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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, 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 Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Trans ...
is a dilatation (also known as dilation), and the dilatations can also form part of a larger
conformal symmetry In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry ...
. *In mathematics, scale invariance usually refers to an invariance of individual functions or
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
s. A closely related concept is
self-similarity __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s of
random process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appe ...
es to display this kind of scale invariance or self-similarity. *In
classical field theory A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
, 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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, scale invariance has an interpretation in terms of
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved. *In
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, scale invariance is a feature of
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
s. 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 A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
are scale invariant. The analogous concept in
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
are
standardized moment In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant ...
s, 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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
or
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
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 :f(\lambda x)=\lambda^f(x) for some choice of exponent , and for all dilations . This is equivalent to   being a
homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
of degree . Examples of scale-invariant functions are the
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer exponent ...
s f(x)=x^n, for which , in that clearly :f(\lambda x) = (\lambda x)^n = \lambda^n f(x)~. An example of a scale-invariant curve is the
logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). Mor ...
, a kind of curve that often appears in nature. In
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
, the spiral can be written as :\theta = \frac \ln(r/a)~. Allowing for rotations of the curve, it is invariant under all rescalings ; that is, is identical to a rotated version of .


Projective geometry

The idea of scale invariance of a monomial generalizes in higher dimensions to the idea of a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
, and more generally to a
homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
. Homogeneous functions are the natural denizens of
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
, and homogeneous polynomials are studied as
projective varieties In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
in
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
. 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 In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
.


Fractals

It is sometimes said that
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
s 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 The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...
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, expected power at frequency , then noise scales as :P(f) = \lambda^ P(\lambda f) with = 0 for
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
, = −1 for
pink noise Pink noise or noise is a signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequency of the signal. In pink noise, each octave interval (halving ...
, and = −2 for
Brownian noise ] In science, Brownian noise, also known as Brown noise or red noise, is the type of signal noise produced by Brownian motion, hence its alternative name of random walk noise. The term "Brown noise" does not come from the color, but after ...
(and more generally,
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
). 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 In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
. Examples of scale-invariant distributions are the
Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actua ...
and the Zipfian distribution.


Scale invariant Tweedie distributions

Tweedie distributions In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the cla ...
are a special case of
exponential dispersion model In probability and statistics, the class of exponential dispersion models (EDM) is a set of probability distributions that represents a generalisation of the natural exponential family.Jørgensen, B. (1987). Exponential dispersion models (with dis ...
s, a class of statistical models used to describe error distributions for the
generalized linear model In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a ''link function'' and b ...
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 In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
,
Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
and
gamma distribution In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...
, as well as more unusual distributions like the compound Poisson-gamma distribution, positive
stable distribution In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stab ...
s, and extreme stable distributions. Consequent to their inherent scale invariance Tweedie
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s ''Y'' demonstrate a
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
var(''Y'') to
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithme ...
E(''Y'') power law: : \text\,(Y) = a text\,(Y)p, 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 Taylor's power law is an empirical law in ecology that relates the variance of the number of individuals of a species per unit area of habitat to the corresponding mean by a power law relationship. It is named after the ecologist who first propos ...
. Random sequences, governed by the Tweedie distributions and evaluated by the method of expanding bins exhibit a
biconditional In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective (\leftrightarrow) used to conjoin two statements and to form the statement " if and only if ", where is known as t ...
relationship between the variance to mean power law and power law
autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
s. The
Wiener–Khinchin theorem In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary ...
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 In statistics, the variance function is a smooth function which depicts the variance of a random quantity as a function of its mean. The variance function is a measure of heteroscedasticity and plays a large role in many settings of statisti ...
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 In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as beco ...
, and the Tweedie distributions become foci of convergence for a wide range of data types. Much as the
central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...
requires certain kinds of random variables to have as a focus of convergence the Gaussian distribution and express
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
, the Tweedie convergence theorem requires certain non-Gaussian random variables to express ''1/f'' noise and fluctuation scaling.


Cosmology

In
physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
, the power spectrum of the spatial distribution of the
cosmic microwave background In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
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 Primordial fluctuations are density variations in the early universe which are considered the seeds of all structure in the universe. Currently, the most widely accepted explanation for their origin is in the context of cosmic inflation. According ...
as a function of
wave number In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
, , is approximately constant, i.e. a flat spectrum. This pattern is consistent with the proposal of
cosmic inflation In physical cosmology, cosmic inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the early universe. The inflationary epoch lasted from  seconds after the conjectured Big Bang singularity ...
.


Scale invariance in classical field theory

Classical field theory A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
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 In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
for ''φ'', and these equations are known as
field equation In theoretical physics and applied mathematics, a field equation is a partial differential equation which determines the dynamics of a physical field, specifically the time evolution and spatial distribution of the field. The solutions to the equ ...
s. 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, :x\rightarrow\lambda x~, :\varphi\rightarrow\lambda^\varphi~. The parameter ''Δ'' is known as the
scaling dimension In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
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''(''x''), one always has other solutions of the form :\lambda^\varphi(\lambda x).


Scale invariance of field configurations

For a particular field configuration, ''φ''(''x''), to be scale-invariant, we require that :\varphi(x)=\lambda^\varphi(\lambda x) where ''Δ'' is, again, the
scaling dimension In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
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 In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
with no charges or currents. The fields are the electric and magnetic fields, E(x,''t'') and B(x,''t''), while their field equations are
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
. With no charges or currents, these field equations take the form of
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
s :\nabla^2 \mathbf = \frac \frac :\nabla^2\mathbf = \frac \frac where ''c'' is the speed of light. These field equations are invariant under the transformation :x\rightarrow\lambda x, :t\rightarrow\lambda t. Moreover, given solutions of Maxwell's equations, E(x, ''t'') and B(x, ''t''), it holds that E(λx, λ''t'') and B(λx, λ''t'') are also solutions.


Massless scalar field theory

Another example of a scale-invariant classical field theory is the massless
scalar field In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
(note that the name
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
is unrelated to scale invariance). 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, :\frac \frac-\nabla^2 \varphi = 0, and is invariant under the transformation :x\rightarrow\lambda x, :t\rightarrow\lambda t. The name massless refers to the absence of a term \propto m^2\varphi 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 :L=\frac, and so it should not be surprising that massive scalar field theory is ''not'' scale-invariant.


φ4 theory

The field equations in the examples above are all
linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
in the fields, which has meant that the
scaling dimension In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
, , has not been so important. However, one usually requires that the scalar field
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
is dimensionless, and this fixes the
scaling dimension In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
of . In particular, :\Delta=\frac, where is the combined number of spatial and time dimensions. Given this scaling dimension for , there are certain nonlinear modifications of massless scalar field theory which are also scale-invariant. One example is massless φ4 theory for =4. The field equation is :\frac \frac-\nabla^2 \varphi+g\varphi^3=0. (Note that the name 4 derives from the form of the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
, which contains the fourth power of .) When =4 (e.g. three spatial dimensions and one time dimension), the scalar field scaling dimension is =1. The field equation is then invariant under the transformation :x\rightarrow\lambda x, :t\rightarrow\lambda t, :\varphi (x)\rightarrow\lambda^\varphi(x). The key point is that the parameter must be dimensionless, otherwise one introduces a fixed length scale into the theory: For 4 theory, this is only the case in =4. Note that under these transformations the argument of the function is unchanged.


Scale invariance in quantum field theory

The scale-dependence of a
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
(QFT) is characterised by the way its coupling parameters depend on the energy-scale of a given physical process. This energy dependence is described by the
renormalization group In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the ...
, and is encoded in the
beta-function In theoretical physics, specifically quantum field theory, a beta function, ''β(g)'', encodes the dependence of a coupling parameter, ''g'', on the energy scale, ''μ'', of a given physical process described by quantum field theory. It is ...
s 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 as fixed points of the corresponding renormalization group flow.


Quantum electrodynamics

A simple example of a scale-invariant QFT is the quantized electromagnetic field without charged particles. This theory actually has no coupling parameters (since
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
s 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 as
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
s. The QFT describing the interactions of photons and charged particles is
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(QED), and this theory is not scale-invariant. We can see this from the QED beta-function. This tells us that the
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
(which is the coupling parameter in the theory) increases with increasing energy. Therefore, while the quantized electromagnetic field without charged particles is scale-invariant, QED is not scale-invariant.


Massless scalar field theory

Free, massless quantized 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 the
Gaussian fixed point A Gaussian fixed point is a fixed point of the renormalization group flow which is noninteracting in the sense that it is described by a free field theory. The word Gaussian comes from the fact that the probability distribution is Gaussian at the ...
. However, even though the classical massless ''φ''4 theory is scale-invariant in ''D''=4, the quantized version is not scale-invariant. We can see this from the
beta-function In theoretical physics, specifically quantum field theory, a beta function, ''β(g)'', encodes the dependence of a coupling parameter, ''g'', on the energy scale, ''μ'', of a given physical process described by quantum field theory. It is ...
for the coupling parameter, ''g''. Even though the quantized massless ''φ''4 is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the Gaussian fixed point. One example is the Wilson–Fisher fixed point, below.


Conformal field theory

Scale-invariant QFTs are almost always invariant under the full
conformal symmetry In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry ...
, and the study of such QFTs is
conformal field theory A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes ...
(CFT).
Operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
in a CFT have a well-defined
scaling dimension In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
, analogous to the
scaling dimension In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
, ''∆'', 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 as
anomalous scaling dimension In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
s.


Scale and conformal anomalies

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 be anomalous. 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 called
cosmic inflation In physical cosmology, cosmic inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the early universe. The inflationary epoch lasted from  seconds after the conjectured Big Bang singularity ...
, as long as the theory can be studied through
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
.


Phase transitions

In
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, as a system undergoes a
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
, its fluctuations are described by a scale-invariant
statistical field theory Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industria ...
. For a system in equilibrium (i.e. time-independent) in spatial dimensions, the corresponding statistical field theory is formally similar to a -dimensional CFT. The scaling dimensions in such problems are usually referred to as
critical exponent Critical or Critically may refer to: *Critical, or critical but stable, medical states **Critical, or intensive care medicine *Critical juncture, a discontinuous change studied in the social sciences. *Critical Software, a company specializing in ...
s, and one can in principle compute these exponents in the appropriate CFT.


The Ising model

An example that links together many of the ideas in this article is the phase transition of the
Ising model The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
, a simple model of
ferromagnet Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials a ...
ic 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 a -dimensional periodic lattice. Associated with each lattice site is a
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets ...
, or
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
, 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, ,
spontaneous magnetization Spontaneous magnetization is the appearance of an ordered spin state ( magnetization) at zero applied magnetic field in a ferromagnetic or ferrimagnetic material below a critical point called the Curie temperature or . Overview Heated to tempe ...
is said to occur. This means that below 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 distance . This has the generic behaviour: :G(r)\propto\frac, for some particular value of \eta, which is an example of a critical exponent.


CFT description

The fluctuations at temperature 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 the Wilson–Fisher fixed point, a particular scale-invariant
scalar field theory In theoretical physics, scalar field theory can refer to a relativistically invariant classical or quantum theory of scalar fields. A scalar field is invariant under any Lorentz transformation. The only fundamental scalar quantum field that has b ...
. In this context, is understood as a
correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables rep ...
of scalar fields, :\langle\phi(0)\phi(r)\rangle\propto\frac. Now we can fit together a number of the ideas seen already. From the above, one sees that the critical exponent, , for this phase transition, is also an anomalous dimension. This is because the classical dimension of the scalar field, :\Delta=\frac is modified to become :\Delta=\frac, where is the number of dimensions of the Ising model lattice. So this anomalous dimension in the conformal field theory is the ''same'' as a particular critical exponent of the Ising model phase transition. Note that for dimension , can be calculated approximately, using the epsilon expansion, and one finds that :\eta=\frac+O(\epsilon^3). In the physically interesting case of three spatial dimensions, we have =1, and so this expansion is not strictly reliable. However, a semi-quantitative prediction is that 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 the minimal models, a family of well-understood CFTs, and it is possible to compute (and the other critical exponents) exactly, :\eta_=\frac.


Schramm–Loewner evolution

The anomalous dimensions in certain two-dimensional CFTs can be related to the typical
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
s of random walks, where the random walks are defined via
Schramm–Loewner evolution In probability theory, the Schramm–Loewner evolution with parameter ''κ'', also known as stochastic Loewner evolution (SLE''κ''), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensiona ...
(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 2''d'' critical Ising model and the more general 2''d'' critical
Potts model In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenome ...
. Relating other 2''d'' CFTs to SLE is an active area of research.


Universality

A phenomenon known as universality 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: * The
Ising model The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
phase transition, described above. * The
liquid A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...
-
vapour In physics, a vapor (American English) or vapour (British English and Canadian English; see spelling differences) is a substance in the gas phase at a temperature lower than its critical temperature,R. H. Petrucci, W. S. Harwood, and F. G. Herr ...
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 or critical 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 a
universality class In statistical mechanics, a universality class is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite s ...
. Other examples of systems which belong to a universality class are: *
Avalanche An avalanche is a rapid flow of snow down a slope, such as a hill or mountain. Avalanches can be set off spontaneously, by such factors as increased precipitation or snowpack weakening, or by external means such as humans, animals, and earth ...
s 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 of
network outage The term downtime is used to refer to periods when a system is unavailable. The unavailability is the proportion of a time-span that a system is unavailable or offline. This is usually a result of the system failing to function because of an un ...
s on the
Internet The Internet (or internet) is the global system of interconnected computer networks that uses the Internet protocol suite (TCP/IP) to communicate between networks and devices. It is a '' network of networks'' that consists of private, pub ...
, 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. * 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. * The
electrical breakdown Electrical breakdown or dielectric breakdown is a process that occurs when an electrical insulating material, subjected to a high enough voltage, suddenly becomes an electrical conductor and electric current flows through it. All insulating mate ...
of
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
s, which resemble cracks and tears. * The
percolation Percolation (from Latin ''percolare'', "to filter" or "trickle through"), in physics, chemistry and materials science, refers to the movement and filtering of fluids through porous materials. It is described by Darcy's law. Broader applicatio ...
of fluids through disordered media, such as
petroleum Petroleum, also known as crude oil, or simply oil, is a naturally occurring yellowish-black liquid mixture of mainly hydrocarbons, and is found in geological formations. The name ''petroleum'' covers both naturally occurring unprocessed crud ...
through fractured rock beds, or water through filter paper, such as in
chromatography In chemical analysis, chromatography is a laboratory technique for the separation of a mixture into its components. The mixture is dissolved in a fluid solvent (gas or liquid) called the ''mobile phase'', which carries it through a system (a ...
. Power-law scaling connects the rate of flow to the distribution of fractures. * The diffusion of molecules in Solution (chemistry), solution, and the phenomenon of diffusion-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 a
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
, and that the language of statistical mechanics and scale-invariant
statistical field theory Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industria ...
may be applied to describe them.


Other examples of scale invariance


Newtonian fluid mechanics with no applied forces

Under certain circumstances, fluid mechanics is a scale-invariant classical field theory. The fields are the velocity of the fluid flow, \mathbf(\mathbf,t), the fluid density, \rho(\mathbf,t), and the fluid pressure, P(\mathbf,t). These fields must satisfy both the Navier–Stokes equation and the continuity equation#Fluid dynamics, continuity equation. For a Newtonian fluid these take the respective forms \rho\frac+\rho\mathbf\cdot\nabla \mathbf = -\nabla P+\mu \left(\nabla^2 \mathbf+\frac\nabla\left(\nabla\cdot\mathbf\right)\right) :\frac+\nabla\cdot \left(\rho\mathbf\right)=0 where \mu is the dynamic viscosity#Viscosity .28dynamic viscosity.29: .CE.BC, dynamic viscosity. In order to deduce the scale invariance of these equations we specify an equation 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 the isothermal ideal gas, which satisfies :P=c_s^2\rho, where 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 :x\rightarrow\lambda x, :t\rightarrow\lambda^2 t, :\rho\rightarrow\lambda^ \rho, :\mathbf\rightarrow\lambda^\mathbf. Given the solutions \mathbf(\mathbf,t) and \rho(\mathbf,t), we automatically have that \lambda\mathbf(\lambda\mathbf,\lambda^2 t) and \lambda\rho(\lambda\mathbf,\lambda^2 t) are also solutions.


Hidden scale invariance in liquids and solids

Certain models studied by Molecular dynamics, Molecular Dynamics computer simulations, including the Lennard-Jones potential, Lennard-Jones and Yukawa potential, Yukawa pair-potential models, have a potential-energy function U(\mathbf) that to a good approximation obeys the "hidden scale invariance" criterion U(\mathbf_) Here \mathbf_and \mathbf_are the full spatial coordinates of two same-density configuration and \lambda is a parameter that uniformly scales the configurations to a different density. Hidden scale invariance means that the ordering of configurations at one density according to their potential energy is maintained if these are scaled uniformly to a different density. This only applies rigorously for systems with an Euler-homogeneous potential-energy function, e.g., systems of particles interacting with inverse-power-law pair potentials. When hidden scale invariance implies for most configurations, this implies the existence of lines in the thermodynamic phase diagram, so-called "isomorphs", along which structure and dynamics in reduced units are invariant to a good approximation. It is believe that most metals and van der Waals bonded systems obey this approximate symmetry in the liquid and solid phases, whereas systems with strong directional bonds like covalently or hydrogen-bonded systems do not; ionic and dipolar systems constitute a class in-between. Most systems do not obey hidden scale invariance in the gas phase. Since an isomorph is line of constant excess entropy, the existence of isomorphs explains to a large extent the ''excess-entropy scaling'' discovered by Rosenfeld in 1977 and why this applies also to mixtures, confined systems, molecular systems, etc.


Computer vision

In computer vision and biological 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.Lindeberg, T. (2013) Invariance of visual operations at the level of receptive fields, PLoS ONE 8(7):e66990.
/ref> Detecting local maxima over scales of normalized derivative responses provides a general framework for obtaining scale invariance from image data.T. Lindeberg (2014

/ref> Examples of applications include blob detection, corner detection, ridge detection, and object recognition via the scale-invariant feature transform.


See also

*Inverse square potential *Power law *Scale-free network


References


Further reading

* Extensive discussion of scale invariance in quantum and statistical field theories, applications to critical phenomena and the epsilon expansion and related topics. * *{{cite book , first=G. , last=Mussardo , title=Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics , publisher=Oxford University Press , year=2010 Symmetry Scaling symmetries Conformal field theory Critical phenomena Scale-invariant systems,