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In quantum mechanics, potential barriers are no longer as tough as they are in classical mechanics. Instead, any physical system can tunnel through a potential barrier with some probability.

Instantons are sequences of field configurations that describe how a field tunnels through a potential barrier. Such potential barriers exist, for example, when there is not only one state with minimum energy but many. Between these possible ground states, we usually have a potential barrier. However, in a quantum theory the field can transform itself from one ground state configuration into another ground state configuration and such a process is called an instanton.


In contrast to other solitons like, for example, monopoles, instantons can not be interpreted as "particle-like". Instead instantons is a continuous set of field configurations that describe how the field tunnels from one vacuum configuration into another. Nevertheless, we call instantons also topological solitons, because they describe field configurations with finite (Euclidean) energy.

Such processes cannot be described by perturbation theory, but instead only with the help of non-perturbative methods. This follows since the wave function of tunnel processes is proportional to $e^{1/x}$ or $e^{1/x^2}$ and the Taylor expansion of such functions vanishes. Hence such effects do not appear in a perturbative expansion also, of course, these effects exist.

The ground state of, for example, QCD consists of an infinite number of degenerate states that are separated by a finite energy barrier. An instanton is a description how the field tunnels (not meant in a spatial sense) through one of these barriers into another vacuum. During the tunnel process the field, also in the ground state at the beginning and end of the process, goes continuously through a set of field configurations that do not correspond to a ground state, i.e. non-zero field energy. This is meant when we say that an instanton "has" finite field energy.

A detailed discussion of instantons written with the needs of students in mind can be found here.

Instantons occur in pure Yang-Mills theory with a non-abelian gauge group, for example, $SU(2)$. In contrast to Dirac monopoles and 't Hooft-Polyakov monopoles, Instantons do care about time. That's where their name comes from: they are localized in space and time. Therefore, this time we must consider their behavior in space and time. For reasons explained here, we describe tunnel processes in Euclidean spacetime, instead of Minkowski spacetime. Thus, for instantons, we investigate their behavior in $\mathbb{R}^4$.

Our demand for a finite field energy is translated mathematically into a description on $S^4$ instead of $\mathbb{R}^4$, because $\mathbb{R}^4 \cup \{ \infty \} \simeq S^4$.

As for the monopoles, the presence of an instanton makes itself felt through the fact that we must consider two patches for the gauge potential and need a transition function in the overlap region. This time the minimal overlap region, the higher-dimensional "equator" of $S^4$, is $S^3$. As for the Dirac monopole, where we also considered a pure gauge theory, our transition function is a map from the overlap region $S^3$ to the gauge group $SU(2)$. $SU(2)$ as a manifold is $S^3$ and thus our transition function is a map $S^3 \to S^3$.

The topological conclusion will be that instantons are characterized by the third homotopy class of $SU(2) \simeq S^3$, which is again simply $\mathbb{Z}$, the set of integers.

Minimas of the "Yang-Mills energy" correspond to pure gauge configurations. A pure gauge configuration of a field is a field configuration that can be written as a gauge transformation of $A_\mu=0$:

\begin{equation} G_{\mu}^{\left( pg\right) }=\frac{\pi}{g}U\partial_{\mu}U^{\dagger} \end{equation}

An example is the famous BPST instanton (source):

$$ G_\mu(x) = \frac{2}{g} \frac{\eta_{\mu\nu} x_\nu}{x^2+\rho^2} ,$$

where $\rho$ is an arbitrary parameter that characterizes the size of the instanton.

Equally, one can derive the distribution of field strength, for example, for an anti-instanton:

$$ G_{\mu\nu}(x) = \frac{\eta_{\mu\nu}192 \rho^4 }{(x^2+\rho^2)^4} .$$

Thus, we can see that the field strength is localized in space.

[An instanton] is much like a topological soliton in field theory, except that it is localized in time rather than in space

  • A great short introduction can be found in section 13.7 "Yang-Mills Instantons" in the book Gauge Field Theories by Guidry
  • The best book in the topic is Solitons and Instantons by R. Rajaraman

The standard references are

  • Aspects of Symmetry by S. Coleman
  • and ABC of Instantons by M. Shifman et al.


One way to write the instanton potential is

$$ A_1 = \frac{r}{r^2+c^2}\gamma^{-1}d\gamma , $$ where $$ \gamma = \frac{1}{r}\left( x_0 + i \sum_i \sigma_i x_i \right),$$ where $\sigma_i$ are the Pauli matrices.

This potential is regular at $x=0$, but decays only as $r^{-1}$ for $r \to \infty$.

We can perform a gauge transformation of $A_1$ to get

$$ A_2 = \gamma A_1 \gamma^{-1} +\gamma d\gamma= \frac{c^2}{r^2+c^2}\gamma d\gamma^{-1} . $$

Now, $A_2$ is singular at $x=0$, but decays sufficiently fast (like $r^{-3}$) as $r \to \infty$.

In this sense, we can not get a global description of the instanton potential, but instead must use the two local descriptions $A_1$ and $A_2$ that are valid in different domain. $A_1$ is valid on $S^4 - \{ \text{south pole} \}=U_1$ and it is the slow decay as $r \to \infty$ that prevents us from using $A_1$ everywhere. Analogously, we can use $A_2$ on $S^4 - \{ \text{north pole} \} =U_2$, because $A_2$ decays sufficiently fast, but is singular at $x=0$.

In the overlap region $U_1 \cup U_2$ the two descriptions $A_1$ and $A_2$ are related by a gauge transformation.

$A_1$ and $A_2$ are local descriptions of a connection on an $SU(2)$ principal bundle over $S^4$ whose transition function is $\gamma$. The total space of the bundle is $S^7$ and thus, we can say that an instanton is described by the Hopf map

$$ S^7 \to S^4 . $$

The physics of isotopic spin led Yang and Mills to propose certain differ- ential equations (about which we will have a bit more to say shortly) that the potential functions B μ should satisfy. In 1975, Belavin, Polyakov, Schwartz and Tyupkin [BPST] found a number of remarkable solutions to these equa- tions that they christened “pseudoparticles”. More remarkable still is the fact that these solutions formally coincide with the pullbacks (6.1.14) to R 4 of connections on the Hopf bundle (only the n = 0 case appears explicitly in [BPST]). This observation was made explicit and generalized by Trautman [Trau] and further generalized by Nowakowski and Trautman [NT]. Topology, Geometry and Gauge Fields: Foundations by Naber

The classical mechanics of an electron propagating in an electromagnetic field on a spacetime $X$ is all encoded in a differential 2-form on $X$, called the Faraday tensor $F$, which encodes the classical Lorentz force that the electromagnetic field exerts on the electron. But this data is insufficient for passing to the quantum theory of the electron: locally, on a coordinate chart $U$, what the quantum electron really couples to is the vector potential, a differential 1-form $A_U$ on $U$, such that $d A_U = F|_U$. But globally such a vector potential may not exist. Dirac realized that what it takes to define the quantized electron globally is, in modern language, a lift of the locally defined vector potentials to an $(\mathbb{R}/\mathbb{Z})$-principal connection on a $(\mathbb{R}/\mathbb{Z})$-principal bundle over spacetime. The first Chern class of this principal bundle is quantized, and this is identified with the quantization of the magnetic charge whose induced force the electron feels. This quantization effect, which needs to be present before the quantization of the dynamics of the electron itself even makes sense globally, is an example of pre-quantization.

A variant of this example occupies particle physics these days. As we pass attention from electrons to quarks, these couple to the weak and strong nuclear force, and this coupling is, similarly, locally described by a 1-form $A_U$, but now with values in a Lie algebra $\mathfrak{su}(n)$, from which the strength of the nuclear force field is encoded by the 2-form $F|_U := d A_U + \tfrac{1}{2}[A_U \wedge A_U]$. For the consistency of the quantization of quarks, notably for the consistent global definition of Wilson loop observables, this local data must be lifted to an $\mathrm{SU}(n)$-principal connection on a $\mathrm{SU}(n)$-principal bundle over spacetime. The second Chern class of this bundle is quantized, and is physically interpreted as the number of instantons. (Strictly speaking, the term "instanton" refers to a principal connection that in addition to having non-trivial topological charge also minimizes Euclidean energy. Here we are just concerned with the nontrivial topological charge, which in particular is insensitive to and independent of any "Wick rotation".) In the physics literature instantons are expressed via Chern-Simons 3-forms, mathematically these constitute the pre-quantization of the 4-form $\mathrm{tr}(F \wedge F)$ to a 2-gerbe with 2-connection, more on this in a moment. [.,.]

A well-kept secret of the traditional formulation of variational calculus on jet bundles is that it does not in fact allow to properly formulate global aspects of local gauge theory. Namely the only way to make the fields of gauge theory be sections of a traditional field bundle is to fix the instanton number (Chern class) of the gauge field configuration. The gauge fields then are taken to be connections on that fixed bundle. One may easily see [Sch14e] that it is impossible to have a description of gauge fields as sections of a field bundle that is both local and respects the gauge principle. However, this is possible with a higher field bundle. Indeed, the natural choice of the field bundle for gauge fields has as typical fiber the smooth moduli stack of principal connections. Formulated this way, not only does the space of all field configurations then span all instanton sectors, but it also has the gauge transformations between gauge field configurations built into it. In fact it is then the globalized (integrated) version of what in the physics literature is known as the (off-shell) BRST complex of gauge theory.

  • See section 6.3 in Topology, Geometry and Gauge Fields: Foundations by Naber for a quick overview
  • Freed, D.S., and K.K. Uhlenbeck, Instantons and Four-Manifolds, MSRI Publications, Springer-Verlag, New York, Berlin, 1984
  • Lawson, H.B., The Theory of Gauge Fields in Four Dimensions, Regional Conference Series in Mathematics # 58, Amer. Math. Soc., Providence, RI, 1985

Why is it interesting?

The vacuum which we inhabit is filled with such instantons at a density of the order of one instanton per femtometer in every direction. (The precise quantitative theoretical predictions of this [ScSh98] suffer from an infrared regularization ambiguity, but numerical simulations demonstrate the phenomenon [Gru13].) This “instanton sea” that fills spacetime governs the mass of the η′-particle [Wit79, Ven79] as well as other non-perturbative chromodynamical phenomena, such as the quark-gluon plasma seen in experiment [Shu01]. It is also at the heart of the standard hypothesis for the mechanism of primordial baryogenesis [Sak67, ’tHo76, RiTr99], the fundamental explanation of a universe filled with matter.

Of all the solutions, the instantons have interested mathematicians most; for physicists they give a semi-classical understanding of some of the topological effects that are present in Yang-Mills theory.

Topological Investigations of Quantized Gauge Theories, by R. Jackiw (1983)

advanced_notions/quantum_field_theory/instantons.txt · Last modified: 2021/10/01 11:46 by