Instanton


An instanton is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean spacetime.
In such quantum theories, solutions to the equations of motion may be thought of as critical points of the action. The critical points of the action may be local maxima of the action, local minima, or saddle points. Instantons are important in quantum field theory because:
  • they appear in the path integral as the leading quantum corrections to the classical behavior of a system, and
  • they can be used to study the tunneling behavior in various systems such as a Yang–Mills theory.
Relevant to dynamics, families of instantons 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 is believed to be the explanation of the noise-induced chaotic phase known as self-organized criticality.

Mathematics

Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory. Instantons are topologically nontrivial solutions of Yang–Mills equations that absolutely minimize the energy functional within their topological type. The first such solutions were discovered in the case of four-dimensional Euclidean space compactified to the four-dimensional sphere, and turned out to be localized in space-time, prompting the names pseudoparticle and instanton.
Yang–Mills instantons have been explicitly constructed in many cases by means of twistor theory, which relates them to algebraic vector bundles on algebraic surfaces, and via the ADHM construction, or hyperkähler reduction, a geometric invariant theory procedure. The groundbreaking work of Simon Donaldson, for which he was later awarded the Fields medal, used the moduli space of instantons over a given four-dimensional differentiable manifold as a new invariant of the manifold that depends on its differentiable structure and applied it to the construction of homeomorphic but not diffeomorphic four-manifolds. Many methods developed in studying instantons have also been applied to monopoles. This is because magnetic monopoles arise as solutions of a dimensional reduction of the Yang–Mills equations.

Quantum mechanics

An instanton can be used to calculate the transition probability for a quantum mechanical particle tunneling through a potential barrier. One example of a system with an instanton effect is a particle in a double-well potential. In contrast to a classical particle, there is non-vanishing probability that it crosses a region of potential energy higher than its own energy.

Motivation of considering instantons

Consider the quantum mechanics of a single particle motion inside the double-well potential
The potential energy takes its minimal value at, and these are called classical minima because the particle tends to lie in one of them in classical mechanics. There are two lowest energy states in classical mechanics.
In quantum mechanics, we solve the Schrödinger equation
to identify the energy eigenstates. If we do this, we will find only the unique lowest-energy state instead of two states. The ground-state wave function localizes at both of the classical minima instead of only one of them because of the quantum interference or quantum tunneling.
Instantons are the tool to understand why this happens within the semi-classical approximation of the path-integral formulation in Euclidean time. We will first see this by using the WKB approximation that approximately computes the wave function itself, and will move on to introduce instantons by using the path integral formulation.

WKB approximation

One way to calculate this probability is by means of the semi-classical WKB approximation, which requires the value of to be small. The time independent Schrödinger equation for the particle reads
If the potential were constant, the solution would be a plane wave, up to a proportionality factor,
with
This means that if the energy of the particle is smaller than the potential energy, one obtains an exponentially decreasing function. The associated tunneling amplitude is proportional to
where a and b are the beginning and endpoint of the tunneling trajectory.

Path integral interpretation via instantons

Alternatively, the use of path integrals allows an instanton interpretation and the same result can be obtained with this approach. In path integral formulation, the transition amplitude can be expressed as
Following the process of Wick rotation to Euclidean spacetime, one gets
with the Euclidean action
The potential energy changes sign under the Wick rotation and the minima transform into maxima, thereby exhibits two "hills" of maximal energy.
Let us now consider the local minimum of the Euclidean action with the double-well potential, and we set just for simplicity of computation. Since we want to know how the two classically lowest energy states are connected, let us set and.
For and, we can rewrite the Euclidean action as
The above inequality is saturated by the solution of with the condition and. Such solutions exist, and the solution takes the simple form when and. The explicit formula for the instanton solution is given by
Here is an arbitrary constant. Since this solution jumps from one classical vacuum to another classical vacuum instantaneously around, it is called an instanton.

Explicit formula for double-well potential

The explicit formula for the eigenenergies of the Schrödinger equation with double-well potential has been given by Müller–Kirsten with derivation by both a perturbation method applied to the Schrödinger equation, and explicit derivation from the path integral. The result is the following. Defining parameters of the Schrödinger equation and the potential by the equations
and
the eigenvalues for are found to be:
Clearly these eigenvalues are asymptotically degenerate as expected as a consequence of the harmonic part of the potential.

Results

Results obtained from the mathematically well-defined Euclidean path integral may be Wick-rotated back and give the same physical results as would be obtained by appropriate treatment of the 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 in the Euclidean path integral. 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 -dimensional field theory - first quantized quantum mechanical description - allows to be interpreted as a tunneling effect between the two vacua of the physical Minkowskian system. In the case of the double-well potential written
the instanton, i.e. solution of
, is
where is the Euclidean time.
Note that a naïve perturbation theory around one of those two vacua alone would never show this non-perturbative tunneling effect, dramatically changing the picture of the vacuum structure of this quantum mechanical system. In fact the naive perturbation theory has to be supplemented by boundary conditions, and these supply the nonperturbative effect, as is evident from the above explicit formula and analogous calculations for other potentials such as a cosine potential or other periodic potentials and irrespective of whether one uses the Schrödinger equation or the path integral.
Therefore, the perturbative approach may not completely describe the vacuum structure of a physical system. This may have important consequences, for example, in the theory of "axions" where the non-trivial QCD vacuum effects spoil the Peccei–Quinn symmetry explicitly and transform massless Nambu–Goldstone bosons into massive pseudo-Nambu–Goldstone ones.

Periodic instantons

In one-dimensional field theory or quantum mechanics one defines as "instanton" a field configuration which is a solution of the classical equation of motion with Euclidean time and finite Euclidean action. In the context of soliton theory the corresponding solution is known as a kink. In view of their analogy with the behaviour of classical particles such configurations or solutions, as well as others, are collectively known as pseudoparticles or pseudoclassical configurations. The "instanton" solution is accompanied by another solution known as "anti-instanton", and instanton and anti-instanton are distinguished by "topological charges" +1 and −1 respectively, but have the same Euclidean action.
"Periodic instantons" are a generalization of instantons. In explicit form they are expressible in terms of Jacobian elliptic functions which are periodic functions. 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 and periodic potentials these equations were discovered to be Lamé equations, see Lamé function. 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.