Instanton
Encyclopedia
An instanton is a notion appearing in theoretical and mathematical physics
. 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.
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 a theory, 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 point
s. Instantons are important in quantum field theory
because (a) they appear in the path integral
as the leading quantum corrections to the classical behavior of a system, and (b) they can be used to study the tunneling behavior in various systems such as a Yang–Mills theory.
, which requires the value of
to be small. The Schrödinger equation
for the particle reads
If the potential were constant, the solution would (up to proportionality) be a plane wave,
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.
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
(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. 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" of the double-well potential, turn into hills in the Euclideanized version of the problem. Thus, the instanton field solution of the (1 + 1)-dimensional field theory (first quantized quantum mechanical system) allows to be interpreted as a tunneling effect between the two vacua of the physical Minkowskian system.
Note that a naive perturbation theory around one of those two vacua would never show this non-perturbative tunneling effect, dramatically changing the picture of the vacuum structure of this quantum mechanical system.
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
). A certain topological vacuum (a "sector" of the true vacuum) is labelled by a topological invariant, the Pontryagin index. As the third homotopy group of 
has been found to be the set of integer
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 a whole 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
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:
Gerard 't Hooft
first performed the field theoretic computation of the effects of the BPST instanton in a theory coupled to fermions in http://www.slac.stanford.edu/spires/find/hep/www?j=PHRVA,D14,3432. 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.
with structure group G, base M, connection
A, and curvature
(Yang–Mills field tensor) F is
where
is the volume form
on
. If the inner product on 
, the Lie algebra
of
in which 
takes values, is given by the Killing form
on
, then this may be denoted as 
, since
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
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
. But notice how similar these equations are; they differ by a Hodge star. Thus a solution to the simpler first order (non-linear) equation
is automatically also a solution of the Yang–Mills equation. 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
. Furthermore, the Bianchi identity
is satisfied.
In quantum field theory
, an instanton is a topologically
nontrivial field configuration in four-dimension
al Euclidean space
(considered as the Wick rotation
of Minkowski spacetime). Specifically, it refers to a Yang–Mills
gauge field A which locally approaches pure gauge at spatial infinity
. This means the field strength defined by 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.
Instantons may be easier to visualise in two dimensions than in four. In the simplest case the gauge group
is U(1). In this case the field can be visualised as an arrow at each point in two-dimensional spacetime. An instanton is a configuration where, for example, the arrows point away from a central point (i.e., a "hedgehog" state). More complicated configurations are also possible.
The field configuration of an instanton is very different from that of the vacuum
. Because of this instantons cannot be studied by using Feynman diagram
s, which only include perturbative
effects. Instantons are fundamentally non-perturbative.
The Yang–Mills energy is given by
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
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,
for all real θ. So, this means
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.
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 monopole's 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.
s, 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 in 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. Quantitative agreement has been demonstrated in many cases between Seiberg–Witten and conventional field-theoretic saddle point calculations.
In N = 4 supersymmetric gauge theories the instantons do not lead to quantum corrections for the metric on the moduli space of vacua.
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
in a principal bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
over a four-dimensional Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
that plays the role of physical space-time in non-abelian gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a 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
Hypersphere
In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...
, 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
Twistor theory
In theoretical and mathematical physics, twistor theory maps the geometric objects of conventional 3+1 space-time into geometric objects in a 4 dimensional space with metric signature...
, which relates them to algebraic vector bundles on algebraic surfaces, and via the ADHM construction
ADHM construction
The ADHM construction or monad construction is the construction of all instantons using method of linear algebra by Michael Atiyah, Vladimir G. Drinfel'd, Nigel. J. Hitchin, Yuri I...
, or hyperkähler reduction (see hyperkähler manifold
Hyperkähler manifold
In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k) In differential geometry, a hyperkähler...
), a sophisticated linear algebra procedure. The groundbreaking work of Simon Donaldson
Simon Donaldson
Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
, for which he was later awarded the Fields medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
, used the moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
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
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
but not diffeomorphic
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
four-manifolds. Many methods developed in studying instantons have also been applied to monopoles.
Physical description
An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
or in quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
.
More precisely, it is a solution to the equations of motion of the classical field theory
Classical field theory
A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ....
on a Euclidean
Euclidean
Euclidean relates to Euclid , a town or others. It may refer to:Geometry...
spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. In such a theory, solutions to the equations of motion may be thought of as critical points
Critical point (mathematics)
In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
of the action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
. The critical points of the action may be local maxima
Maxima and minima
In mathematics, the maximum and minimum of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety .More generally, the...
of the action, local minima
Maxima and minima
In mathematics, the maximum and minimum of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety .More generally, the...
, or saddle point
Saddle point
In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
s. Instantons are important in quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
because (a) they appear in the path integral
Functional integration
Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions...
as the leading quantum corrections to the classical behavior of a system, and (b) they can be used to study the tunneling behavior in various systems such as a Yang–Mills theory.
Quantum mechanics
An instanton can be used to calculate the transition probability for a quantum mechanical particle tunneling through a potential barrier. One of the simplest examples 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. One way to calculate this probability is by means of the semi-classical WKB approximationWKB approximation
In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear partial differential equations with spatially varying coefficients...
, which requires the value of
to be small. The Schrödinger equationSchrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
for the particle reads
If the potential were constant, the solution would (up to proportionality) be a plane wave,
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.
Alternatively, the use of path integrals
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
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
Wick rotation
In physics, Wick rotation, named after 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 substitutes an imaginary-number variable for a real-number variable...
(analytic continuation) to Euclidean spacetime (
), one getswith 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. Results obtained from the mathematically well-defined Euclidean path integralLine integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
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" of the double-well potential, turn into hills in the Euclideanized version of the problem. Thus, the instanton field solution of the (1 + 1)-dimensional field theory (first quantized quantum mechanical system) allows to be interpreted as a tunneling effect between the two vacua of the physical Minkowskian system.Note that a naive perturbation theory around one of those two vacua would never show this non-perturbative tunneling effect, dramatically changing the picture of the vacuum structure of this quantum mechanical system.
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 naive 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
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, which records information about loops in a space...
of SU(2) (whose group manifold is the 3-sphere
). A certain topological vacuum (a "sector" of the true vacuum) is labelled by a topological invariant, the Pontryagin index. As the third homotopy group of has been found to be the set of integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
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 a whole 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 instantonBPST instanton
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 Yang-Mills theory in Euclidean space-time , meaning it describes a transition between two...
after its discoverers Alexander Belavin
Alexander Belavin
Alexander "Sasha" Abramovich Belavin is a Russian physicist, known for his contributions to string theory.He is a professor at the Independent University of Moscow and is researcher at the Landau Institute of Theoretical Physics....
, 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:
Gerard 't Hooft
Gerardus 't Hooft
Gerardus 't Hooft 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...
first performed the field theoretic computation of the effects of the BPST instanton in a theory coupled to fermions in http://www.slac.stanford.edu/spires/find/hep/www?j=PHRVA,D14,3432. 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 bundlePrincipal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
with structure group G, base M, connection
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
A, and curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
(Yang–Mills field tensor) F is
where
is the volume formVolume form
In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero. A manifold has a volume...
on
. If the inner product on , the Lie algebraLie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
of
in which takes values, is given by the Killing formKilling 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...
on
, then this may be denoted as , sinceFor example, in the case of the gauge group U(1), F will be the electromagnetic field tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
. From the principle of stationary action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
, the Yang–Mills equations follow. They are
The first of these is an identity, because dF = d2A = 0, but the second is a second-order partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
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. Thus a solution to the simpler first order (non-linear) equationis automatically also a solution of the Yang–Mills equation. 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 derivativeExterior covariant derivative
In mathematics, the exterior covariant derivative, sometimes also covariant exterior derivative, is a very useful notion for calculus on manifolds, which makes it possible to simplify formulas which use a principal connection....
. Furthermore, the Bianchi identity
is satisfied.
In quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, an instanton is a topologically
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
nontrivial field configuration in four-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
(considered as the Wick rotation
Wick rotation
In physics, Wick rotation, named after 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 substitutes an imaginary-number variable for a real-number variable...
of Minkowski spacetime). Specifically, it refers to a Yang–Mills
Yang–Mills
Yang–Mills theory is a gauge theory based on the SU group. Wolfgang Pauli formulated in 1953 the first consistent generalization of the five-dimensional theory of Kaluza, Klein, Fock and others to a higher dimensional internal space...
gauge field A which locally approaches pure gauge at spatial infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
. This means the field strength defined by 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.
Instantons may be easier to visualise in two dimensions than in four. In the simplest case the gauge group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is U(1). In this case the field can be visualised as an arrow at each point in two-dimensional spacetime. An instanton is a configuration where, for example, the arrows point away from a central point (i.e., a "hedgehog" state). More complicated configurations are also possible.
The field configuration of an instanton is very different from that of the vacuum
Vacuum state
In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
. Because of this instantons cannot be studied by using Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s, which only include perturbative
Perturbation theory (quantum mechanics)
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 which a mathematical solution is known, and add an...
effects. Instantons are fundamentally non-perturbative.
The Yang–Mills energy is given by
where ∗ is the Hodge dual
Hodge dual
In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.-Dimensions and algebra:...
. If we insist that the solutions to the Yang–Mills equations have finite energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
, then the curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
of the solution at infinity (taken as a limit
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
) 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
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
, to taking the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
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,
for all real θ. So, this means
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
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
(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
Chiral anomaly
A chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.The non-conservation happens...
of QCD. Note that there is sometimes also a corresponding soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
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
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, 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
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
characteristic class
Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
. If the gauge symmetry is a unitary group
Unitary group
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
or special unitary group
Special unitary group
The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
then this characteristic class is the second Chern class
Chern class
In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...
, 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 monopole's 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
QED
QED may refer to:*Q.E.D., from the Latin quod erat demonstrandum, used at the end of a definitive proof*Quantum electrodynamics, a field of physics...
coupled to a scalar field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
lead to a mass for the photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
.
In 2-dimensional abelian gauge theories worldsheet instantons are magnetic vortices
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...
. They are responsible for many nonperturbative effects in string theory, playing a central role in mirror symmetry
Mirror symmetry
In physics and mathematics, mirror symmetry is a relation that can exist between two Calabi-Yau manifolds. It happens, usually for two such six-dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they are equivalent if they are employed as hidden...
.
In 1-dimensional quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, instantons describe tunneling, which is invisible in perturbation theory.
4d supersymmetric gauge theories
Supersymmetric gauge theories often obey nonrenormalization theoremSupersymmetry nonrenormalization theorems
In theoretical physics a nonrenormalization theorem is a limitation on how a certain quantity in the classical description of a quantum field theory may be modified by renormalization in the full quantum theory...
s, which restrict the kinds of quantum corrections which are allowed. Many of these theorems only apply to corrections calculable in perturbation theory
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
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
Superpotential
Superpotential is a concept from particle physics' supersymmetry.-Example of superpotentiality:Let's look at the example of a one dimensional nonrelativistic particle with a 2D internal degree of freedom called "spin"...
, sometimes lifting all of the vacua. In 1984 Ian Affleck, Michael Dine and Nathan Seiberg
Nathan Seiberg
Nathan "Nati" Seiberg is an Israeli American theoretical physicist who works on string theory. He was recipient of a 1996 MacArthur Fellowship and the Dannie Heineman Prize for Mathematical Physics in 1998. He is currently a professor at the Institute for Advanced Study in Princeton, New Jersey, USA...
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
Chiral superfield
In theoretical physics, one often analyzes theories with supersymmetry in which chiral superfields play an important role. In four dimensions, the minimal N=1 supersymmetry may be written using the notion of superspace...
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 in 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
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
of vacua from instantons was calculated in a series of papers. First, the one instanton correction was calculated by Nathan Seiberg
Nathan Seiberg
Nathan "Nati" Seiberg is an Israeli American theoretical physicist who works on string theory. He was recipient of a 1996 MacArthur Fellowship and the Dannie Heineman Prize for Mathematical Physics in 1998. He is currently a professor at the Institute for Advanced Study in Princeton, New Jersey, USA...
in Supersymmetry and Nonperturbative beta Functions. The full set of corrections for SU(2) Yang–Mills theory was calculated by Nathan Seiberg
Nathan Seiberg
Nathan "Nati" Seiberg is an Israeli American theoretical physicist who works on string theory. He was recipient of a 1996 MacArthur Fellowship and the Dannie Heineman Prize for Mathematical Physics in 1998. He is currently a professor at the Institute for Advanced Study in Princeton, New Jersey, USA...
and Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
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. Quantitative agreement has been demonstrated in many cases between Seiberg–Witten and conventional field-theoretic saddle point calculations.
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 fluidInstanton fluidIn quantum chromodynamics, the instanton fluid model is a model of Wick rotated Euclidean quantum chromodynamics. If we examine the path integral of the action, we find that it has infinitely many local minima, corresponding to varying instanton numbers...
- CaloronCaloronIn mathematical physics, a caloron is the finite temperature generalization of an instanton.-Finite temperature and instantons:At zero temperature, instantons are the name given to solutions of the classical equations of motion of the Euclidean version of the theory under consideration, and which...
- Sidney ColemanSidney ColemanSidney Richard Coleman was an American theoretical physicist who studied under Murray Gell-Mann.- Life and work :Sidney Coleman grew up on the Far North Side of Chicago...
(Physicist and Mathematician) - Holstein-Herring Method