An instanton (or pseudoparticle) is a notion appearing in theoretical and
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
. An instanton is a classical solution to
equations of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
with a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
,
non-zero action, either in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
or in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
. More precisely, it is a solution to the equations of motion of the
classical field theory
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
on a
Euclidean spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
.
In such quantum theories, solutions to the equations of motion may be thought of as
critical points of the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
. The critical points of the action may be
local maxima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
of the action,
local minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
, or
saddle point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...
s. Instantons are important in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
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
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using th ...
.
Relevant to
dynamics, families of instantons permit that instantons, i.e. different critical points of the equation of motion, be related to one another. 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
Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. Their macroscopic behavior thus displays the spatial or temporal scale-invariance characteristic of the critical point of a phase ...
.
Mathematics
Mathematically, a ''Yang–Mills instanton'' is a self-dual or anti-self-dual
connection in a
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...
over a four-dimensional
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
that plays the role of physical
space-time
In physics, spacetime is a mathematical model that combines the three-dimensional space, three dimensions of space and one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visualize S ...
in
non-abelian gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
. 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
In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a branch of theoretical and mathematical physics. Penrose proposed that twistor space should be the basic arena ...
, which relates them to algebraic
vector bundles
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
on
algebraic surfaces
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
, and via the
ADHM construction
In mathematical physics and gauge theory, the ADHM construction or monad construction is the construction of all instantons using methods of linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel Hitchin, Yuri I. Manin in their paper "Constru ...
, or hyperkähler reduction (see
hyperkähler manifold
In differential geometry, a hyperkähler manifold is a Riemannian manifold (M, g) endowed with three integrable almost complex structures I, J, K that are Kähler with respect to the Riemannian metric g and satisfy the quaternionic relations I^2 ...
), a sophisticated linear algebra procedure. The groundbreaking work of
Simon Donaldson
Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. He i ...
, for which he was later awarded the
Fields medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
, 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 In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for diff ...
and applied it to the construction of
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
but not
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
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 The so-called double-well potential is one of a number of Quartic function, quartic potentials of considerable interest in quantum mechanics, in quantum field theory and elsewhere for the exploration of various physical phenomena or mathematical pro ...
. 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
Consider the quantum mechanics of a single particle motion inside the double-well potential
The potential energy takes its minimal value at
, 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
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
:
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
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
In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mecha ...
, which requires the value of
to be small. The
time independent Schrödinger equation for the particle reads
:
If the potential were constant, the solution would be a plane wave, up to a proportionality factor,
:
with
:
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
:
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
:
Following the process of
Wick rotation
In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that sub ...
(analytic continuation) to Euclidean spacetime (
), one gets
:
with the Euclidean action
:
The potential energy changes sign
under the Wick rotation and the minima transform into maxima, thereby
exhibits two "hills" of maximal energy.
Let us now consider the local minimum of the Euclidean action
with the double-well potential
, and we set
just for simplicity of computation. Since we want to know how the two classically lowest energy states
are connected, let us set
and
.
For
and
, we can rewrite the Euclidean action as
:
:
:
The above inequality is saturated by the solution of
with the condition
and
. Such solutions exist, and the solution takes the simple form when
and
. The explicit formula for the instanton solution is given by
:
Here
is an arbitrary constant. Since this solution jumps from one classical vacuum
to another classical vacuum
instantaneously around
, 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 The so-called double-well potential is one of a number of Quartic function, quartic potentials of considerable interest in quantum mechanics, in quantum field theory and elsewhere for the exploration of various physical phenomena or mathematical pro ...
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
:
and
:
the eigenvalues for
are found to be:
:
:
Clearly these eigenvalues are asymptotically (
) 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 (
) 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 The so-called double-well potential is one of a number of Quartic function, quartic potentials of considerable interest in quantum mechanics, in quantum field theory and elsewhere for the exploration of various physical phenomena or mathematical pro ...
, 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
:
the instanton, i.e. solution of
:
(i.e. with energy
), is
:
where
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
In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation
:
\frac + (a - 2q\cos(2x))y = 0,
where a and q are parameters. They were first introduced by Émile Léonard Mathieu, ...
) or other periodic potentials (cf. e.g.
Lamé function
In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper . Lamé's equation appears in the method of separation of variable ...
and
spheroidal wave function
Spheroidal wave functions are solutions of the Helmholtz equation that are found by writing the equation in spheroidal coordinates and applying the technique of separation of variables, just like the use of spherical coordinates lead to spherical h ...
) 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
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
theory the corresponding solution is known as a
kink
Kink or KINK may refer to:
Common uses
* Kink (sexuality), a colloquial term for non-normative sexual behavior
* Kink, a curvature, bend, or twist
Geography
* Kink, Iran, a village in Iran
* The Kink, a man-made geographic feature in remote east ...
. 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 function
In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper . Lamé's equation appears in the method of separation of variable ...
. 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
A potential energy surface (PES) describes the energy of a system, especially a collection of atoms, in terms of certain parameters, normally the positions of the atoms. The surface might define the energy as a function of one or more coordinates; ...
(PES). The thermal rate constant
can then be related to the imaginary part of the free energy
by
whereby
is the canonical partition function which is calculated by taking the trace of the Boltzmann operator in the position representation.
Using a wick rotation and identifying the Euclidean time with
one obtains a path integral representation for the partition function in mass weighted coordinates
:
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
where
is a periodic instanton and
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
:
the eigenvalues as given by Müller-Kirsten are, for
:
The imaginary part of this expression agrees with the well known result of Bender and Wu. In their notation
Quantum field theory
In studying
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
(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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
vacua
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
".
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,
[See also: ]Non-abelian gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
a
Yang–Mills theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using th ...
. For a Yang–Mills theory these inequivalent sectors can be (in an appropriate gauge) classified by the third
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
of
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
(whose group manifold is the
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
). A certain topological vacuum (a "sector" of the true vacuum) is labelled by an
unaltered transform, the
Pontryagin index In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four.
Definition
Given a real vector bundl ...
. As the third homotopy group of
has been found to be the set of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s,
:
there are infinitely many topologically inequivalent vacua, denoted by
, where
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,
. One can imagine an ''instanton'' with index
to quantify tunneling between topological vacua
and
. If ''Q'' = 1, the configuration is named
BPST instanton
In theoretical physics, the BPST instanton is the instanton with winding number 1 found by Alexander Belavin, Alexander Polyakov, Albert Schwarz and Yu. S. Tyupkin. It is a classical solution to the equations of motion of SU(2) Yang–Mills the ...
after its discoverers
Alexander Belavin
Alexander "Sasha" Abramovich Belavin (russian: Алекса́ндр Абра́мович Бела́вин, born 1942) is a Russian physicist, known for his contributions to string theory.
He is a professor at the Independent University of Moscow a ...
,
Alexander Polyakov,
Albert S. Schwarz and
Yu. S. Tyupkin. The true vacuum of the theory is labelled by an "angle" theta and is an overlap of the topological sectors:
:
Gerard 't Hooft
Gerardus (Gerard) 't Hooft (; born July 5, 1946) is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G. Veltman "for elucidating the ...
first performed the field theoretic computation of the effects of the BPST instanton in a theory coupled to fermions i
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
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...
with structure group ''G'', base ''M'',
connection ''A'', and
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
(Yang–Mills field tensor) ''F'' is
:
where
is the
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the ...
on
. If the inner product on
, the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of
in which
takes values, is given by the
Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show ...
on
, then this may be denoted as
, since
:
For example, in the case of the
gauge group
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie group ...
U(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
, ''F'' will be the electromagnetic field
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
. From the
principle of stationary action, the Yang–Mills equations follow. They are
:
The first of these is an identity, because d''F'' = d
2''A'' = 0, but the second is a second-order
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
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
. But notice how similar these equations are; they differ by a
Hodge star
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
. Thus a solution to the simpler first order (non-linear) equation
:
is automatically also a solution of the Yang–Mills equation. This simplification occurs on 4 manifolds with :
so that
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,
and
where D is the
exterior covariant derivative
In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection.
Definition
Let ''G' ...
. Furthermore, the
Bianchi identity In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let ''G'' be a Lie group with Lie alge ...
:
is satisfied.
In
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, an ''instanton'' is a
topologically nontrivial field configuration in four-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
(considered as the
Wick rotation
In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that sub ...
of
Minkowski spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inert ...
). Specifically, it refers to a
Yang–Mills gauge field
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
''A'' which approaches
pure gauge
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie group ...
at
spatial infinity. This means the field strength
:
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
In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
may be easier to visualise because it admits the simplest case of the gauge
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, namely U(1), that is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
. 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,
, abelian instantons are impossible.
The field configuration of an instanton is very different from that of the
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
. Because of this instantons cannot be studied by using
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s, which only include
perturbative
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for whi ...
effects. Instantons are fundamentally
non-perturbative
In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function
: f(x) = e^,
which does not have a Taylor series at ''x'' = 0. Every coefficient of the Taylor ...
.
The Yang–Mills energy is given by
: