Scale invariance

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 is a dilatation. Dilatations can form part of a larger conformal symmetry.

In mathematics, scale invariance usually refers to an invariance of individual functions or curves. 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 distributions of random processes 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 transitions. The key observation is that near a phase transition or 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 theories, and are formally very similar to scale-invariant quantum field theories.
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 moments, which are scale-invariant statistics of a variable, while the unstandardized moments are not.
Scale-invariant curves and self-similarity

In mathematics, one can consider the scaling properties of a function or curve under rescalings of the variable. That is, one is interested in the shape of for some scale factor, which can be taken to be a length or size rescaling. The requirement for to be invariant under all rescalings is usually taken to be
for some choice of exponent Δ, and for all dilations. This is equivalent to being a homogeneous function of degree Δ.
Examples of scale-invariant functions are the monomials, for which, in that clearly
An example of a scale-invariant curve is the logarithmic spiral, a kind of curve that often appears in nature. In polar coordinates, the spiral can be written as
Allowing for rotations of the curve, it is invariant under all rescalings ; that is, is identical to a rotated version.

Projective geometry

The idea of scale invariance of a monomial generalizes in higher dimensions to the idea of a homogeneous polynomial, and more generally to a homogeneous function. Homogeneous functions are the natural denizens of projective space, and homogeneous polynomials are studied as projective varieties in projective geometry. Projective geometry is a particularly rich field of mathematics; in its most abstract forms, the geometry of schemes, it has connections to various topics in string theory.

Fractals

It is sometimes said that fractals are scale-invariant, although more precisely, one should say that they are self-similar. A fractal is equal to itself typically for only a discrete set of values, and even then a translation and rotation may have to be applied to match the fractal up to itself.
Thus, for example, the Koch curve scales with, but the scaling holds only for values of for integer. In addition, the Koch curve scales not only at the origin, but, in a certain sense, "everywhere": miniature copies of itself can be found all along the curve.
Some fractals may have multiple scaling factors at play at once; such scaling is studied with multi-fractal analysis.
Periodic external and internal rays are invariant curves.

Scale invariance in stochastic processes

If is the average power at frequency, then noise scales as
with Δ = 0 for white noise, Δ = −1 for pink noise, and Δ = −2 for Brownian noise.
More precisely, scaling in stochastic systems concerns itself with the likelihood of choosing a particular configuration out of the set of all possible random configurations. This likelihood is given by the probability distribution.
Examples of scale-invariant distributions are the Pareto distribution and the Zipfian distribution.

Scale-invariant Tweedie distributions

Tweedie distributions are a special case of exponential dispersion models, a class of statistical models used to describe error distributions for the generalized linear model and characterized by closure under additive and reproductive convolution as well as under scale transformation. These include a number of common distributions: the normal distribution, Poisson distribution and gamma distribution, as well as more unusual distributions like the compound Poisson-gamma distribution, positive stable distributions, and extreme stable distributions.
Consequent to their inherent scale invariance Tweedie random variables Y demonstrate a variance var to mean E power law:
where a and p are positive constants. This variance to mean power law is known in the physics literature as fluctuation scaling, and in the ecology literature as Taylor's law.
Random sequences, governed by the Tweedie distributions and evaluated by the method of expanding bins exhibit a biconditional relationship between the variance to mean power law and power law autocorrelations. The Wiener–Khinchin theorem further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest 1/f noise.
The Tweedie convergence theorem provides a hypothetical explanation for the wide manifestation of fluctuation scaling and 1/f noise. It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a variance function that comes within the domain of attraction of a Tweedie model. Almost all distribution functions with finite cumulant generating functions qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form. Hence many probability distributions have variance functions that express this asymptotic behavior, and the Tweedie distributions become foci of convergence for a wide range of data types.
Much as the central limit theorem requires certain kinds of random variables to have as a focus of convergence the Gaussian distribution and express white noise, the Tweedie convergence theorem requires certain non-Gaussian random variables to express 1/f noise and fluctuation scaling.

Cosmology

In physical cosmology, the power spectrum of the spatial distribution of the cosmic microwave background is near to being a scale-invariant function. Although in mathematics this means that the spectrum is a power-law, in cosmology the term "scale-invariant" indicates that the amplitude,, of primordial fluctuations as a function of wave number,, is approximately constant, i.e. a flat spectrum. This pattern is consistent with the proposal of cosmic inflation.

Scale invariance in classical field theory

is generically described by a field, or set of fields, φ, that depend on coordinates, x. Valid field configurations are then determined by solving differential equations for φ, and these equations are known as field equations.
For a theory to be scale-invariant, its field equations should be invariant under a rescaling of the coordinates, combined with some specified rescaling of the fields,
The parameter Δ is known as the scaling dimension of the field, and its value depends on the theory under consideration. Scale invariance will typically hold provided that no fixed length scale appears in the theory. Conversely, the presence of a fixed length scale indicates that a theory is not scale-invariant.
A consequence of scale invariance is that given a solution of a scale-invariant field equation, we can automatically find other solutions by rescaling both the coordinates and the fields appropriately. In technical terms, given a solution, φ, one always has other solutions of the form

Scale invariance of field configurations

For a particular field configuration, φ, to be scale-invariant, we require that
where Δ is, again, the scaling dimension of the field.
We note that this condition is rather restrictive. In general, solutions even of scale-invariant field equations will not be scale-invariant, and in such cases the symmetry is said to be spontaneously broken.

Classical electromagnetism

An example of a scale-invariant classical field theory is electromagnetism with no charges or currents. The fields are the electric and magnetic fields, E and B, while their field equations are Maxwell's equations.
With no charges or currents, these field equations take the form of wave equations
where c is the speed of light.
These field equations are invariant under the transformation
Moreover, given solutions of Maxwell's equations, E and B, it holds that
E and B are also solutions.

Massless scalar field theory

Another example of a scale-invariant classical field theory is the massless scalar field. The scalar field, is a function of a set of spatial variables, x, and a time variable,.
Consider first the linear theory. Like the electromagnetic field equations above, the equation of motion for this theory is also a wave equation,
and is invariant under the transformation
The name massless refers to the absence of a term in the field equation. Such a term is often referred to as a `mass' term, and would break the invariance under the above transformation. In relativistic field theories, a mass-scale, is physically equivalent to a fixed length scale through
and so it should not be surprising that massive scalar field theory is not scale-invariant.