Instanton
An instanton (or pseudoparticle
In such quantum theories, solutions to the equations of motion may be thought of as
- 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
Mathematics
Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual
Yang–Mills instantons have been explicitly constructed in many cases by means of
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.[4]
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 (analytic continuation) to Euclidean spacetime (), one gets
with the Euclidean action
The potential energy changes sign under the Wick rotation and the minima transform into maxima, thereby exhibits two "hills" of maximal energy.
Let us now consider the local minimum of the Euclidean action with the double-well potential , and we set just for simplicity of computation. Since we want to know how the two classically lowest energy states are connected, let us set and . For and , we can rewrite the Euclidean action as
The above inequality is saturated by the solution of with the condition and . Such solutions exist, and the solution takes the simple form when and . The explicit formula for the instanton solution is given by
Here is an arbitrary constant. Since this solution jumps from one classical vacuum to another classical vacuum instantaneously around , it is called an instanton.
Explicit formula for double-well potential
The explicit formula for the eigenenergies of the Schrödinger equation with double-well potential 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
and
the eigenvalues for are found to be:
Clearly these eigenvalues are asymptotically () degenerate as expected as a consequence of the harmonic part of the potential.
Results
Results obtained from the mathematically well-defined Euclidean path integral may be Wick-rotated back and give the same physical results as would be obtained by appropriate treatment of the (potentially divergent) Minkowskian path integral. As can be seen from this example, calculating the transition probability for the particle to tunnel through a classically forbidden region () with the Minkowskian path integral corresponds to calculating the transition probability to tunnel through a classically allowed region (with potential −V(X)) in the Euclidean path integral (pictorially speaking – in the Euclidean picture – this transition corresponds to a particle rolling from one hill of a double-well potential standing on its head to the other hill). This classical solution of the Euclidean equations of motion is often named "kink solution" and is an example of an instanton. In this example, the two "vacua" (i.e. ground states) of the double-well potential, turn into hills in the Euclideanized version of the problem.
Thus, the instanton field solution of the (Euclidean, i. e., with imaginary time) (1 + 1)-dimensional field theory – first quantized quantum mechanical description – allows to be interpreted as a tunneling effect between the two vacua (ground states – higher states require periodic instantons) of the physical (1-dimensional space + real time) Minkowskian system. In the case of the double-well potential written
the instanton, i.e. solution of
(i.e. with energy ), is
where is the Euclidean time.
Note that a naïve perturbation theory around one of those two vacua alone (of the Minkowskian description) would never show this non-perturbative tunneling effect, dramatically changing the picture of the vacuum structure of this quantum mechanical system. In fact the naive perturbation theory has to be supplemented by boundary conditions, and these supply the nonperturbative effect, as is evident from the above explicit formula and analogous calculations for other potentials such as a cosine potential (cf. Mathieu function) or other periodic potentials (cf. e.g. Lamé function and spheroidal wave function) and irrespective of whether one uses the Schrödinger equation or the path 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
Periodic instantons
In one-dimensional field theory or quantum mechanics one defines as "instanton" a field configuration which is a solution of the classical (Newton-like) equation of motion with Euclidean time and finite Euclidean action. In the context of soliton 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" (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.
The stability of these pseudoclassical configurations can be investigated by expanding the Lagrangian defining the theory around the pseudoparticle configuration and then investigating the equation of small fluctuations around it. For all versions of quartic potentials (double-well, inverted double-well) and periodic (Mathieu) potentials these equations were discovered to be Lamé equations, see Lamé function.[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]
Instantons in reaction rate theory
In the context of reaction rate theory periodic instantons are used to calculate the rate of tunneling of atoms in chemical reactions. The progress of a chemical reaction can be described as the movement of a pseudoparticle on a high dimensional potential energy surface (PES). The thermal rate constant can then be related to the imaginary part of the free energy by
whereby is the canonical partition function which is calculated by taking the trace of the Boltzmann operator in the position representation.
Using a wick rotation and identifying the Euclidean time with one obtains a path integral representation for the partition function in mass weighted coordinates
The path integral is then approximated via a steepest descent integration which takes only into account the contributions from the classical solutions and quadratic fluctuations around them. This yields for the rate constant expression in mass weighted coordinates
where is a periodic instanton and is the trivial solution of the pseudoparticle at rest which represents the reactant state configuration.
Inverted double-well formula
As for the double-well potential one can derive the eigenvalues for the inverted double-well potential. In this case, however, the eigenvalues are complex. Defining parameters by the equations
the eigenvalues as given by Müller-Kirsten are, for
The imaginary part of this expression agrees with the well known result of Bender and Wu.[11] In their notation
Quantum field theory
Hypersphere | |
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In studying
A well understood and illustrative example of an instanton and its interpretation can be found in the context of a QFT with a
there are infinitely many topologically inequivalent vacua, denoted by , where is their corresponding Pontryagin index. An instanton is a field configuration fulfilling the classical equations of motion in Euclidean spacetime, which is interpreted as a tunneling effect between these different topological vacua. It is again labelled by an integer number, its Pontryagin index, . One can imagine an instanton with index to quantify tunneling between topological vacua and . If Q = 1, the configuration is named
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
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
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
is automatically also a solution of the Yang–Mills equation. This simplification occurs on 4 manifolds with : so that on 2-forms. Such solutions usually exist, although their precise character depends on the dimension and topology of the base space M, the principal bundle P, and the gauge group G.
In nonabelian Yang–Mills theories, and where D is the
is satisfied.
In
vanishes at infinity. The name instanton derives from the fact that these fields are localized in space and (Euclidean) time – in other words, at a specific instant.
The case of instantons on the two-dimensional space may be easier to visualise because it admits the simplest case of the gauge group, namely U(1), that is an abelian group. In this case the field A can be visualised as simply a vector field. An instanton is a configuration where, for example, the arrows point away from a central point (i.e., a "hedgehog" state). In Euclidean four dimensions, , abelian instantons are impossible.
The field configuration of an instanton is very different from that of the
The Yang–Mills energy is given by
where ∗ is the
This is a homotopy invariant and it tells us which
Since the integral of a nonnegative
for all real θ. So, this means
If this bound is saturated, then the solution is a
In the Standard Model instantons are expected to be present both in the
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
In 3-dimensional gauge theories with
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
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
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. For gauge theories with gauge group U(N) the Seiberg–Witten geometry has been derived from gauge theory using Nekrasov partition functions in 2003 by Nikita Nekrasov and Andrei Okounkov and independently by Hiraku Nakajima and Kota Yoshioka.
In N = 4 supersymmetric gauge theories the instantons do not lead to quantum corrections for the metric on the moduli space of vacua.
Explicit solutions on R4
An ansatz provided by Corrigan and Fairlie provides a solution to the anti-self dual Yang–Mills equations with gauge group SU(2) from any harmonic function on .[13][14] The ansatz gives explicit expressions for the gauge field and can be used to construct solutions with arbitrarily large instanton number.
Defining the antisymmetric -valued objects as
In four dimensions, the fundamental solution to Laplace's equation is for any fixed . Superposing of these gives -soliton solutions of the form
See also
- 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 systems
- 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
References and notes
- Notes
- ^ Because this projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect <0,0,0,1> have infinite radius (= straight line).
- Non-abelian gauge theory
- ^ See also: Pseudo-Goldstone boson
- Citations
- ^ Instantons in Gauge Theories. Edited by Mikhail A. Shifman. World Scientific, 1994.
- ^ Interactions Between Charged Particles in a Magnetic Field. By Hrachya Nersisyan, Christian Toepffer, Günter Zwicknagel. Springer, Apr 19, 2007. Pg 23
- ^ Large-Order Behaviour of Perturbation Theory. Edited by J.C. Le Guillou, J. Zinn-Justin. Elsevier, Dec 2, 2012. Pg. 170.
- ^ ISSN 0038-5670.
- ^ "Yang-Mills instanton in nLab". ncatlab.org. Retrieved 2023-04-11.
- ^ See, for instance, Nigel Hitchin's paper "Self-Duality Equations on Riemann Surface".
- ISBN 978-981-4397-73-5; formula (18.175b), p. 525.
- ISBN 978-981-4397-73-5.
- ^ a b Harald J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (Singapore, 2012).
- ISSN 0370-2693.
- ISSN 0556-2821.
- S2CID 229220708.
- .
- ISBN 9780198570639.
- General
- Instantons in Gauge Theories, a compilation of articles on instantons, edited by doi:10.1142/2281
- Solitons and Instantons, R. Rajaraman (Amsterdam: North Holland, 1987), ISBN 0-444-87047-4
- The Uses of Instantons, by 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.
- The Geometry of Four-Manifolds, S.K. Donaldson, P.B. Kronheimer, Oxford University Press, 1990, ISBN 0-19-853553-8.
External links
- The dictionary definition of instanton at Wiktionary