Scale invariance
That is, one is interested in the shape of f (λx) for some scale factor λ, which can be taken to be a length or size rescaling.
The requirement for f (x) to be invariant under all rescalings is usually taken to be for some choice of exponent Δ, and for all dilations λ.
Consequent to their inherent scale invariance Tweedie random variables Y demonstrate a variance var(Y) to mean E(Y) power law: where a and p are positive constants.
The Wiener–Khinchin theorem further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest 1/f noise.
[4] The Tweedie convergence theorem provides a hypothetical explanation for the wide manifestation of fluctuation scaling and 1/f noise.
[5] It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a variance function that comes within the domain of attraction of a Tweedie model.
[4] Much as the central limit theorem requires certain kinds of random variables to have as a focus of convergence the Gaussian distribution and express white noise, the Tweedie convergence theorem requires certain non-Gaussian random variables to express 1/f noise and fluctuation scaling.
[4] In physical cosmology, the power spectrum of the spatial distribution of the cosmic microwave background is near to being a scale-invariant function.
However, one usually requires that the scalar field action is dimensionless, and this fixes the scaling dimension of φ.
Given this scaling dimension for φ, there are certain nonlinear modifications of massless scalar field theory which are also scale-invariant.
The field equation is (Note that the name φ4 derives from the form of the Lagrangian, which contains the fourth power of φ.)
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 described by the renormalization group, and 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.
[6] A simple example of a scale-invariant QFT is the quantized electromagnetic field without charged particles.
The QFT describing the interactions of photons and charged particles is quantum electrodynamics (QED), and this theory is not scale-invariant.
However, the scaling dimensions of operators in a CFT typically differ from those of the fields in the corresponding classical 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.
This is a statistical mechanics model, which also has a description in terms of conformal field theory.
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.
This means that below Tc 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
In this context, G(r) is understood as a correlation function of scalar fields, Now we can fit together a number of the ideas seen already.
From the above, one sees that the critical exponent, η, for this phase transition, is also an anomalous dimension.
So this anomalous dimension in the conformal field theory is the same as a particular critical exponent of the Ising model phase transition.
In particular, it is equivalent to one of the minimal models, a family of well-understood CFTs, and it is possible to compute η (and the other critical exponents) exactly, The anomalous dimensions in certain two-dimensional CFTs can be related to the typical fractal dimensions of random walks, where the random walks are defined via Schramm–Loewner evolution (SLE).
It expresses the idea that different microscopic physics can give rise to the same scaling behaviour at a phase transition.
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.
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.
