Non-perturbative

 ( Perturbation Theory ) 

In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function

$f(x)\; =\; e^\{-1/x^2\},$

which does not equal its own Taylor series 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, 't Hooft–Polyakov monopoles, , , and are examples.

A concrete, physical example is given by the Schwinger effect, 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,

$\backslash Gamma\; =\; \backslash frac\{\; (e\; E)^2\; \}\{\; 4\; \backslash pi^3\}\; \backslash mathrm\{e\}^\{-\backslash frac\{\backslash pi\; m^2\}\{eE\}\}$

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 $c=\backslash hbar=1$.

In theoretical physics, 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.

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See also

• Lattice QCD
• Soliton
• Sphaleron
• Instanton
• BCFW recursion
• Operator product expansion
• Conformal bootstrap
• Loop quantum gravity
• Causal dynamical triangulation

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