Inphysics,mathematics andstatistics,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 thistransformation is adilatation (also known asdilation). Dilatations can form part of a largerconformal symmetry.

• In mathematics, scale invariance usually refers to an invariance of individualfunctions orcurves. A closely related concept isself-similarity, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for theprobability distributions ofrandom processes to display this kind of scale invariance or self-similarity.
• Inclassical 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.
• Inquantum field theory, scale invariance has an interpretation in terms ofparticle physics. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.
• Instatistical mechanics, scale invariance is a feature ofphase transitions. The key observation is that near a phase transition orcritical 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-invariantstatistical 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 instatistics arestandardized moments, which are scale-invariant statistics of a variable, while the unstandardized moments are not.

In mathematics, one can consider the scaling properties of afunction orcurvef (x) under rescalings of the variablex. That is, one is interested in the shape off (λx) for some scale factorλ, which can be taken to be a length or size rescaling. The requirement forf (x) to be invariant under all rescalings is usually taken to be

${\displaystyle f(\lambda x)={\lambda }^{\mathrm{\Delta }}f(x)}$

for some choice of exponent Δ, and for all dilationsλ. This is equivalent tof   being ahomogeneous function of degree Δ.

Examples of scale-invariant functions are themonomials${\displaystyle f(x)=x^n}$, for whichΔ =n, in that clearly

${\displaystyle f(\lambda x)=(\lambda x{)}^n={\lambda }^{n}f(x).}$

An example of a scale-invariant curve is thelogarithmic spiral, a kind of curve that often appears in nature. Inpolar coordinates(r,θ), the spiral can be written as

${\displaystyle \theta =\frac{1}{b}\ln (r/a).}$

Allowing for rotations of the curve, it is invariant under all rescalingsλ; that is,θ(λr) is identical to a rotated version${\displaystyle \theta (r)+\frac{1}{b}\ln \lambda }$.

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

It is sometimes said thatfractals are scale-invariant, although more precisely, one should say that they areself-similar. A fractal is equal to itself typically for only a discrete set of valuesλ, and even then a translation and rotation may have to be applied to match the fractal up to itself.

Thus, for example, theKoch curve scales with∆ = 1, but the scaling holds only for values ofλ = 1/3ⁿ for integern. 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 withmulti-fractal analysis.

Periodicexternal and internal rays are invariant curves .

IfP(f ) is theaverage (expected) power at frequencyf, then noise scales as

${\displaystyle P(f)={\lambda }^{\mathrm{\Delta }}P(\lambda f)}$

with Δ = 0 forwhite noise, Δ = −1 forpink noise, and Δ = −2 forBrownian noise (and more generally,Brownian motion).

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 theprobability distribution.

Examples of scale-invariant distributions are thePareto distribution and theZipfian distribution.

Tweedie distributions are a special case ofexponential dispersion models, a class of statistical models used to describe error distributions for thegeneralized linear model and characterized byclosure under additive and reproductive convolution as well as under scale transformation.^[1] These include a number of common distributions: thenormal distribution,Poisson distribution andgamma distribution, as well as more unusual distributions like the compound Poisson-gamma distribution, positivestable distributions, and extreme stable distributions.Consequent to their inherent scale invariance Tweedierandom variablesY demonstrate avariance var(Y) tomean E(Y) power law:

${\displaystyle \text{var}(Y)=a[\text{E}(Y){]}^p}$,

wherea andp are positive constants. This variance to mean power law is known in the physics literature asfluctuation scaling,^[2] and in the ecology literature asTaylor's law.^[3]

Random sequences, governed by the Tweedie distributions and evaluated by themethod of expanding bins exhibit abiconditional relationship between the variance to mean power law and power lawautocorrelations. TheWiener–Khinchin theorem further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest1/f noise.^[4]

TheTweedie convergence theorem provides a hypothetical explanation for the wide manifestation of fluctuation scaling and1/f noise.^[5] It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express avariance function that comes within thedomain of attraction of a Tweedie model. Almost all distribution functions with finitecumulant 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 thisasymptotic behavior, and the Tweedie distributions become foci of convergence for a wide range of data types.^[4]

Much as thecentral limit theorem requires certain kinds of random variables to have as a focus of convergence theGaussian distribution and expresswhite noise, the Tweedie convergence theorem requires certain non-Gaussian random variables to express1/f noise and fluctuation scaling.^[4]

Inphysical cosmology, the power spectrum of the spatial distribution of thecosmic 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,P(k), ofprimordial fluctuations as a function ofwave number,k, is approximately constant, i.e. a flat spectrum. This pattern is consistent with the proposal ofcosmic inflation.

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 solvingdifferential equations forφ, and these equations are known asfield 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,

${\displaystyle x\to \lambda x,}$

${\displaystyle \phi \to {\lambda }^{-\mathrm{\Delta }}\phi .}$

The parameter Δ is known as thescaling 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 isnot 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(x), one always has other solutions of the form

${\displaystyle {\lambda }^{\mathrm{\Delta }}\phi (\lambda x).}$

For a particular field configuration,φ(x), to be scale-invariant, we require that

${\displaystyle \phi (x)={\lambda }^{-\mathrm{\Delta }}\phi (\lambda x)}$

where Δ is, again, thescaling dimension of the field.

We note that this condition is rather restrictive. In general, solutions even of scale-invariant field equations willnot be scale-invariant, and in such cases the symmetry is said to bespontaneously broken.

An example of a scale-invariant classical field theory iselectromagnetism with no charges or currents. The fields are the electric and magnetic fields,E(x,t) andB(x,t), while their field equations areMaxwell's equations.

With no charges or currents,these field equations take the form ofwave equations

${\displaystyle \begin{array}{r}{\mathrm{\nabla }}^2\mathbf{E}=\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{E}}{\mathrm{\partial }{t}^2}\\ {\mathrm{\nabla }}^2\mathbf{B}=\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{B}}{\mathrm{\partial }{t}^2}\end{array}}$

wherec is the speed of light.

These field equations are invariant under the transformation

${\displaystyle \begin{array}{r}x\to \lambda x,\\ t\to \lambda t.\end{array}}$

Moreover, given solutions of Maxwell's equations,E(x,t) andB(x,t), it holds thatE(λx,λt) andB(λx,λt) are also solutions.

Another example of a scale-invariant classical field theory is the masslessscalar field (note that the namescalar is unrelated to scale invariance). The scalar field,φ(x,t) is a function of a set of spatial variables,x, and a time variable,t.

Consider first the linear theory. Like the electromagnetic field equations above, the equation of motion for this theory is also a wave equation,

${\displaystyle \frac{1}{c^2}\frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\phi =0,}$

and is invariant under the transformation

${\displaystyle x\to \lambda x,}$

${\displaystyle t\to \lambda t.}$

The name massless refers to the absence of a term${\displaystyle \propto m^2\phi }$ in the field equation. Such a term is often referred to as a `mass' term, and would break the invariance under the above transformation. Inrelativistic field theories, a mass-scale,m is physically equivalent to a fixed length scale through

${\displaystyle L=\frac{\hslash }{mc},}$

and so it should not be surprising that massive scalar field theory isnot scale-invariant.

The field equations in the examples above are alllinear in the fields, which has meant that thescaling dimension, Δ, has not been so important. However, one usually requires that the scalar fieldaction is dimensionless, and this fixes thescaling dimension ofφ. In particular,

${\displaystyle \mathrm{\Delta }=\frac{D-2}{2},}$

whereD is the combined number of spatial and time dimensions.

Given this scaling dimension forφ, there are certain nonlinear modifications of massless scalar field theory which are also scale-invariant. One example is masslessφ⁴ theory forD = 4. The field equation is

${\displaystyle \frac{1}{c^2}\frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\phi +g{\phi }^3=0.}$

(Note that the nameφ⁴ derives from the form of theLagrangian, which contains the fourth power ofφ.)

WhenD = 4 (e.g. three spatial dimensions and one time dimension), the scalar field scaling dimension is Δ = 1. The field equation is then invariant under the transformation

${\displaystyle x\to \lambda x,}$

${\displaystyle t\to \lambda t,}$

${\displaystyle \phi (x)\to {\lambda }^{-1}\phi (x).}$

The key point is that the parameterg must be dimensionless, otherwise one introduces a fixed length scale into the theory: Forφ⁴ theory, this is only the case inD = 4.Note that under these transformations the argument of the functionφ is unchanged.

The scale-dependence of aquantum field theory (QFT) is characterised by the way itscoupling parameters depend on the energy-scale of a given physical process. This energy dependence is described by therenormalization group, and is encoded in thebeta-functions of the theory.

For a QFT to be scale-invariant, its coupling parameters must be independent of the energy-scale, and this is indicated by the vanishing of the beta-functions of the theory. Such theories are also known asfixed points of the corresponding renormalization group flow.^[6]

A simple example of a scale-invariant QFT is the quantized electromagnetic field without charged particles. This theory actually has no coupling parameters (sincephotons are massless and non-interacting) and is therefore scale-invariant, much like the classical theory.

However, in nature the electromagnetic field is coupled to charged particles, such aselectrons. The QFT describing the interactions of photons and charged particles isquantum electrodynamics (QED), and this theory is not scale-invariant. We can see this from theQED beta-function. This tells us that theelectric charge (which is the coupling parameter in the theory) increases with increasing energy. Therefore, while the quantized electromagnetic field without charged particlesis scale-invariant, QED isnot scale-invariant.

Free, masslessquantized scalar field theory has no coupling parameters. Therefore, like the classical version, it is scale-invariant. In the language of the renormalization group, this theory is known as theGaussian fixed point.

However, even though the classical masslessφ⁴ theory is scale-invariant inD = 4, the quantized version isnot scale-invariant. We can see this from thebeta-function for the coupling parameter,g.

Even though the quantized masslessφ⁴ is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the Gaussian fixed point. One example is theWilson–Fisher fixed point, below.

Scale-invariant QFTs are almost always invariant under the fullconformal symmetry, and the study of such QFTs isconformal field theory (CFT).Operators in a CFT have a well-definedscaling dimension, analogous to thescaling dimension,∆, of a classical field discussed above. However, the scaling dimensions of operators in a CFT typically differ from those of the fields in the corresponding classical theory. The additional contributions appearing in the CFT are known asanomalous scaling dimensions.

The φ⁴ theory example above demonstrates that the coupling parameters of a quantum field theory can be scale-dependent even if the corresponding classical field theory is scale-invariant (or conformally invariant). If this is the case, the classical scale (or conformal) invariance is said to beanomalous. A classically scale-invariant field theory, where scale invariance is broken by quantum effects, provides an explication of the nearly exponential expansion of the early universe calledcosmic inflation, as long as the theory can be studied throughperturbation theory.^[7]

Instatistical mechanics, as a system undergoes aphase transition, its fluctuations are described by a scale-invariantstatistical field theory. For a system in equilibrium (i.e. time-independent) inD spatial dimensions, the corresponding statistical field theory is formally similar to aD-dimensional CFT. The scaling dimensions in such problems are usually referred to ascritical exponents, and one can in principle compute these exponents in the appropriate CFT.

An example that links together many of the ideas in this article is the phase transition of theIsing model, a simple model offerromagnetic substances. This is a statistical mechanics model, which also has a description in terms of conformal field theory. The system consists of an array of lattice sites, which form aD-dimensional periodic lattice. Associated with each lattice site is amagnetic moment, orspin, and this spin can take either the value +1 or −1. (These states are also called up and down, respectively.)

The key point is that the Ising model has a spin-spin interaction, making it energetically favourable for two adjacent spins to be aligned. On the other hand, thermal fluctuations typically introduce a randomness into the alignment of spins. At some critical temperature,T_c ,spontaneous magnetization is said to occur. This means that belowT_c the spin-spin interaction will begin to dominate, and there is some net alignment of spins in one of the two directions.

An example of the kind of physical quantities one would like to calculate at this critical temperature is the correlation between spins separated by a distancer. This has the generic behaviour:

${\displaystyle G(r)\propto \frac{1}{r^{D-2+\eta }},}$

for some particular value of${\displaystyle \eta }$, which is an example of a critical exponent.

The fluctuations at temperatureT_c are scale-invariant, and so the Ising model at this phase transition is expected to be described by a scale-invariant statistical field theory. In fact, this theory is theWilson–Fisher fixed point, a particular scale-invariantscalar field theory.

In this context,G(r) is understood as acorrelation function of scalar fields,

${\displaystyle ⟨\varphi (0)\varphi (r)⟩\propto \frac{1}{r^{D-2+\eta }}.}$

Now we can fit together a number of the ideas seen already.

From the above, one sees that the critical exponent,η, for this phase transition, is also ananomalous dimension. This is because the classical dimension of the scalar field,

${\displaystyle \mathrm{\Delta }=\frac{D-2}{2}}$

is modified to become

${\displaystyle \mathrm{\Delta }=\frac{D-2+\eta }{2},}$

whereD is the number of dimensions of the Ising model lattice.

So thisanomalous dimension in the conformal field theory is thesame as a particular critical exponent of the Ising model phase transition.

Note that for dimensionD ≡ 4−ε,η can be calculated approximately, using theepsilon expansion, and one finds that

${\displaystyle \eta =\frac{{ϵ}^2}{54}+O({ϵ}^3)}$.

In the physically interesting case of three spatial dimensions, we haveε=1, and so this expansion is not strictly reliable. However, a semi-quantitative prediction is thatη is numerically small in three dimensions.

On the other hand, in the two-dimensional case the Ising model is exactly soluble. In particular, it is equivalent to one of theminimal models, a family of well-understood CFTs, and it is possible to computeη (and the other critical exponents) exactly,

${\displaystyle {\eta }_{{}_{D=2}}=\frac{1}{4}}$.

The anomalous dimensions in certain two-dimensional CFTs can be related to the typicalfractal dimensions of random walks, where the random walks are defined viaSchramm–Loewner evolution (SLE). As we have seen above, CFTs describe the physics of phase transitions, and so one can relate the critical exponents of certain phase transitions to these fractal dimensions. Examples include the 2d critical Ising model and the more general 2d criticalPotts model. Relating other 2d CFTs to SLE is an active area of research.

A phenomenon known asuniversality is seen in a large variety of physical systems. It expresses the idea that different microscopic physics can give rise to the same scaling behaviour at a phase transition. A canonical example of universality involves the following two systems:

• TheIsing model phase transition, described above.
• Theliquid-vapour transition in classical fluids.

Even though the microscopic physics of these two systems is completely different, their critical exponents turn out to be the same. Moreover, one can calculate these exponents using the same statistical field theory. The key observation is that at a phase transition orcritical point, fluctuations occur at all length scales, and thus one should look for a scale-invariant statistical field theory to describe the phenomena. In a sense, universality is the observation that there are relatively few such scale-invariant theories.

The set of different microscopic theories described by the same scale-invariant theory is known as auniversality class. Other examples of systems which belong to a universality class are:

• Avalanches in piles of sand. The likelihood of an avalanche is in power-law proportion to the size of the avalanche, and avalanches are seen to occur at all size scales.
• The frequency ofnetwork outages on theInternet, as a function of size and duration.
• The frequency of citations of journal articles, considered in the network of all citations amongst all papers, as a function of the number of citations in a given paper.^{[citation needed]}
• The formation and propagation of cracks and tears in materials ranging from steel to rock to paper. The variations of the direction of the tear, or the roughness of a fractured surface, are in power-law proportion to the size scale.
• Theelectrical breakdown ofdielectrics, which resemble cracks and tears.
• Thepercolation of fluids through disordered media, such aspetroleum through fractured rock beds, or water through filter paper, such as inchromatography. Power-law scaling connects the rate of flow to the distribution of fractures.
• Thediffusion ofmolecules insolution, and the phenomenon ofdiffusion-limited aggregation.
• The distribution of rocks of different sizes in an aggregate mixture that is being shaken (with gravity acting on the rocks).

The key observation is that, for all of these different systems, the behaviour resembles aphase transition, and that the language of statistical mechanics and scale-invariantstatistical field theory may be applied to describe them.

Under certain circumstances,fluid mechanics is a scale-invariant classical field theory. The fields are the velocity of the fluid flow,${\displaystyle \mathbf{u}(\mathbf{x},t)}$, the fluid density,${\displaystyle \rho (\mathbf{x},t)}$, and the fluid pressure,${\displaystyle P(\mathbf{x},t)}$. These fields must satisfy both theNavier–Stokes equation and thecontinuity equation. For aNewtonian fluid these take the respective forms

$${\displaystyle \rho \frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+\rho \mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}=-\mathrm{\nabla }P+\mu \left({\mathrm{\nabla }}^2\mathbf{u}+\frac{1}{3}\mathrm{\nabla }\left(\mathrm{\nabla }\cdot \mathbf{u}\right)\right)}$$

${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \left(\rho \mathbf{u}\right)=0}$

where${\displaystyle \mu }$ is thedynamic viscosity.

In order to deduce the scale invariance of these equations we specify anequation of state, relating the fluid pressure to the fluid density. The equation of state depends on the type of fluid and the conditions to which it is subjected. For example, we consider theisothermalideal gas, which satisfies

${\displaystyle P=c_s^2\rho ,}$

where${\displaystyle c_s}$ is the speed of sound in the fluid. Given this equation of state, Navier–Stokes and the continuity equation are invariant under the transformations

${\displaystyle x\to \lambda x,}$

${\displaystyle t\to {\lambda }^{2}t,}$

${\displaystyle \rho \to {\lambda }^{-1}\rho ,}$

${\displaystyle \mathbf{u}\to {\lambda }^{-1}\mathbf{u}.}$

Given the solutions${\displaystyle \mathbf{u}(\mathbf{x},t)}$ and${\displaystyle \rho (\mathbf{x},t)}$, we automatically have that${\displaystyle \lambda \mathbf{u}(\lambda \mathbf{x},{\lambda }^{2}t)}$ and${\displaystyle \lambda \rho (\lambda \mathbf{x},{\lambda }^{2}t)}$ are also solutions.

Incomputer vision andbiological vision, scaling transformations arise because of the perspective image mapping and because of objects having different physical size in the world. In these areas, scale invariance refers to local image descriptors or visual representations of the image data that remain invariant when the local scale in the image domain is changed.^[8] Detecting local maxima over scales of normalized derivative responses provides a general framework for obtaining scale invariance from image data.^[9][10]Examples of applications includeblob detection,corner detection,ridge detection, and object recognition via thescale-invariant feature transform.

• Invariant (mathematics)
• Inverse square potential
• Power law
• Scale-free network

• Zinn-Justin, Jean (2002).Quantum Field Theory and Critical Phenomena. Oxford University Press. Extensive discussion of scale invariance in quantum and statistical field theories, applications to critical phenomena and the epsilon expansion and related topics.
• DiFrancesco, P.; Mathieu, P.; Senechal, D. (1997).Conformal Field Theory. Springer-Verlag.
• Mussardo, G. (2010).Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics. Oxford University Press.

• Symmetry
• Scaling symmetries
• Conformal field theory
• Critical phenomena
• Scale-invariant systems

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