Editing Scale invariance (section)

{{Short description|Features that do not change if length or energy scales are multiplied by a common factor}}
[[File:Wiener process animated.gif|thumb|right|500px|The [[Wiener process]] is scale-invariant]]

In [[physics]], [[mathematics]] and [[statistics]], '''scale invariance''' is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.

The technical term for this [[transformation (mathematics)|transformation]] is a '''dilatation''' (also known as '''dilation'''). Dilatations can form part of a larger [[conformal symmetry]].

*In mathematics, scale invariance usually refers to an invariance of individual [[function (mathematics)|functions]] or [[curve]]s. A closely related concept is [[self-similarity]], where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the [[probability distribution]]s of [[random process]]es to display this kind of scale invariance or self-similarity.
*In [[classical field theory]], scale invariance most commonly applies to the invariance of a whole theory under dilatations. Such theories typically describe classical physical processes with no characteristic length scale.
*In [[quantum field theory]], scale invariance has an interpretation in terms of [[particle physics]]. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.
*In [[statistical mechanics]], scale invariance is a feature of [[phase transition]]s. The key observation is that near a phase transition or [[critical point (thermodynamics)|critical point]], fluctuations occur at all length scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena. Such theories are scale-invariant [[statistical field theory|statistical field theories]], and are formally very similar to scale-invariant quantum field theories.
*[[universality (dynamical systems)|Universality]] is the observation that widely different microscopic systems can display the same behaviour at a phase transition. Thus phase transitions in many different systems may be described by the same underlying scale-invariant theory.
*In general, [[dimensionless quantities]] are scale-invariant. The analogous concept in [[statistics]] are [[standardized moment]]s, which are scale-invariant statistics of a variable, while the unstandardized moments are not.