Encyclopedia of Nonlinear Science
A remarkable feature of the Yang-Mills action is that there are finite-action topological soliton solutions to the classical field equations. These solitons are known as instantons or, in early papers, as pseudoparticles. The Yang-Mills action is important in particle physics; in particular, it describes the behavior of gluons, the particles that carry the strong nuclear force. Before instanton solutions were discovered in 1975 by Belavin et al. (1975), the Yang-Mills theory of the strong force appeared to have a symmetry not found in nature. This was known as the axial U(1) problem and was solved by ’t Hooft who realized that one effect of the instanton solutions was to break this unwanted symmetry. This was the first example of an extended classical solution having a physical consequence in a field theory of particle physics.
In four Euclidean dimensions, the pure SU(2) Yang-Mills action is
where µ and ν are space indices running from 1 to 4 and repeated indices are summed. Fμν is a field strength tensor, a skew Hermitian 2×2 matrix field in four-dimensional space, related to a gauge potential Aμ by
The action is invariant under local gauge transformations
where g is a special unitary matrix function in four dimensions. Under this transformation 
One key consequence of this is that Fμν not only vanishes when Aμ=0, but also, whenever Aμ is a gauge transformation of zero. This means that the asymptotic condition required for finiteness of the action is that Aμ approaches a gauge transformation of zero:
for 
Now, the group of special unitary 2×2 matrices is topologically a three-sphere, and so the gauge potential on the large three-sphere at infinity gives a map from that three-sphere to the three-sphere of values of g. Maps between three-sphere are classified topologically by a single integer: the winding number. This integer, therefore, classifies finite-action field configurations; in fact, it is the only gauge-invariant information determined by the asymptotic field behavior. The topologically nontrivial stationary points of the action S[A] are instantons.
Since they are stationary points of the action, instantons obey the Euler-Lagrange equation for the action:
This is known as the Yang-Mills equation. It can be shown that minimal action solutions obey a first-order equation called the self-dual equation:
where εµνλσ is totally skewsymmetric in its indices with ε1234=1. Solutions to this equation obey the second-order Yang-Mills equation. A solution with winding number one is given by
where R is an arbitrary scale and g has the hedgehog form
where I2 is the 2×2 identity matrix and the σi’s are the Pauli matrices. Allowing for gauge equivalence, there is an eight-dimensional space of winding number one instantons, four of these dimensions correspond to the choice of position in space, one to the choice of scale, and three to an overall group orientation. Although these results are particular to SU(2), other groups can be studied; the main difference is in the number of overall group orientation parameters.
The self-dual equations are integrable. In fact, many of the integrable equations of mathematical physics can be derived from the self-dual equations by demanding that the solution has some translational or rotational symmetry. Because of this integrability, solutions to the self-dual equations with winding numbers greater than one may be constructed, in principle, by twistor theory and by algebraic methods (Christ et al., 1978). There is also a simple ansatz (Jackiw et al., 1977), but this does not give a general solution.
In the 1980s, the self-dual equations revolutionized the study of smooth four-dimensional manifolds: by studying the space of solutions to the self-dual equations over different manifolds, it is possible to derive invariants, known as the Donaldson invariants (Donaldson & Kronheimer, 1990). This approach has since been largely superceded by Seiberg-Witten theory.
In the path integral approach to quantum field theory, physical values are derived by certain weighted integrations over all possible field configurations. These path integrals are often intractable, and it is common to expand the integration about the stationary points of the action. This works because of the way the action appears in the integrand of the path integral. Calculations of this type are known as semi-classical calculations because they are effectively an expansion in Planck’s constant, ћ. This expansion must include a sum over all possible stationary points, and so, in Yang-Mills theory, it includes a sum over instanton configurations. By studying the symmetry properties of the measure in quantum chromodynamics (QCD), the Yang-Mills theory describing the strong nuclear force, it can be shown that terms in this expansion break the axial U(1) symmetry. This symmetry is unbroken if the instanton terms are omitted. The symmetry breaking allows processes that violate baryon and lepton conservation; however, the amplitudes for these effects are highly suppressed. One useful approach is to consider these processes as tunneling events between different vacua, in fact, instanton calculations in quantum field theory are very similar to Wentzel-Kramers-Brillouin (WKB) calculations in quantum mechanics.
While instantons provide a qualitative explanation for a host of phenomena in quantum chromodynamics, useful quantitative results are not available within the semi-classical approach, and even qualitatively, instanton calculations are not rigorous since they relate only to the semi-classical approximation, a truncation of the full quantum theory. The modern approach to quantum chromodynamics is lattice QCD. It is possible within lattice QCD to verify the original ideas about the role of instantons in the physics of the strong force; however, there are limits to the precision with which the lattice and semi-classical approaches can be compared (Negele, 1998).
Finite action soliton solutions in other equation systems are sometimes referred to as instantons. Examples include the finite action solutions to the forced Burgers equation, which arises in the study of turbulence, and the vortex-like solutions in the abelian Higgs model, which is related to condensed matter physics. Instantons have many similarities to the lump solitons found in certain sigma models.
CONOR HOUGHTON
See also Burgers equation; Higgs boson; Particles and antiparticles; Quantum field theory; Solitons; Yang-Mills theory
Further Reading
Belavin, A.A., Polyakov, A.M., Schwarz, A.S. & Tyupkin, Yu.S. 1975. Pseudoparticle solutions of the Yang-Mills equations. Physics Letters B, 59:85–87
Christ, N.H., Weinberg, E.J. & Stanton, N.K. 1978. General self-dual Yang-Mills solutions. Physical Review D, 18:2013–2025 (Includes a review of the complete solution to the self-dual equations originally due to Atiyah, Hitchin, Drinfeld, and Manin)
Coleman, S. 1985. Aspects of Symmetries, Cambridge and New York: Cambridge University Press (Chapter 7 gives a celebrated description of instantons in QCD)
Donaldson, S.K. & Kronheimer, P.B. 1990. The Geometry of Four-manifolds, Oxford: Clarendon Press and New York: Oxford University Press
Jackiw, R., Nohl, C. & Rebbi, C. 1977. Conformal properties of pseudoparticle configurations. Physical Review D, 15:1642–1646 (A large, but incomplete, family of solutions to the self-dual equations.)
Negele, J.W. 1998. Instantons, the QCD vacuum, and hadronic physics. Nuclear Physics B Proceedings Supplements, 73: 92–104
’t Hooft, G. 1976. Symmetry breaking through Bell-Jackiw anomalies. Physical Review Letters, 37:8–11
Weinberg, S. 1996. Quantum Field Theory, vol. 2, Cambridge and New York: Cambridge University Press (Chapter 23 concerns extended field configurations in particle physics and has a very clear treatment of instantons.)
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