Scale invariance
Encyclopedia
In physics
and mathematics
, 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. The technical term for this transformation
is a dilatation (also known as dilation), and the dilatations can also form part of a larger conformal symmetry
.
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
, and for all dilations 
.
Examples of scale-invariant functions are the monomial
s
, for which one has 
, 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 (r, θ) 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 of 
.
, 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
.
It is sometimes said that fractal
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 must be applied to match up to the fractal to itself. Thus, for example the Koch curve scales with 
, but the scaling holds only for values of 
for integer n. 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.
is the average, expected power at frequency 
, then noise scales as
with
for white noise
,
for pink noise
, and
for Brownian 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 the probability distribution
. Examples of scale-invariant distributions are the Pareto distribution and the Zipfian distribution.
, 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, P(k), of primordial fluctuations
as a function of wave number, k, is approximately constant, i.e. a flat spectrum. This pattern is consistent with the proposal of cosmic inflation
.
is generically described by a field, or set of fields,
, that depend on coordinates, x. Valid field configurations are then determined by solving differential equations for 
, and these equations are known as field equation
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:
The parameter
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,
, one always has other solutions of the form 
.
, to be scale-invariant, we require that
where
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.
with no charges or currents. The fields are the electric and magnetic fields,
and 
, while their field equations are Maxwell's equations
. With no charges or currents, these field equations take the form of wave equation
s
where c is the speed of light.
These field equations are invariant under the transformation
Moreover, given solutions of Maxwell's equations,
and 
, we have that 
and 
are also solutions.
(note that the name scalar
is unrelated to scale invariance). The scalar field,
is a function of a set of spatial variables, 
, and a time variable, t. We first consider the linear theory. Much 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 via
and so it should not be surprising that massive scalar field theory is not scale-invariant.
in the fields, which has meant that the scaling dimension,
, has not been so important. However, one usually requires that the scalar field action
is dimensionless, and this fixes the scaling dimension of
. In particular,
where D 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 
. The field equation is
(Note that the name
derives from the form of the Lagrangian, which contains the fourth power of 
.)
When D=4 (e.g. three spatial dimensions and one time dimension), the scalar field scaling dimension is
. The field equation is then invariant under the transformation
The key point is that the parameter g must be dimensionless, otherwise one introduces a fixed length scale into the theory. For φ4 theory this is only the case in
.
(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
, and is encoded in the beta-function
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 point
s of the corresponding renormalization group flow.
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
s. The QFT describing the interactions of photons and charged particles is quantum electrodynamics
(QED), and this theory is not scale-invariant. We can see this from the QED beta-function. This tells us that the electric charge
(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.
.
However, even though the classical massless φ4 theory is scale-invariant in
, the quantized version is not scale-invariant. We can see this from the beta-function
for the coupling parameter, g.
Even though the quantized massless φ4 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.
, and the study of such QFTs is conformal field theory
(CFT). Operators
in a CFT have a well-defined scaling dimension, analogous to the scaling dimension,
, of a classical field discussed above. However, the scaling dimensions of operators in a CFT typically differ from the those of the fields in the corresponding classical theory. The additional contributions appearing in the CFT are known as anomalous scaling dimension
s.
.
, as a system undergoes a phase transition
, its fluctuations are described by a scale-invariant statistical field theory
. For a system in equilibrium (i.e. time-independent) in D spatial dimensions, the corresponding statistical field theory is formally similar to a D-dimensional CFT. The scaling dimensions in such problems are usually referred to as critical exponent
s, and one can in principle compute these exponents in the appropriate CFT.
, a crude model of ferromagnetic 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 D-dimensional periodic lattice. Associated with each lattice site is a magnetic moment
, or spin
, 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
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 r. This has the generic behaviour:
for some particular value of
, which is an example of a critical exponent.
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 this context, 
is understood as a correlation function
of scalar fields:
Now we can fit together a number of the ideas we've seen already. From the above we can see that the critical exponent,
, for this phase transition, is also an anomalous dimension. This is because the classical dimension of the scalar field
is modified to become
where D 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.
We note that for dimension
, 
can be calculated approximately, using the epsilon expansion, and one finds that
.
In the physically interesting case of three spatial dimensions we have
, 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 model
s, a family of well-understood CFTs, and it is possible to compute
(and the other critical exponents) exactly:
.
s of random walks, where the random walks are defined via Schramm–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 critical Potts model
. Relating other 2d CFTs to SLE is an active area of research.
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:
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. Other examples of systems which belong to a universality class are:
The key observation is that, for all of these different systems, the behaviour resembles a phase transition
, and that the language of statistical mechanics and scale-invariant statistical field theory
may be applied to describe them.
is a scale-invariant classical field theory. The fields are the velocity of the fluid flow,
, the fluid density, 
, and the fluid pressure, 
. These fields must satisfy both the Navier–Stokes equation and the continuity equation. For a Newtonian fluid
these take the respective forms
where
is the 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
where
is the speed of sound in the fluid. Given this equation of state, Navier–Stokes and the continuity equation are invariant under the transformations
Given the solutions
and 
, we automatically have that
and 
are also solutions.
, scale invariance refers to a local image description that remains invariant when the scale of the image is changed. A general framework for obtaining scale invariance in practice is by detecting local maxima over scales of normalized derivative responses -- see the article on scale-space for a brief introduction to the general theory and references. Examples of scale invariant blob detectors and ridge detectors are given in the articles on blob detection
and ridge detection
. An example of the application of scale invariance to object recognition is given in the article on the scale-invariant feature transform
.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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. The technical term for this transformation
Transformation (mathematics)
In mathematics, a transformation could be any function mapping a set X on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure.Examples include...
is a dilatation (also known as dilation), and the dilatations can also form part of a larger conformal symmetry
Conformal symmetry
In theoretical physics, conformal symmetry is a symmetry under dilatation and under the special conformal transformations...
.
- In mathematics, scale invariance usually refers to an invariance of individual functionsFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
or curveCurveIn mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
s. A closely related concept is self-similaritySelf-similarityIn mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...
, where a function or curve is invariant under a discrete subset of the dilatations. It is also possible for the probability distributionProbability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s of random processes to display this kind of scale invariance or self-similarity.
- In classical field theoryClassical field theoryA classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ....
, 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 theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, scale invariance has an interpretation in terms of particle physicsParticle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.
- In statistical mechanicsStatistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, scale invariance is a feature of phase transitionPhase transitionA phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
s. The key observation is that near a phase transition or critical pointCritical point (thermodynamics)In physical chemistry, thermodynamics, chemistry and condensed matter physics, a critical point, also called a critical state, specifies the conditions at which a phase boundary ceases to exist...
, 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 theoriesStatistical field theoryA statistical field theory is any model in statistical mechanics where the degrees of freedom comprise a field or fields. In other words, the microstates of the system are expressed through field configurations...
, and are formally very similar to scale-invariant quantum field theories.
- UniversalityUniversality (dynamical systems)In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together...
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 statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
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 functionFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
or curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
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 befor some choice of exponent
, and for all dilations .Examples of scale-invariant functions are the monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...
s
, for which one has , in that clearlyAn example of a scale-invariant curve is the logarithmic spiral
Logarithmic spiral
A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous...
, a kind of curve that often appears in nature. In polar coordinates (r, θ) 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 of .Projective geometry
The idea of scale invariance of a monomial generalizes in higher dimensions to the idea of a homogeneous polynomialHomogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...
, and more generally to a homogeneous function
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
. Homogeneous functions are the natural denizens of projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
, and homogeneous polynomials are studied as projective varieties in projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
. Projective geometry is a particularly rich field of mathematics; in its most abstract forms, the geometry of schemes
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
, it has connections to various topics in string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
.
Fractals
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
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 must be applied to match up to the fractal to itself. Thus, for example the Koch curve scales with , but the scaling holds only for values of for integer n. 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.
Scale invariance in stochastic processes
If is the average, expected power at frequency , then noise scales aswith
for white noiseWhite noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
,
for pink noisePink noise
Pink noise or 1/ƒ noise is a signal or process with a frequency spectrum such that the power spectral density is inversely proportional to the frequency. In pink noise, each octave carries an equal amount of noise power...
, and
for Brownian noiseBrownian noise
In science, Brownian noise , also known as Brown noise or red noise, is the kind of signal noise produced by Brownian motion, hence its alternative name of random walk noise...
(and more generally, Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
).
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
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
. Examples of scale-invariant distributions are the Pareto distribution and the Zipfian distribution.
Cosmology
In physical cosmologyPhysical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
, 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, P(k), of primordial fluctuations
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...
as a function of wave number, k, is approximately constant, i.e. a flat spectrum. This pattern is consistent with the proposal of cosmic inflation
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...
.
Scale invariance in classical field theory
Classical field theoryClassical field theory
A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ....
is generically described by a field, or set of fields,
, that depend on coordinates, x. Valid field configurations are then determined by solving differential equations for , and these equations are known as field equationField equation
A field equation is an equation in a physical theory that describes how a fundamental force interacts with matter...
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:
The parameter
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,
, one always has other solutions of the form .Scale invariance of field configurations
For a particular field configuration,, to be scale-invariant, we require thatwhere
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 electromagnetismElectromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
with no charges or currents. The fields are the electric and magnetic fields,
and , while their field equations are Maxwell's equationsMaxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
. With no charges or currents, these field equations take the form of wave equation
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
s
where c is the speed of light.
These field equations are invariant under the transformation
Moreover, given solutions of Maxwell's equations,
and , we have that and are also solutions.Massless scalar field theory
Another example of a scale-invariant classical field theory is the massless scalar fieldScalar field theory
In theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a "scalar", in contrast to a vector or tensor field...
(note that the name scalar
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...
is unrelated to scale invariance). The scalar field,
is a function of a set of spatial variables, , and a time variable, t. We first consider the linear theory. Much like the electromagnetic field equations above, the equation of motion for this theory is also a wave equationand 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 viaand 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 linearLinear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
in the fields, which has meant that the scaling dimension,
, has not been so important. However, one usually requires that the scalar field actionAction (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
is dimensionless, and this fixes the scaling dimension of
. In particular,where D 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 . The field equation is(Note that the name
derives from the form of the Lagrangian, which contains the fourth power of .)When D=4 (e.g. three spatial dimensions and one time dimension), the scalar field scaling dimension is
. The field equation is then invariant under the transformationThe key point is that the parameter g must be dimensionless, otherwise one introduces a fixed length scale into the theory. For φ4 theory this is only the case in
.Scale invariance in quantum field theory
The scale-dependence of a quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
(QFT) is characterised by the way its coupling parameters
Coupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
depend on the energy-scale of a given physical process. This energy dependence is described by the renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
, and is encoded in the beta-function
Beta-function
In theoretical physics, specifically quantum field theory, a beta function β encodes the dependence of a coupling parameter, g, on the energy scale, \mu of a given physical process....
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 point
Fixed point
"Fixed point" has many meanings in science, most of them mathematical.* Fixed point * Fixed-point combinator* Fixed-point arithmetic, a manner of doing arithmetic on computers* Benchmark , fixed points used by geodesists...
s 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 photonPhoton
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
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
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s. The QFT describing the interactions of photons and charged particles is quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics 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 special relativity is achieved...
(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
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
(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 pointGaussian 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 Gaussian fixed point. This means that Gaussian fixed...
.
However, even though the classical massless φ4 theory is scale-invariant in
, the quantized version is not scale-invariant. We can see this from the beta-functionBeta-function
In theoretical physics, specifically quantum field theory, a beta function β encodes the dependence of a coupling parameter, g, on the energy scale, \mu of a given physical process....
for the coupling parameter, g.
Even though the quantized massless φ4 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.
Conformal field theory
Scale-invariant QFTs are almost always invariant under the full conformal symmetryConformal symmetry
In theoretical physics, conformal symmetry is a symmetry under dilatation and under the special conformal transformations...
, and the study of such QFTs is conformal field theory
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
(CFT). Operators
Operator (physics)
In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
in a CFT have a well-defined scaling dimension, analogous to the scaling dimension,
, of a classical field discussed above. However, the scaling dimensions of operators in a CFT typically differ from the those of the fields in the corresponding classical theory. The additional contributions appearing in the CFT are known as anomalous scaling dimensionAnomalous scaling dimension
In theoretical physics, by anomaly one usually means that the symmetry remains broken when the symmetry-breaking factor goes to zero.When the symmetry which is broken is scale invariance, then true power laws usually cannot be found from dimensional reasoning like in turbulence or quantum field...
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 anomalousConformal anomaly
Conformal anomaly is an anomaly i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory.A classically conformal theory is a theory which, when placed on a surface with arbitrary background metric, has an action that is invariant under rescalings of the background metric...
.
Phase transitions
In statistical mechanicsStatistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, as a system undergoes a phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
, its fluctuations are described by a scale-invariant statistical field theory
Statistical field theory
A statistical field theory is any model in statistical mechanics where the degrees of freedom comprise a field or fields. In other words, the microstates of the system are expressed through field configurations...
. For a system in equilibrium (i.e. time-independent) in D spatial dimensions, the corresponding statistical field theory is formally similar to a D-dimensional CFT. The scaling dimensions in such problems are usually referred to as critical exponent
Critical exponent
Critical exponents describe the behaviour of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e...
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 modelIsing model
The Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
, a crude model of ferromagnetic 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 D-dimensional periodic lattice. Associated with each lattice site is a magnetic moment
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
, or spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
, 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 magnetizationSpontaneous magnetization
Spontaneous magnetization is the term used to describe the appearance of an ordered spin state at zero applied magnetic field in a ferromagnetic or ferrimagnetic material below a critical point called the Curie temperature or .-Overview:...
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 r. This has the generic behaviour:
for some particular value of
, 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 this context, is understood as a correlation functionCorrelation function
A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...
of scalar fields:
Now we can fit together a number of the ideas we've seen already. From the above we can see that the critical exponent,
, for this phase transition, is also an anomalous dimension. This is because the classical dimension of the scalar fieldis modified to become
where D 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.
We note that for dimension
, can be calculated approximately, using the epsilon expansion, and one finds that.In the physically interesting case of three spatial dimensions we have
, 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 modelMinimal model
*In theoretical physics, the term minimal model usually refers to a special class of conformal field theories that generalize the Ising model, or to some closely related representations of the Virasoro algebra...
s, a family of well-understood CFTs, and it is possible to compute
(and the other critical exponents) exactly:.Schramm–Loewner evolution
The anomalous dimensions in certain two-dimensional CFTs can be related to the typical fractal dimensionFractal dimension
In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
s of random walks, where the random walks are defined via Schramm–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 critical Potts model
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 phenomena of solid state physics...
. Relating other 2d CFTs to SLE is an active area of research.
Universality
A phenomenon known as universalityUniversality (dynamical systems)
In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together...
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 modelIsing modelThe Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
phase transition, described above. - The liquidLiquidLiquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...
-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 or critical point
Critical point (thermodynamics)
In physical chemistry, thermodynamics, chemistry and condensed matter physics, a critical point, also called a critical state, specifies the conditions at which a phase boundary ceases to exist...
, 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. Other examples of systems which belong to a universality class are:
- AvalancheAvalancheAn avalanche is a sudden rapid flow of snow down a slope, occurring when either natural triggers or human activity causes a critical escalating transition from the slow equilibrium evolution of the snow pack. Typically occurring in mountainous terrain, an avalanche can mix air and water with the...
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 outages on the InternetInternetThe Internet is a global system of interconnected computer networks that use the standard Internet protocol suite to serve billions of users worldwide...
, 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 breakdownElectrical breakdownThe term electrical breakdown or electric breakdown has several similar but distinctly different meanings. For example, the term can apply to the failure of an electric circuit....
of dielectricDielectricA dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
s, which resemble cracks and tears. - The percolationPercolationIn physics, chemistry and materials science, percolation concerns the movement and filtering of fluids through porous materials...
of fluids through disordered media, such as petroleumPetroleumPetroleum or crude oil is a naturally occurring, flammable liquid consisting of a complex mixture of hydrocarbons of various molecular weights and other liquid organic compounds, that are found in geologic formations beneath the Earth's surface. Petroleum is recovered mostly through oil drilling...
through fractured rock beds, or water through filter paper, such as in chromatographyChromatographyChromatography is the collective term for a set of laboratory techniques for the separation of mixtures....
. Power-law scaling connects the rate of flow to the distribution of fractures. - The diffusionDiffusionMolecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
of moleculeMoleculeA molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s in solutionSolutionIn chemistry, a solution is a homogeneous mixture composed of only one phase. In such a mixture, a solute is dissolved in another substance, known as a solvent. The solvent does the dissolving.- Types of solutions :...
, and the phenomenon of diffusion-limited aggregationDiffusion-limited aggregationDiffusion-limited aggregation is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by Witten and Sander in 1981, is applicable to aggregation in any system where diffusion is the primary means...
. - 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
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
, and that the language of statistical mechanics and scale-invariant statistical field theory
Statistical field theory
A statistical field theory is any model in statistical mechanics where the degrees of freedom comprise a field or fields. In other words, the microstates of the system are expressed through field configurations...
may be applied to describe them.
Newtonian fluid mechanics with no applied forces
Under certain circumstances, fluid mechanicsFluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
is a scale-invariant classical field theory. The fields are the velocity of the fluid flow,
, the fluid density, , and the fluid pressure, . These fields must satisfy both the Navier–Stokes equation and the continuity equation. For a Newtonian fluidNewtonian fluid
A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...
these take the respective forms
where
is the dynamic viscosity.In order to deduce the scale invariance of these equations we specify an equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
, 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
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
, which satisfies
where
is the speed of sound in the fluid. Given this equation of state, Navier–Stokes and the continuity equation are invariant under the transformationsGiven the solutions
and , we automatically have that and are also solutions.Computer vision
In computer visionComputer vision
Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...
, scale invariance refers to a local image description that remains invariant when the scale of the image is changed. A general framework for obtaining scale invariance in practice is by detecting local maxima over scales of normalized derivative responses -- see the article on scale-space for a brief introduction to the general theory and references. Examples of scale invariant blob detectors and ridge detectors are given in the articles on blob detection
Blob detection
In the area of computer vision, blob detection refers to visual modules that are aimed at detecting points and/or regions in the image that differ in properties like brightness or color compared to the surrounding...
and ridge detection
Ridge detection
The ridges of a smooth function of two variables is a set of curves whose points are, in one or more ways to be made precise below, local maxima of the function in at least one dimension. For a function of N variables, its ridges are a set of curves whose points are local maxima in N-1 dimensions...
. An example of the application of scale invariance to object recognition is given in the article on the scale-invariant feature transform
Scale-invariant feature transform
Scale-invariant feature transform is an algorithm in computer vision to detect and describe local features in images. The algorithm was published by David Lowe in 1999....
.
See also
- Scale spaceScale spaceScale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision...
, in computer visionComputer visionComputer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...
and image processingImage processingIn electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image... - Scale relativityScale relativityScale relativity is a theory of space-time initially developed by Laurent Nottale, working at the French observatory of Meudon, near Paris. It is an extension of the concept of relativity found the special relativity and general relativity to physical scales . If scales in nature are always...
, physics