
gauge group, a Yang–Mills theory. For a
Yang–Mills theory these inequivalent sectors
can be (in an appropriate gauge) classified
by the third homotopy group of SU(2) (whose
group manifold is the 3-sphere
S
3
{\displaystyle S^{3}}
). A certain topological vacuum (a "sector"
of the true vacuum) is labelled by an unaltered
transform, the Pontryagin index. As the third
homotopy group of
S
3
{\displaystyle S^{3}}
has been found to be the set of integers,
π
3
{\displaystyle \pi _{3}}
(
S
3
)
=
{\displaystyle (S^{3})=}
Z
{\displaystyle \mathbb {Z} \,}
there are infinitely many topologically inequivalent
vacua, denoted by
|
N
⟩
{\displaystyle |N\rangle }
, where
N
{\displaystyle N}
is their corresponding Pontryagin index. An
instanton is a field configuration fulfilling
the classical equations of motion in Euclidean
spacetime, which is interpreted as a tunneling
effect between these different topological
vacua. It is again labelled by an integer
number, its Pontryagin index,
Q
{\displaystyle Q}
. One can imagine an instanton with index
Q
{\displaystyle Q}
to quantify tunneling between topological
vacua
|
N
⟩
{\displaystyle |N\rangle }
and
|
N
+
Q
⟩
{\displaystyle |N+Q\rangle }
. If Q = 1, the configuration is named BPST
instanton after its discoverers Alexander
Belavin, Alexander Polyakov, Albert S. Schwartz
and Yu. S. Tyupkin. The true vacuum of the
theory is labelled by an "angle" theta and
is an overlap of the topological sectors:
|
θ
⟩
=
∑
N
=
−
∞
N
=
+
∞
e
i
θ
N
|
N
⟩
.
{\displaystyle |\theta \rangle =\sum _{N=-\infty
}^{N=+\infty }e^{i\theta N}|N\rangle .}
Gerard 't Hooft first performed the field
theoretic computation of the effects of the
BPST instanton in a theory coupled to fermions
in [1]. He showed that zero modes of the Dirac
equation in the instanton background lead
to a non-perturbative multi-fermion interaction
in the low energy effective action.
== Yang–Mills theory ==
The classical Yang–Mills action on a principal
bundle with structure group G, base M, connection
A, and curvature (Yang–Mills field tensor)
F is
S
Y
M
=
∫
M
|
F
|
2
d
v
o
l
M
,
{\displaystyle S_{YM}=\int _{M}\left|F\right|^{2}d\mathrm
{vol} _{M},}
where
d
v
o
l
M
{\displaystyle d\mathrm {vol} _{M}}
is the volume form on
M
{\displaystyle M}
. If the inner product on
g
{\displaystyle {\mathfrak {g}}}
, the Lie algebra of
G
{\displaystyle G}
in which
F
{\displaystyle F}
takes values, is given by the Killing form
on
g
{\displaystyle {\mathfrak {g}}}
, then this may be denoted as
∫
M
T
r
(
F
∧
∗
F
)
{\displaystyle \int _{M}\mathrm {Tr} (F\wedge
*F)}
, since
F
∧
∗
F
=
⟨
F
,
F
⟩
d
v
o
l
M
.
{\displaystyle F\wedge *F=\langle F,F\rangle
d\mathrm {vol} _{M}.}
For example, in the case of the gauge group
U(1), F will be the electromagnetic field
tensor. From the principle of stationary action,
the Yang–Mills equations follow. They are
d
F
=
0
,
d
∗
F
=
0.
{\displaystyle \mathrm {d} F=0,\quad \mathrm
{d} {*F}=0.}
The first of these is an identity, because
dF = d2A = 0, but the second is a second-order
partial differential equation for the connection
A, and if the Minkowski current vector does
not vanish, the zero on the rhs. of the second
equation is replaced by
J
{\displaystyle \mathbf {J} }
. But notice how similar these equations are;
they differ by a Hodge star. Thus a solution
to the simpler first order (non-linear) equation
∗
F
=
±
F
{\displaystyle {*F}=\pm F\,}
is automatically also a solution of the Yang–Mills
equation. This simplification occurs on 4
manifolds with :
s
=
1
{\displaystyle s=1}
so that
∗
2
=
+
1
{\displaystyle *^{2}=+1}
on 2-forms. Such solutions usually exist,
although their precise character depends on
the dimension and topology of the base space
M, the principal bundle P, and the gauge group
G.
In nonabelian Yang–Mills theories,
D
F
=
0
{\displaystyle DF=0}
and
D
∗
F
=
0
{\displaystyle D*F=0}
where D is the exterior covariant derivative.
Furthermore, the Bianchi identity
D
F
=
d
F
+
A
∧
F
−
F
∧
A
=
d
(
d
A
+
A
∧
A
)
+
A
∧
(
d
A
+
A
∧
A
)
−
(
d
A
+
A
∧
A
)
∧
A
=
0
{\displaystyle DF=dF+A\wedge F-F\wedge A=d(dA+A\wedge
A)+A\wedge (dA+A\wedge A)-(dA+A\wedge A)\wedge
A=0}
is satisfied.
In quantum field theory, an instanton is a
topologically nontrivial field configuration
in four-dimensional Euclidean space (considered
as the Wick rotation of Minkowski spacetime).
Specifically, it refers to a Yang–Mills
gauge field A which approaches pure gauge
at spatial infinity. This means the field
strength
F
=
d
A
+
A
∧
A
{\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf
{A} \wedge \mathbf {A} }
vanishes at infinity. The name instanton derives
from the fact that these fields are localized
in space and (Euclidean) time – in other
words, at a specific instant.
The case of instantons on the two-dimensional
space may be easier to visualise because it
admits the simplest case of the gauge group,
namely U(1), that is an abelian group. In
this case the field A can be visualised as
simply a vector field. An instanton is a configuration
where, for example, the arrows point away
from a central point (i.e., a "hedgehog" state).
In Euclidean four dimensions,
R
4
{\displaystyle \mathbb {R} ^{4}}
, abelian instantons are impossible.
The field configuration of an instanton is
very different from that of the vacuum. Because
of this instantons cannot be studied by using
Feynman diagrams, which only include perturbative
effects. Instantons are fundamentally non-perturbative.
The Yang–Mills energy is given by
1
2
∫
R
4
Tr
⁡
[
∗
F
∧
F
]
{\displaystyle {\frac {1}{2}}\int _{\mathbb
{R} ^{4}}\operatorname {Tr} [*\mathbf {F}
\wedge \mathbf {F} ]}
where ∗ is the Hodge dual. If we insist
that the solutions to the Yang–Mills equations
have finite energy, then the curvature of
the solution at infinity (taken as a limit)
has to be zero. This means that the Chern–Simons
invariant can be defined at the 3-space boundary.
This is equivalent, via Stokes' theorem, to
taking the integral
∫
R
4
Tr
⁡
[
F
∧
F
]
.
{\displaystyle \int _{\mathbb {R} ^{4}}\operatorname
{Tr} [\mathbf {F} \wedge \mathbf {F} ].}
This is a homotopy invariant and it tells
us which homotopy class the instanton belongs
to.
Since the integral of a nonnegative integrand
is always nonnegative,
0
≤
1
2
∫
R
4
Tr
⁡
[
(
∗
F
+
e
−
i
θ
F
)
∧
(
F
+
e
i
θ
∗
F
)
]
=
∫
R
4
Tr
⁡
[
∗
F
∧
F
+
cos
⁡
θ
F
∧
F
]
{\displaystyle 0\leq {\frac {1}{2}}\int _{\mathbb
{R} ^{4}}\operatorname {Tr} [(*\mathbf {F}
+e^{-i\theta }\mathbf {F} )\wedge (\mathbf
{F} +e^{i\theta }*\mathbf {F} )]=\int _{\mathbb
{R} ^{4}}\operatorname {Tr} [*\mathbf {F}
\wedge \mathbf {F} +\cos \theta \mathbf {F}
\wedge \mathbf {F} ]}
for all real θ. So, this means
1
2
∫
R
4
Tr
⁡
[
∗
F
∧
F
]
≥
1
2
|
∫
R
4
Tr
⁡
[
F
∧
F
]
|
.
{\displaystyle {\frac {1}{2}}\int _{\mathbb
{R} ^{4}}\operatorname {Tr} [*\mathbf {F}
\wedge \mathbf {F} ]\geq {\frac {1}{2}}\left|\int
_{\mathbb {R} ^{4}}\operatorname {Tr} [\mathbf
{F} \wedge \mathbf {F} ]\right|.}
If this bound is saturated, then the solution
is a BPS state. For such states, either ∗F
= F or ∗F = − F depending on the sign
of the homotopy invariant.
Instanton effects are important in understanding
the formation of condensates in the vacuum
of quantum chromodynamics (QCD) and in explaining
the mass of the so-called 'eta-prime particle',
a Goldstone-boson which has acquired mass
through the axial current anomaly of QCD.
Note that there is sometimes also a corresponding
soliton in a theory with one additional space
dimension. Recent research on instantons links
them to topics such as D-branes and Black
holes and, of course, the vacuum structure
of QCD. For example, in oriented string theories,
a Dp brane is a gauge theory instanton in
the world volume (p + 5)-dimensional U(N)
gauge theory on a stack of N
D(p + 4)-branes.
== Various numbers of dimensions ==
Instantons play a central role in the nonperturbative
dynamics of gauge theories. The kind of physical
excitation that yields an instanton depends
on the number of dimensions of the spacetime,
but, surprisingly, the formalism for dealing
with these instantons is relatively dimension-independent.
In 4-dimensional gauge theories, as described
in the previous section, instantons are gauge
bundles with a nontrivial four-form characteristic
class. If the gauge symmetry is a unitary
group or special unitary group then this characteristic
class is the second Chern class, which vanishes
in the case of the gauge group U(1). If the
gauge symmetry is an orthogonal group then
this class is the first Pontrjagin class.
In 3-dimensional gauge theories with Higgs
fields, 't Hooft–Polyakov monopoles play
the role of instantons. In his 1977 paper
Quark Confinement and Topology of Gauge Groups,
Alexander Polyakov demonstrated that instanton
effects in 3-dimensional QED coupled to a
scalar field lead to a mass for the photon.
In 2-dimensional abelian gauge theories worldsheet
instantons are magnetic vortices. They are
responsible for many nonperturbative effects
in string theory, playing a central role in
mirror symmetry.
In 1-dimensional quantum mechanics, instantons
describe tunneling, which is invisible in
perturbation theory.
== 4d supersymmetric gauge theories ==
Supersymmetric gauge theories often obey nonrenormalization
theorems, which restrict the kinds of quantum
corrections which are allowed. Many of these
theorems only apply to corrections calculable
in perturbation theory and so instantons,
which are not seen in perturbation theory,
provide the only corrections to these quantities.
Field theoretic techniques for instanton calculations
in supersymmetric theories were extensively
studied in the 1980s by multiple authors.
Because supersymmetry guarantees the cancellation
of fermionic vs. bosonic non-zero modes in
the instanton background, the involved 't
Hooft computation of the instanton saddle
point reduces to an integration over zero
modes.
In N = 1 supersymmetric gauge theories instantons
can modify the superpotential, sometimes lifting
all of the vacua. In 1984, Ian Affleck, Michael
Dine and Nathan Seiberg calculated the instanton
corrections to the superpotential in their
paper Dynamical Supersymmetry Breaking in
Supersymmetric QCD. More precisely, they were
only able to perform the calculation when
the theory contains one less flavor of chiral
matter than the number of colors in the special
unitary gauge group, because in the presence
of fewer flavors an unbroken nonabelian gauge
symmetry leads to an infrared divergence and
in the case of more flavors the contribution
is equal to zero. For this special choice
of chiral matter, the vacuum expectation values
of the matter scalar fields can be chosen
to completely break the gauge symmetry at
weak coupling, allowing a reliable semi-classical
saddle point calculation to proceed. By then
considering perturbations by various mass
terms they were able to calculate the superpotential
in the presence of arbitrary numbers of colors
and flavors, valid even when the theory is
no longer weakly coupled.
In N = 2 supersymmetric gauge theories the
superpotential receives no quantum corrections.
However the correction to the metric of the
moduli space of vacua from instantons was
calculated in a series of papers. First, the
one instanton correction was calculated by
Nathan Seiberg in Supersymmetry and Nonperturbative
beta Functions. The full set of corrections
for SU(2) Yang–Mills theory was calculated
by Nathan Seiberg and Edward Witten in "Electric
– magnetic duality, monopole condensation,
and confinement in N=2 supersymmetric Yang–Mills
theory," in the process creating a subject
that is today known as Seiberg–Witten theory.
They extended their calculation to SU(2) gauge
theories with fundamental matter in Monopoles,
duality and chiral symmetry breaking in N=2
supersymmetric QCD. These results were later
extended for various gauge groups and matter
contents, and the direct gauge theory derivation
was also obtained in most cases. For gauge
theories with gauge group U(N) the Seiberg-Witten
geometry has been derived from gauge theory
using Nekrasov partition functions in 2003
by Nikita Nekrasov and Andrei Okounkov and
independently by Hiraku Nakajima and Kota
Yoshioka.
In N = 4 supersymmetric gauge theories the
instantons do not lead to quantum corrections
for the metric on the moduli space of vacua.
== See also ==
Instanton fluid
Caloron
Sidney Coleman
Holstein–Herring method
Gravitational instanton
== References and notes ==
Notes
Citations
GeneralInstantons in Gauge Theories, a compilation
of articles on instantons, edited by Mikhail
A. Shifman, doi:10.1142/2281
Solitons and Instantons, R. Rajaraman (Amsterdam:
North Holland, 1987), ISBN 0-444-87047-4
The 
Uses of Instantons, by Sidney Coleman in Proc.
Int. School of Subnuclear Physics, (Erice,
1977); and in Aspects of Symmetry p. 265,
Sidney Coleman, Cambridge University Press,
1985, ISBN 0-521-31827-0; and in Instantons
in Gauge Theories
Solitons, Instantons and Twistors. M. Dunajski,
Oxford University Press. ISBN 978-0-19-857063-9.
The Geometry of Four-Manifolds, S.K. Donaldson,
P.B. Kronheimer, Oxford University Press,
1990, ISBN 0-19-853553-8.
== External links ==
The dictionary definition of instanton at
Wiktionary
