Non-perturbative - Citizendia

In Mathematics and Physics, a non-perturbative function or process is one that cannot be accurately described by Perturbation theory. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function This article describes perturbation theory as a general mathematical method An example is the function

$f(x)=e^{-1/x^2}$ .

The function e^-1/x². The Taylor series is 0, but the function is not.

The Taylor series for this function is exactly zero to all orders in perturbation theory, but the function is non-zero if x ≠ 0. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

The implication of this for Physics is that there are some phenomena which are impossible to understand by perturbation theory, however many orders of perturbation theory we use. Instantons are an example. An instanton or pseudoparticle is a notion appearing in theoretical and Mathematical physics.

Therefore; in Theoretical Physics, a non-perturbative solution or theory is one that does not require perturbation theory to explicate, or does not simply describe the dynamics of perturbations around some fixed background. Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world For this reason; non-perturbative solutions and theories yield insights into areas and subjects perturbative methods cannot reveal.

See also