Scale invariance

In physics and mathematics, scale invariance is a feature of objects or laws that do not change if length scales (or energy scales) are multiplied by a common factor. The technical term for this transformation is a dilatation, and the dilatations can also form part of a larger conformal symmetry.

Scale invariance most commonly applies to the invariance of a whole theory under dilatations, but can sometimes refer to an invariance of individual functions or field configurations. A closely related concept is self-similarity, where a function or field configuration is invariant under a subset of the dilatations.

Scale invariance in classical field theory

Classical field theory is generically described by a field, or set of fields, $\phi$, which depend on coordinates, $x$. Valid field configurations are then determined by solving differential equations for $\phi (x)$, and these equations are known as field equations.

For a theory to be scale-invariant, its field equations should be invariant under a rescaling of the coordinates, combined with some specified rescaling of the fields:

$x\rightarrow \lambda x$,

$\phi \rightarrow \lambda ^{-\Delta }\phi$.

The parameter $\Delta$ 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, $\phi (x)$, one always has other solutions of the form $\lambda ^{-\Delta }\phi (\lambda x)$.

Scale invariance of field configurations

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

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

where $\Delta$ 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.

Examples of scale-invariant theories

Classical electromagnetism

In electromagnetism, the fields are the electric and magnetic fields, $\mathbf {E} (\mathbf {x} ,t)$ and $\mathbf {B} (\mathbf {x} ,t)$, while their field equations are Maxwell's equations. With no charges or currents, these field equations take the form of wave equations:

$\nabla ^{2}\mathbf {E} ={\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}$

$\nabla ^{2}\mathbf {B} ={\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}$

where c is the velocity 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, $\mathbf {E} (\mathbf {x} ,t)$ and $\mathbf {B} (\mathbf {x} ,t)$, we have that $\mathbf {E} (\lambda \mathbf {x} ,\lambda t)$ and $\mathbf {B} (\lambda \mathbf {x} ,\lambda t)$ are also solutions.

Newtonian fluid mechanics with no applied forces

In fluid mechanics, the fields are the velocity of the fluid flow, $\mathbf {u} (\mathbf {x} ,t)$, the fluid density, $\rho (\mathbf {x} ,t)$, and the fluid pressure, $P(\mathbf {x} ,t)$. These fields must satisfy both the Navier-Stokes equation and the continuity equation. For a Newtonian fluid these take the respective forms

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

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

where $\mu$ 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

$P=c_{s}^{2}\rho$,

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

$x\rightarrow \lambda x,$

$t\rightarrow \lambda t,$

$\rho \rightarrow \lambda \rho ,$

$\mathbf {u} \rightarrow \mathbf {u} .$

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

Scale invariance in quantum field theory

The scale-dependence of a quantum field theory (QFT) is characterised by the way its coupling parameters depend on the energy-scale of a given physical process. This energy-dependence is encoded in the beta-functions of the theory.

For a QFT to be scale-invariant, its coupling parameters must be independent of the energy-scale, and this is indicated by the vanishing of the beta-functions of the theory. A simple example is the quantized electromagnetic field, without charged particles. This theory actually has no coupling parameters (since photons are massless and non-interacting) and is therefore scale-invariant.

However, in the case of the electromagnetic field, adding charged particles breaks this scale invariance. The coupling parameter in quantum electrodynamics (QED) is the electric charge, and the QED beta-function tells us that this coupling increases with increasing energy. Therefore, QED is not scale-invariant.

We note that the coupling parameters of a quantum field theory can be scale-dependent even if the corresponding classical field theory is scale-invariant. In this case, the classical scale invariance is said to be anomalous.

Conformal field theory

Scale-invariant QFTs are almost always invariant under the full conformal symmetry, 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, $\Delta$, 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 dimensions.

Phase transitions

In statistical mechanics, as a system undergoes a phase transition, its fluctuations can be described by a scale-invariant statistical field theory, and such theories are very closely related to conformal field theories. A typical phenomenon is critical opalescence, where a fluid becomes opaque due to density fluctuations at all size scales. The scaling dimensions in such problems are usually referred to as critical exponents, and one can in principle compute these exponents in the appropriate CFT.

The Ising model

A canonical example is the phase transition of the Ising model, a crude model of a ferromagnetic substance. 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, $T_{c}$, spontaneous magnetization is said to occur. This means that below $T_{c}$ the spin-spin interaction will begin to dominate, and there is some net alignment of spins in one of the two directions. An example of the kind of quantities one would like to calculate is the net magnetization, M, as a function of temperature near the critical point $T=T_{c}$:

$M\propto (T_{c}-T)^{\beta }$

In particular, $\beta$ is an example of a critical exponent (but should not be confused with beta-functions).

The fluctuations at temperature $T_{c}$ are scale-invariant, and so the Ising model at this phase transition is expected to be described by a scale-invariant statistical field theory. In fact, this theory is the Wilson-Fisher fixed point, a particular scale-invariant scalar field theory. The magnetization, M is understood as the expectation value of the scalar field, and $\beta$ can be calculated approximately using the epsilon expansion.

For D=2, 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 its critical exponents exactly.

Scale-invariant functions and self-similarity

In mathematics, one can consider the scaling properties of a function, $f(x)$ under rescalings of the variable $x$. The requirement for $f(x)$ to be invariant under all rescalings is usually taken to be

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

for some choice of $\Delta$, and for all dilatations $\lambda$. This is quite analogous to the condition for a classical field configuration to be scale-invariant.

An example is the logarithmic spiral, a kind of curve that often appears in nature. In polar coordinates (r, θ) the spiral can be written as

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

Allowing for rotations of the curve, it is self-similar for all scales $\lambda$; that is $\theta (\lambda r)$ is identical to a rotated version of $\theta (r)$.

Fractals

Fractals are a class of shapes that exhibit self-similarity, in addition to a number of other properties. They appear naturally in many problems of classical chaos and quantum chaos, such as, for example, turbulent flow in fluid dynamics. In such cases, the scaling exponents of physical quantities can be related to a fractal dimension. (See also the Hausdorff dimension and Kolmogorov dimension.)

Fractional scaling is also frequently seen in networks, such as the connectivity of the Internet, the network of citations made by journal articles to other journal articles, or the network of sand grains that are in contact with one another.

The classical fractals, such as the Koch curve, have a discrete set of symmetries, which are combinations of dilations, translations and rotations. In particular, they are self similar only for a discrete set of dilations (ignoring the translations and rotations). For example, the Koch curve satisfies $\mathbf {x} (s)=\lambda ^{-1}\mathbf {x} (\lambda s)$, for $\lambda =3^{n}$ ($n$ takes integer values).

More complex fractals usually do not have a single scaling exponent, but a range of them that are active at once. Such systems can sometimes be decomposed into pieces, each of which does have a single fixed scaling exponent. Such a decomposition into distinct scaling regimes is known as multifractal analysis. An example of a classical fractal which has a multitude of scaling exponents at work is the Minkowski question mark function. In physics, such multi-scaling behaviours have been described by P. W. Anderson and others for examples such as dripping water faucets.

Other examples of scale invariance

• In physical cosmology, the power spectrum of the cosmic microwave background is near to scale-invariant. This means that the amplitude, $P(k)$, of primordial fluctuations as a function of wave number, $k$, is approximately constant. This pattern is consistent with the proposal of cosmic inflation.

• In computer vision, scale invariance refers to a local image description that remains invariant when the scale of the image is changed. Such a representation is provided by the SIFT operator.

See also

• Conformal symmetry
• Conformal field theory
• Self-similarity