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In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a
nonperturbative functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
or process is one that cannot be accurately described by
perturbation theoryPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
. An example is the function
.
The
Taylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
for this function is exactly zero to all orders in perturbation theory, but the function is nonzero if
x ≠ 0.
The implication of this for physics is that there are some phenomena which are impossible to understand by perturbation theory, regardless of how many orders of perturbation theory we use.
InstantonAn instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a selfdual or antiselfdual connection in a principal bundle over a fourdimensional Riemannian manifold that plays the role of physical spacetime in nonabelian gauge theory...
s are an example.
Therefore, in
theoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, a
nonperturbative 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. For this reason, nonperturbative solutions and theories yield insights into areas and subjects perturbative methods cannot reveal.
See also
 Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
 Soliton
In mathematics and physics, a soliton is a selfreinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
 Instanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a selfdual or antiselfdual connection in a principal bundle over a fourdimensional Riemannian manifold that plays the role of physical spacetime in nonabelian gauge theory...
 Loop quantum gravity
Loop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...
 Causal dynamical triangulation