Thedx¹⊗σ₃ coefficient of aBPST instanton on the(x¹,x²)-slice ofR⁴ whereσ₃ is the thirdPauli matrix (top left). Thedx²⊗σ₃ coefficient (top right). These coefficients determine the restriction of the BPST instantonA withg=2,ρ=1,z=0 to this slice. The corresponding field strength centered aroundz=0 (bottom left). A visual representation of the field strength of a BPST instanton with centerz on thecompactificationS⁴ ofR⁴ (bottom right). The BPST instanton is a classical instanton solution to theYang–Mills equations onR⁴.

Aninstanton (orpseudoparticle^[1][2][3]) is a notion appearing in theoretical andmathematical physics. An instanton is a classical solution toequations of motion with a finite,non-zero action, either inquantum mechanics or inquantum field theory. More precisely, it is a solution to the equations of motion of theclassical field theory on aEuclideanspacetime.^[4]

In such quantum theories, solutions to the equations of motion may be thought of ascritical points of theaction. The critical points of the action may belocal maxima of the action,local minima, orsaddle points. Instantons are important inquantum field theory because:

• they appear in thepath 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 aYang–Mills theory.

Relevant todynamics, 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 thenoise-induced chaotic phase known asself-organized criticality.

Mathematically, aYang–Mills instanton is a self-dual or anti-self-dualconnection in aprincipal bundle over a four-dimensionalRiemannian manifold that plays the role of physicalspace-time innon-abeliangauge theory. Instantons are topologically nontrivial solutions ofYang–Mills equations that absolutely minimize the energy functional within their topological type.^[5] The first such solutions were discovered in the case of four-dimensional Euclidean space compactified to thefour-dimensional sphere, and turned out to be localized in space-time, prompting the namespseudoparticle andinstanton.

Yang–Mills instantons have been explicitly constructed in many cases by means oftwistor theory, which relates them to algebraicvector bundles onalgebraic surfaces, and via theADHM construction, or hyperkähler reduction (seehyperkähler manifold), a geometric invariant theory procedure. The groundbreaking work ofSimon Donaldson, for which he was later awarded theFields medal, used themoduli space of instantons over a given four-dimensional differentiable manifold as a new invariant of the manifold that depends on itsdifferentiable structure and applied it to the construction ofhomeomorphic but notdiffeomorphic four-manifolds. Many methods developed in studying instantons have also been applied tomonopoles. This is because magnetic monopoles arise as solutions of a dimensional reduction of the Yang–Mills equations.^[6]

Aninstanton can be used to calculate the transition probability for a quantum mechanical particle tunneling through a potential barrier. One example of a system with aninstanton effect is a particle in adouble-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.^[4]

Consider the quantum mechanics of a single particle motion inside the double-well potential${\displaystyle V(x)=\frac{1}{4}(x^2-1{)}^2.}$The potential energy takes its minimal value at${\displaystyle x=\pm 1}$, and these are called classical minima because the particle tends to lie in one of them in classical mechanics. There are two lowest energy states in classical mechanics.

In quantum mechanics, we solve theSchrödinger equation

${\displaystyle -\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\psi (x)+V(x)\psi (x)=E\psi (x),}$

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${\displaystyle x=\pm 1}$ 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.

One way to calculate this probability is by means of the semi-classicalWKB approximation, which requires the value of${\displaystyle \hslash }$ to be small. Thetime independent Schrödinger equation for the particle reads

${\displaystyle \frac{d^2\psi }{d{x}^2}=\frac{2m(V(x)-E)}{{\hslash }^2}\psi .}$

If the potential were constant, the solution would be a plane wave, up to a proportionality factor,

${\displaystyle \psi =\exp (-\mathrm{i}kx)}$

with

${\displaystyle k=\frac{\sqrt{2m(E-V)}}{\hslash }.}$

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

${\displaystyle e^{-\frac{1}{\hslash }{\int }_a^b\sqrt{2m(V(x)-E)}dx},}$

wherea andb are the beginning and endpoint of the tunneling trajectory.

Alternatively, the use ofpath integrals allows aninstanton interpretation and the same result can be obtained with this approach. In path integral formulation, the transition amplitude can be expressed as

${\displaystyle K(a,b;t)=⟨x=a|e^{-\frac{i\mathbb{H}t}{\hslash }}|x=b⟩=\int d[x(t)]e^{\frac{iS[x(t)]}{\hslash }}.}$

Following the process ofWick rotation (analytic continuation) to Euclidean spacetime (${\displaystyle it\to \tau }$), one gets

${\displaystyle K_E(a,b;\tau )=⟨x=a|e^{-\frac{\mathbb{H}\tau }{\hslash }}|x=b⟩=\int d[x(\tau )]e^{-\frac{S_E[x(\tau )]}{\hslash }},}$

with the Euclidean action

${\displaystyle S_E={\int }_{{\tau }_a}^{{\tau }_b}\left(\frac{1}{2}m{\left(\frac{dx}{d\tau }\right)}^2+V(x)\right)d\tau .}$

The potential energy changes sign${\displaystyle V(x)\to -V(x)}$ under the Wick rotation and the minima transform into maxima, thereby${\displaystyle V(x)}$ exhibits two "hills" of maximal energy.

Let us now consider the local minimum of the Euclidean action${\displaystyle S_E}$ with the double-well potential${\displaystyle V(x)=\frac{1}{4}(x^2-1{)}^2}$, and we set${\displaystyle m=1}$ just for simplicity of computation. Since we want to know how the two classically lowest energy states${\displaystyle x=\pm 1}$ are connected, let us set${\displaystyle a=-1}$ and${\displaystyle b=1}$. For${\displaystyle a=-1}$ and${\displaystyle b=1}$, we can rewrite the Euclidean action as

${\displaystyle S_E={\int }_{{\tau }_a}^{{\tau }_b}d\tau \frac{1}{2}{\left(\frac{dx}{d\tau }-\sqrt{2V(x)}\right)}^2+\sqrt{2}{\int }_{{\tau }_a}^{{\tau }_b}d\tau \frac{dx}{d\tau }\sqrt{V(x)}}$

${\displaystyle ={\int }_{{\tau }_a}^{{\tau }_b}d\tau \frac{1}{2}{\left(\frac{dx}{d\tau }-\sqrt{2V(x)}\right)}^2+{\int }_{-1}^{1}dx\frac{1}{\sqrt{2}}(1-x^2).}$

${\displaystyle \ge \frac{2\sqrt{2}}{3}.}$

The above inequality is saturated by the solution of${\displaystyle \frac{dx}{d\tau }=\sqrt{2V(x)}}$ with the condition${\displaystyle x({\tau }_a)=-1}$ and${\displaystyle x({\tau }_b)=1}$. Such solutions exist, and the solution takes the simple form when${\displaystyle {\tau }_a=-\mathrm{\infty }}$ and${\displaystyle {\tau }_b=\mathrm{\infty }}$. The explicit formula for the instanton solution is given by

${\displaystyle x(\tau )=\tanh \left(\frac{1}{\sqrt{2}}(\tau -{\tau }_0)\right).}$

Here${\displaystyle {\tau }_0}$ is an arbitrary constant. Since this solution jumps from one classical vacuum${\displaystyle x=-1}$ to another classical vacuum${\displaystyle x=1}$ instantaneously around${\displaystyle \tau ={\tau }_0}$, it is called an instanton.

The explicit formula for the eigenenergies of the Schrödinger equation withdouble-well potential has been given by Müller–Kirsten^[7] 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

${\displaystyle \frac{d^{2}y(z)}{d{z}^2}+[E-V(z)]y(z)=0,}$

and

${\displaystyle V(z)=-\frac{1}{4}{z}^2{h}^4+\frac{1}{2}{c}^2{z}^4,c^2>0,h^4>0,}$

the eigenvalues for${\displaystyle q_0=1,3,5,...}$ are found to be:

${\displaystyle E_{\pm }(q_0,h^2)=-\frac{h^8}{2^5{c}^2}+\frac{1}{\sqrt{2}}{q}_0{h}^2-\frac{c^2(3q_0^2+1)}{2h^4}-\frac{\sqrt{2}{c}^4{q}_0}{8h^{10}}(17q_0^2+19)+O(\frac{1}{h^{16}})}$

${\displaystyle \mp \frac{2^{q_0+1}{h}^2(h^6/2c^2{)}^{q_0/2}}{\sqrt{\pi }{2}^{q_0/4}[(q_0-1)/2]!}{e}^{-h^6/6\sqrt{2}{c}^2}.}$

Clearly these eigenvalues are asymptotically (${\displaystyle h^2\to \mathrm{\infty }}$) degenerate as expected as a consequence of the harmonic part of the potential.

Results obtained from the mathematically well-defined Euclideanpath 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 (${\displaystyle V(x)}$) 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 aninstanton. In this example, the two "vacua" (i.e. ground states) of thedouble-well potential, turn into hills in the Euclideanized version of the problem.

Thus, theinstanton 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

${\displaystyle V(\varphi )=\frac{m^4}{2g^2}{\left(1-\frac{g^2{\varphi }^2}{m^2}\right)}^2}$

the instanton, i.e. solution of

${\displaystyle \frac{d^2\varphi }{d{\tau }^2}=V^{\prime }(\varphi ),}$

(i.e. with energy${\displaystyle E_{cl}=0}$), is

${\displaystyle {\varphi }_c(\tau )=\frac{m}{g}\tanh \left[m(\tau -{\tau }_0)\right],}$

where${\displaystyle \tau =it}$ is the Euclidean time.

Note that a naïve perturbation theory around one of those two vacua alone (of the Minkowskian description) would never show thisnon-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 andspheroidal wave function) and irrespective of whether one uses the Schrödinger equation or thepath integral.^[8]

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 theinstantons) spoil thePeccei–Quinn symmetry explicitly and transform masslessNambu–Goldstone bosons into massivepseudo-Nambu–Goldstone ones.

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 ofsoliton theory the corresponding solution is known as akink. In view of their analogy with the behaviour of classical particles such configurations or solutions, as well as others, are collectively known aspseudoparticles 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.^[9] In explicit form they are expressible in terms ofJacobian 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, seeLamé function.^[10] 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.^[9]

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 a pseudoparticle on a high dimensionalpotential energy surface (PES). The thermal rate constant${\displaystyle k}$ can then be related to the imaginary part of the free energy${\displaystyle F}$ by^[11]

${\displaystyle k(\beta )=-\frac{2}{\hslash }\text{Im}\mathrm{F}=\frac{2}{\beta \hslash }\text{Im}\text{ln}(Z_k)\approx \frac{2}{\hslash \beta }\frac{\text{Im}{Z}_k}{\text{Re}{Z}_k},\text{Re}{Z}_k\gg \text{Im}{Z}_k}$

whereby${\displaystyle Z_k}$ is the canonical partition function, which is calculated by taking the trace of the Boltzmann operator in the position representation.

${\displaystyle Z_k=\text{Tr}(e^{-\beta \hat{H}})=\int d\mathbf{x}⟨\mathbf{x}\left|e^{-\beta \hat{H}}\right|\mathbf{x}⟩}$

Using a Wick rotation and identifying the Euclidean time with${\displaystyle \hslash \beta =1/(k_{b}T)}$, one obtains a path integral representation for the partition function in mass-weighted coordinates:^[12]

${\displaystyle Z_k=\oint \mathcal{D}\mathbf{x}(\tau )e^{-S_E[\mathbf{x}(\tau )]/\hslash },S_E={\int }_0^{\beta \hslash }\left({\frac{\dot{\mathbf{x}}}{2}}^2+V(\mathbf{x}(\tau ))\right)d\tau }$

The path integral is then approximated via a steepest descent integration, which takes into account only the contributions from the classical solutions and quadratic fluctuations around them. This yields for the rate constant expression in mass-weighted coordinates

${\displaystyle k(\beta )=\frac{2}{\beta \hslash }{\left(\frac{\text{det}\left[-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\tau }^2}+{\mathbf{V}}^{″}(x_{\text{RS}}(\tau ))\right]}{\text{det}\left[-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\tau }^2}+{\mathbf{V}}^{″}(x_{\text{Inst}}(\tau ))\right]}\right)}^{\frac{1}{2}}\exp \left(\frac{-S_E[x_{\text{inst}}(\tau )+S_E[x_{\text{RS}}(\tau )]}{\hslash }\right)}$

where${\displaystyle {\mathbf{x}}_{\text{Inst}}}$ is a periodic instanton and${\displaystyle {\mathbf{x}}_{\text{RS}}}$ is the trivial solution of the pseudoparticle at rest which represents the reactant state configuration.

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

${\displaystyle \frac{d^{2}y}{d{z}^2}+[E-V(z)]y(z)=0,V(z)=\frac{1}{4}{h}^4{z}^2-\frac{1}{2}{c}^2{z}^4,}$

the eigenvalues as given by Müller-Kirsten are, for${\displaystyle q_0=1,3,5,...,}$

${\displaystyle E=\frac{1}{2}{q}_0{h}^2-\frac{3c^2}{4h^4}(q_0^2+1)-\frac{q_0{c}^4}{h^{10}}(4q_0^2+29)+O(\frac{1}{h^{16}})\pm i\frac{2^{q_0}{h}^2(h^6/2c^2{)}^{q_0/2}}{(2\pi {)}^{1/2}[(q_0-1)/2]!}{e}^{-h^6/6c^2}.}$

The imaginary part of this expression agrees with the well known result of Bender and Wu.^[13] In their notation${\displaystyle \hslash =1,q_0=2K+1,h^6/2c^2=ϵ.}$

Hypersphere${\displaystyle S^3}$

In studyingquantum 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 "topologicalvacua".

A well understood and illustrative example of aninstanton and its interpretation can be found in the context of a QFT with anon-abelian gauge group,^{[note 2]} aYang–Mills theory. For a Yang–Mills theory these inequivalent sectors can be (in an appropriate gauge) classified by the thirdhomotopy group ofSU(2) (whose group manifold is the3-sphere${\displaystyle S^3}$). A certain topological vacuum (a "sector" of the true vacuum) is labelled by anunaltered transform, thePontryagin index. As the third homotopy group of${\displaystyle S^3}$ has been found to be the set ofintegers,

${\displaystyle {\pi }_3}$${\displaystyle (S^3)=}$${\displaystyle \mathbb{Z}}$

there are infinitely many topologically inequivalent vacua, denoted by${\displaystyle N⟩}$, where${\displaystyle N}$ is their corresponding Pontryagin index. Aninstanton 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,${\displaystyle Q}$. One can imagine aninstanton with index${\displaystyle Q}$ to quantify tunneling between topological vacua${\displaystyle |N⟩}$ and${\displaystyle |N+Q⟩}$. IfQ = 1, the configuration is namedBPST instanton after its discoverersAlexander Belavin,Alexander Polyakov,Albert S. Schwarz andYu. S. Tyupkin. The true vacuum of the theory is labelled by an "angle" theta and is an overlap of the topological sectors:

${\displaystyle |\theta ⟩=\sum _{N=-\mathrm{\infty }}^{N=+\mathrm{\infty }}{e}^{i\theta N}|N⟩.}$

Gerard 't Hooft first performed the field theoretic computation of the effects of the BPST instanton in a theory coupled to fermions in[1]^{[dead link‍]}. 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.

The classical Yang–Mills action on aprincipal bundle with structure groupG, baseM,connectionA, andcurvature (Yang–Mills field tensor)F is

${\displaystyle S_{YM}={\int }_M{\left|F\right|}^{2}d{\mathrm{v}\mathrm{o}\mathrm{l}}_M,}$

where${\displaystyle d{\mathrm{v}\mathrm{o}\mathrm{l}}_M}$ is thevolume form on${\displaystyle M}$. If the inner product on${\displaystyle \mathfrak{g}}$, theLie algebra of${\displaystyle G}$ in which${\displaystyle F}$ takes values, is given by theKilling form on${\displaystyle \mathfrak{g}}$, then this may be denoted as${\displaystyle {\int }_M\mathrm{T}\mathrm{r}(F\wedge \ast F)}$, since

${\displaystyle F\wedge \ast F=⟨F,F⟩d{\mathrm{v}\mathrm{o}\mathrm{l}}_M.}$

For example, in the case of thegauge groupU(1),F will be the electromagnetic fieldtensor. From theprinciple of stationary action, the Yang–Mills equations follow. They are

${\displaystyle \mathrm{d}F=0,\mathrm{d}\ast F=0.}$

The first of these is an identity, because dF = d²A = 0, but the second is a second-orderpartial differential equation for the connectionA, and if the Minkowski current vector does not vanish, the zero on the rhs. of the second equation is replaced by${\displaystyle \mathbf{J}}$. But notice how similar these equations are; they differ by aHodge star. Thus a solution to the simpler first order (non-linear) equation

${\displaystyle \ast F=\pm F}$

is automatically also a solution of the Yang–Mills equation. This simplification occurs on 4 manifolds with :${\displaystyle s=1}$ so that${\displaystyle {\ast }^2=+1}$ 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,${\displaystyle DF=0}$ and${\displaystyle D\ast F=0}$ where D is theexterior covariant derivative. Furthermore, theBianchi identity

${\displaystyle DF=dF+A\wedge F-F\wedge A=d(dA+A\wedge A)+A\wedge (dA+A\wedge A)-(dA+A\wedge A)\wedge A=0}$

is satisfied.

Inquantum field theory, aninstanton is atopologically nontrivial field configuration in four-dimensionalEuclidean space (considered as theWick rotation ofMinkowski spacetime). Specifically, it refers to aYang–Millsgauge fieldA which approachespure gauge atspatial infinity. This means the field strength

${\displaystyle \mathbf{F}=d\mathbf{A}+\mathbf{A}\wedge \mathbf{A}}$

vanishes at infinity. The nameinstanton 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 thetwo-dimensional space may be easier to visualise because it admits the simplest case of the gaugegroup, namely U(1), that is anabelian group. In this case the fieldA can be visualised as simply avector field. An instanton is a configuration where, for example, the arrows point away from a central point (i.e., a "hedgehog" state). In Euclideanfour dimensions,${\displaystyle {\mathbb{R}}^4}$, abelian instantons are impossible.

The field configuration of an instanton is very different from that of thevacuum. Because of this instantons cannot be studied by usingFeynman diagrams, which only includeperturbative effects. Instantons are fundamentallynon-perturbative.

The Yang–Mills energy is given by

${\displaystyle \frac{1}{2}{\int }_{{\mathbb{R}}^4}\mathrm{Tr}[\ast \mathbf{F}\wedge \mathbf{F}]}$

where ∗ is theHodge dual. If we insist that the solutions to the Yang–Mills equations have finiteenergy, then thecurvature of the solution at infinity (taken as alimit) has to be zero. This means that theChern–Simons invariant can be defined at the 3-space boundary. This is equivalent, viaStokes' theorem, to taking theintegral

${\displaystyle {\int }_{{\mathbb{R}}^4}\mathrm{Tr}[\mathbf{F}\wedge \mathbf{F}].}$

This is a homotopy invariant and it tells us whichhomotopy class the instanton belongs to.

Since the integral of a nonnegativeintegrand is always nonnegative,

${\displaystyle 0\le \frac{1}{2}{\int }_{{\mathbb{R}}^4}\mathrm{Tr}[(\ast \mathbf{F}+e^{-i\theta }\mathbf{F})\wedge (\mathbf{F}+e^{i\theta }\ast \mathbf{F})]={\int }_{{\mathbb{R}}^4}\mathrm{Tr}[\ast \mathbf{F}\wedge \mathbf{F}+\cos \theta \mathbf{F}\wedge \mathbf{F}]}$

for all real θ. So, this means

${\displaystyle \frac{1}{2}{\int }_{{\mathbb{R}}^4}\mathrm{Tr}[\ast \mathbf{F}\wedge \mathbf{F}]\ge \frac{1}{2}\left|{\int }_{{\mathbb{R}}^4}\mathrm{Tr}[\mathbf{F}\wedge \mathbf{F}]\right|.}$

If this bound is saturated, then the solution is aBPS state. For such states, either ∗F =F or ∗F = −F depending on the sign of thehomotopy invariant.

In the Standard Model instantons are expected to be present both in theelectroweak sector and the chromodynamic sector, however, their existence has not yet been experimentally confirmed.^[14] Instanton effects are important in understanding the formation of condensates in the vacuum ofquantum chromodynamics (QCD) and in explaining the mass of the so-called 'eta-prime particle', aGoldstone-boson^{[note 3]} which has acquired mass through theaxial current anomaly of QCD. Note that there is sometimes also a correspondingsoliton in a theory with one additional space dimension. Recent research oninstantons links them to topics such asD-branes andBlack holes and, of course, the vacuum structure of QCD. For example, in orientedstring theories, a Dp brane is a gauge theory instanton in the world volume (p + 5)-dimensionalU(N) gauge theory on a stack ofN D(p + 4)-branes.

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 nontrivialfour-formcharacteristic class. If the gauge symmetry is aunitary group orspecial unitary group then this characteristic class is the secondChern 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 firstPontrjagin class.

In 3-dimensional gauge theories withHiggs fields,'t Hooft–Polyakov monopoles play the role of instantons. In his 1977 paperQuark Confinement and Topology of Gauge Groups,Alexander Polyakov demonstrated that instanton effects in 3-dimensionalQED coupled to ascalar field lead to a mass for thephoton.

In 2-dimensional abelian gauge theoriesworldsheet instantons are magneticvortices. They are responsible for many nonperturbative effects in string theory, playing a central role inmirror symmetry.

In 1-dimensionalquantum mechanics, instantons describetunneling, which is invisible in perturbation theory.

Supersymmetric gauge theories often obeynonrenormalization theorems, which restrict the kinds of quantum corrections which are allowed. Many of these theorems only apply to corrections calculable inperturbation 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.

InN = 1 supersymmetric gauge theories instantons can modify thesuperpotential, sometimes lifting all of the vacua. In 1984,Ian Affleck,Michael Dine andNathan Seiberg calculated the instanton corrections to the superpotential in their paperDynamical Supersymmetry Breaking in Supersymmetric QCD. More precisely, they were only able to perform the calculation when the theory contains one less flavor ofchiral 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.

InN = 2 supersymmetric gauge theories the superpotential receives no quantum corrections. However the correction to the metric of themoduli space of vacua from instantons was calculated in a series of papers. First, the one instanton correction was calculated byNathan Seiberg inSupersymmetry and Nonperturbative beta Functions. The full set of corrections for SU(2) Yang–Mills theory was calculated byNathan Seiberg andEdward 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 asSeiberg–Witten theory. They extended their calculation to SU(2) gauge theories with fundamental matter inMonopoles, 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 usingNekrasov partition functions in 2003 byNikita Nekrasov andAndrei Okounkov and independently byHiraku Nakajima andKota Yoshioka.

InN = 4 supersymmetric gauge theories the instantons do not lead to quantum corrections for the metric on the moduli space of vacua.

Anansatz provided byCorrigan andFairlie provides a solution to the anti-self dual Yang–Mills equations with gauge group SU(2) from anyharmonic function on${\displaystyle {\mathbb{R}}^4}$.^[15][16] The ansatz gives explicit expressions for the gauge field and can be used to construct solutions with arbitrarily large instanton number.

Defining the antisymmetric${\displaystyle \mathfrak{s}\mathfrak{u}(2)}$-valued objects${\displaystyle {\sigma }_{\mu \nu }}$ as$${\displaystyle {\sigma }_{ij}={ϵ}_{ijk}{T}_k,{\sigma }_{i4}=-{\sigma }_{4i}=T_i,}$$where Greek indices run from 1 to 4, Latin indices run from 1 to 3, and${\displaystyle T_i}$ is a basis of${\displaystyle \mathfrak{s}\mathfrak{u}(2)}$ satisfying${\displaystyle [T_i,T_j]=-{ϵ}_{ijk}{T}_k}$. Then$${\displaystyle A_{\mu }={\sigma }_{\mu \nu }\frac{{\mathrm{\partial }}_{\nu }\rho }{\rho }={\sigma }_{\mu \nu }{\mathrm{\partial }}_{\nu }\log (\rho )}$$is a solution as long as${\displaystyle \rho :{\mathbb{R}}^4\to \mathbb{R}}$ is harmonic.

In four dimensions, thefundamental solution toLaplace's equation is${\displaystyle |x-y{|}^{-2}}$ for any fixed${\displaystyle y}$. Superposing${\displaystyle N+1}$ of these gives${\displaystyle N}$-soliton solutions of the form$${\displaystyle \rho (x)=\sum _{p=1}^N\frac{{\lambda }_p}{|x-x_p{|}^2}.}$$All solutions of instanton number 1 or 2 are of this form, but for larger instanton number there are solutions not of this form.

• Instanton fluid – Non-perturbative path integral approximation
• Caloron – Finite temperature instanton
• Sidney Coleman – American physicist (1937–2007)
• Holstein–Herring method – effective means of getting the exchange energy splittings of asymptotically degenerate energy states in molecular systemsPages displaying wikidata descriptions as a fallback
• Gravitational instanton – Four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations
• Semiclassical transition state theory – Chemical reaction rate theory
• Yang–Mills equations – Partial differential equations whose solutions are instantons
• Gauge theory (mathematics) – Study of vector bundles, principal bundles, and fibre bundles

Notes

Citations

General

• Instantons in Gauge Theories, a compilation of articles on instantons, edited byMikhail A. Shifman,doi:10.1142/2281
• Solitons and Instantons, R. Rajaraman (Amsterdam: North Holland, 1987),ISBN 0-444-87047-4
• The Uses of Instantons, bySidney Coleman inProc. Int. School of Subnuclear Physics, (Erice, 1977); and inAspects of Symmetry p. 265, Sidney Coleman, Cambridge University Press, 1985,ISBN 0-521-31827-0; and inInstantons 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.

• The dictionary definition ofinstanton at Wiktionary

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