mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, a non-perturbative function or process is one that cannot be described by

perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...

. An example is the function :

f(x) = e^,

which does not equal its own

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

in any neighborhood around ''x'' = 0. Every coefficient of the Taylor expansion around ''x'' = 0 is exactly zero, but the function is non-zero if ''x'' ≠ 0. In physics, such functions arise for phenomena which are impossible to understand by perturbation theory, at any finite order. In

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

, 't Hooft–Polyakov monopoles, domain walls, flux tubes, and

instanton An instanton (or pseudoparticle) 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. M ...

s are examples. A concrete, physical example is given by the

Schwinger effect The Schwinger effect is a predicted physical phenomenon whereby matter is created by a strong electric field. It is also referred to as the Sauter–Schwinger effect, Schwinger mechanism, or Schwinger pair production. It is a prediction of quantum ...

, whereby a strong electric field may spontaneously decay into electron-positron pairs. For not too strong fields, the rate per unit volume of this process is given by, :

\Gamma = \frac \mathrm^

which cannot be expanded in a Taylor series in the electric charge

e

, or the electric field strength

E

. Here

m

is the mass of an electron and we have used units where

c=\hbar=1

. In

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

, a non-perturbative solution is one that cannot be described in terms of perturbations about some simple background, such as empty space. For this reason, non-perturbative solutions and theories yield insights into areas and subjects that perturbative methods cannot reveal.

References

Perturbation theory {{quantum-stub

See also

References