Instanton

An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics.

Physical description

An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean spacetime. In such 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 points. Instantons are important in quantum field theory because:

they appear in the path integral as the leading quantum corrections to the classical behavior of a system, and
they can be used to study the tunneling behavior in various systems such as a Yang–Mills theory.

Mathematics

Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory. Instantons are topologically nontrivial solutions of Yang–Mills equations that absolutely minimize the energy functional within their topological type. The first such solutions were discovered in the case of four-dimensional Euclidean space compactified to the four-dimensional sphere, and turned out to be localized in space-time, prompting the names pseudoparticle and instanton.

Yang–Mills instantons have been explicitly constructed in many cases by means of twistor theory, which relates them to algebraic vector bundles on algebraic surfaces, and via the ADHM construction, or hyperkähler reduction (see hyperkähler manifold), a sophisticated linear algebra procedure. The groundbreaking work of Simon Donaldson, for which he was later awarded the Fields medal, used the moduli space of instantons over a given four-dimensional differentiable manifold as a new invariant of the manifold that depends on its differentiable structure and applied it to the construction of homeomorphic but not diffeomorphic four-manifolds. Many methods developed in studying instantons have also been applied to monopoles.^[why?]

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 approximation, which requires the value of $\hbar$ to be small. The Schrödinger equation for the particle reads

{\frac {d^{2}\psi }{dx^{2}}}={\frac {2m(V(x)-E)}{\hbar ^{2}}}\psi .

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

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

with

k={\frac {\sqrt {2m(E-V)}}{\hbar }}.

This means that if the energy of the particle is smaller than the potential energy, one obtains an exponentially decreasing function. The associated tunneling amplitude is proportional to

e^{-{\frac {1}{\hbar }}\int _{a}^{b}{\sqrt {2m(V(x)-E)}}\,dx},

where a and b are the beginning and endpoint of the tunneling trajectory.

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

K(a,b;t)=\langle x=a|e^{-{\frac {i\mathbb {H} t}{\hbar }}}|x=b\rangle =\int d[x(t)]e^{\frac {iS[x(t)]}{\hbar }}.

Following the process of Wick rotation (analytic continuation) to Euclidean spacetime ( $it\rightarrow \tau$ ), one gets

K_{E}(a,b;\tau )=\langle x=a|e^{-{\frac {\mathbb {H} \tau }{\hbar }}}|x=b\rangle =\int d[x(\tau )]e^{-{\frac {S_{E}[x(\tau )]}{\hbar }}},

with the Euclidean action

S_{E}=\int _{\tau _{a}}^{\tau _{b}}\left({\frac {1}{2}}m\left({\frac {dx}{d\tau }}\right)^{2}+V(x)\right)d\tau .

The potential energy changes sign $V(x)\rightarrow -V(x)$ under the Wick rotation and the minima transform into maxima, thereby $V(x)$ 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 ( $V(x)$ ) with the Minkowskian path integral corresponds to calculating the transition probability to tunnel through a classically allowed region (with potential −V(X)) in the Euclidean path integral (pictorially speaking—in the Euclidean picture—this transition corresponds to a particle rolling from one hill of a double-well potential standing on its head to the other hill). This classical solution of the Euclidean equations of motion is often named "kink solution" and is an example of an instanton. In this example, the two "vacua" 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 of SU(2) (whose group manifold is the 3-sphere $S^{3}$ ). 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 $S^{3}$ has been found to be the set of integers,

\pi _{3}(S^{3})=\mathbb {Z} \,

there are infinitely many topologically inequivalent vacua, denoted by $|N\rangle$ , where $N$ is their corresponding Pontryagin index. An instanton is a field configuration fulfilling the classical equations of motion in Euclidean spacetime, which is interpreted as a tunneling effect between these different topological vacua. It is again labelled by an integer number, its Pontryagin index, $Q$ . One can imagine an instanton with index $Q$ to quantify tunneling between topological vacua $|N\rangle$ and $|N+Q\rangle$ . If Q = 1, the configuration is named BPST instanton after its discoverers Alexander Belavin, Alexander Polyakov, Albert S. Schwartz and Yu. S. Tyupkin. The true vacuum of the theory is labelled by an "angle" theta and is an overlap of the topological sectors:

|\theta \rangle =\sum _{N=-\infty }^{N=+\infty }e^{i\theta N}|N\rangle .

Gerard 't Hooft first performed the field theoretic computation of the effects of the BPST instanton in a theory coupled to fermions in [1]. He showed that zero modes of the Dirac equation in the instanton background lead to a non-perturbative multi-fermion interaction in the low energy effective action.

Yang–Mills theory

The classical Yang–Mills action on a principal bundle with structure group G, base M, connection A, and curvature (Yang–Mills field tensor) F is

S_{YM}=\int _{M}\left|F\right|^{2}d\mathrm {vol} _{M},

where $d\mathrm {vol} _{M}$ is the volume form on $M$ . If the inner product on ${\mathfrak {g}}$ , the Lie algebra of $G$ in which $F$ takes values, is given by the Killing form on ${\mathfrak {g}}$ , then this may be denoted as $\int _{M}\mathrm {Tr} (F\wedge *F)$ , since

F\wedge *F=\langle F,F\rangle d\mathrm {vol} _{M}.

For example, in the case of the gauge group U(1), F will be the electromagnetic field tensor. From the principle of stationary action, the Yang–Mills equations follow. They are

\mathrm {d} F=0,\quad \mathrm {d} {*F}=0.

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

{*F}=\pm F\,

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, $DF=0$ and $D*F=0$ where D is the exterior covariant derivative. Furthermore, the Bianchi identity

DF=dF+A\wedge F-F\wedge A=d(dA+A\wedge A)+A\wedge (dA+A\wedge A)-(dA+A\wedge A)\wedge A=0

is satisfied.

In quantum field theory, an instanton is a topologically nontrivial field configuration in four-dimensional Euclidean space (considered as the Wick rotation of Minkowski spacetime). Specifically, it refers to a Yang–Mills gauge field A which locally approaches pure gauge at spatial infinity. This means the field strength defined by A,

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

vanishes at infinity. The name instanton derives from the fact that these fields are localized in space and (Euclidean) time – in other words, at a specific instant.

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 diagrams, which only include perturbative effects. Instantons are fundamentally non-perturbative.

The Yang–Mills energy is given by

{\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} [*{\mathbf {F}}\wedge {\mathbf {F}}]

where ∗ is the Hodge dual. If we insist that the solutions to the Yang–Mills equations have finite energy, then the curvature of the solution at infinity (taken as a limit) has to be zero. This means that the Chern–Simons invariant can be defined at the 3-space boundary. This is equivalent, via Stokes' theorem, to taking the integral

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

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

Since the integral of a nonnegative integrand is always nonnegative,

0\leq {\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} [(*{\mathbf {F}}+e^{-i\theta }{\mathbf {F}})\wedge ({\mathbf {F}}+e^{i\theta }*{\mathbf {F}})]=\int _{\mathbb {R} ^{4}}\operatorname {Tr} [*{\mathbf {F}}\wedge {\mathbf {F}}+\cos \theta {\mathbf {F}}\wedge {\mathbf {F}}]

for all real θ. So, this means

{\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} [*{\mathbf {F}}\wedge {\mathbf {F}}]\geq {\frac {1}{2}}\left|\int _{\mathbb {R} ^{4}}\operatorname {Tr} [{\mathbf {F}}\wedge {\mathbf {F}}]\right|.

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

Instanton effects are important in understanding the formation of condensates in the vacuum of quantum chromodynamics (QCD) and in explaining the mass of the so-called 'eta-prime particle', a Goldstone-boson which has acquired mass through the axial current anomaly of QCD. Note that there is sometimes also a corresponding soliton in a theory with one additional space dimension. Recent research on instantons links them to topics such as D-branes and Black holes and, of course, the vacuum structure of QCD. For example, in oriented string theories, a Dp brane is a gauge theory instanton in the world volume (p + 5)-dimensional U(N) gauge theory on a stack of N D(p + 4)-branes.

Various numbers of dimensions

Instantons play a central role in the nonperturbative dynamics of gauge theories. The kind of physical excitation that yields an instanton depends on the number of dimensions of the spacetime, but, surprisingly, the formalism for dealing with these instantons is relatively dimension-independent.

In 4-dimensional gauge theories, as described in the previous section, instantons are gauge bundles with a nontrivial four-form characteristic class. If the gauge symmetry is a unitary group or special unitary group then this characteristic class is the second Chern class, which vanishes in the case of the gauge group U(1). If the gauge symmetry is an orthogonal group then this class is the first Pontrjagin class.

In 3-dimensional gauge theories with Higgs fields, 't Hooft–Polyakov 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.

4d supersymmetric gauge theories

Supersymmetric gauge theories often obey nonrenormalization theorems, which restrict the kinds of quantum corrections which are allowed. Many of these theorems only apply to corrections calculable in perturbation theory and so instantons, which are not seen in perturbation theory, provide the only corrections to these quantities.

Field theoretic techniques for instanton calculations in supersymmetric theories were extensively studied in the 1980s by multiple authors. Because supersymmetry guarantees the cancellation of fermionic vs. bosonic non-zero modes in the instanton background, the involved 't Hooft computation of the instanton saddle point reduces to an integration over zero modes.

In N = 1 supersymmetric gauge theories instantons can modify the superpotential, sometimes lifting all of the vacua. In 1984 Ian Affleck, Michael Dine and Nathan Seiberg calculated the instanton corrections to the superpotential in their paper Dynamical Supersymmetry Breaking in Supersymmetric QCD. More precisely, they were only able to perform the calculation when the theory contains one less flavor of chiral matter than the number of colors in the special unitary gauge group, because in the presence of fewer flavors an unbroken nonabelian gauge symmetry leads to an infrared divergence and in the case of more flavors the contribution is equal to zero. For this special choice of chiral matter, the vacuum expectation values of the matter scalar fields can be chosen to completely break the gauge symmetry at weak coupling, allowing a reliable semi-classical saddle point calculation to proceed. By then considering perturbations by various mass terms they were able to calculate the superpotential in the presence of arbitrary numbers of colors and flavors, valid even when the theory is no longer weakly coupled.

In N = 2 supersymmetric gauge theories the superpotential receives no quantum corrections. However the correction to the metric of the moduli space of vacua from instantons was calculated in a series of papers. First, the one instanton correction was calculated by Nathan Seiberg in Supersymmetry and Nonperturbative beta Functions. The full set of corrections for SU(2) Yang–Mills theory was calculated by Nathan Seiberg and Edward Witten in Electric – magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang–Mills theory, in the process creating a subject that is today known as Seiberg-Witten theory. They extended their calculation to SU(2) gauge theories with fundamental matter in Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD. These results were later extended for various gauge groups and matter contents, and the direct gauge theory derivation was also obtained in most cases. 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.

References

Instantons in Gauge Theories, a compilation of articles on instantons, edited by Mikhail A. Shifman
Solitons and Instantons, R. Rajaraman (Amsterdam: North Holland, 1987), ISBN 0-444-87047-4
The Uses of Instantons, by Sidney Coleman in Proc. Int. School of Subnuclear Physics, (Erice, 1977); and in Aspects of Symmetry p. 265, Sidney Coleman, Cambridge University Press, 1985, ISBN 0-521-31827-0; and in Instantons in Gauge Theories
Solitons, Instantons and Twistors. M. Dunajski, Oxford University Press. ISBN 978-0-19-857063-9.