Instanton

The dx¹⊗σ₃ coefficient of a BPST instanton on the (x¹,x²)-slice of R⁴ where σ₃ is the third Pauli matrix (top left). The dx²⊗σ₃ coefficient (top right). These coefficients determine the restriction of the BPST instanton A with g=2,ρ=1,z=0 to this slice. The corresponding field strength centered around z=0 (bottom left). A visual representation of the field strength of a BPST instanton with center z on the compactification S⁴ of R⁴ (bottom right). The BPST instanton is a classical instanton solution to the Yang–Mills equations on R⁴.

An instanton (or pseudoparticle^[1][2][3]) 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.^[4]

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 (and noise-induced anti-instantons) is believed to be the explanation of the noise-induced chaotic phase known as self-organized criticality.