# What is Scale invariance: Definition and 15 Discussions

In physics, mathematics and statistics, 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 is a dilatation (also known as dilation), and the dilatations can also form part of a larger conformal symmetry.

In mathematics, scale invariance usually refers to an invariance of individual functions or curves. A closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the probability distributions of random processes to display this kind of scale invariance or self-similarity.
In classical field theory, scale invariance most commonly applies to the invariance of a whole theory under dilatations. Such theories typically describe classical physical processes with no characteristic length scale.
In quantum field theory, scale invariance has an interpretation in terms of particle physics. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.
In statistical mechanics, scale invariance is a feature of phase transitions. 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 are scale invariant. The analogous concept in statistics are standardized moments, which are scale invariant statistics of a variable, while the unstandardized moments are not.

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1. ### I Plotting polar equations and scale invariance

Hello, In the plane, using Cartesian coordinates ##x## and ##y##, an equation (or a function) is a relationship between the ##x## and ##y## variables expressed as ##y=f(x)##. The variable ##y## is usually the dependent variable while ##x## is the independent variable. The polar coordinates...
2. ### B Scale Invariance and dark matter/dark energy

https://arxiv.org/pdf/1710.11425.pdf The dark matter/dark energy issues has not made any sense to me since 1996 when it was evident that the ratio of dark matter increased as the volume of space measured increased. This was an obvious affect of empty space in the dark matter question that was...
3. ### B Is Scale Invariance the Key to Understanding the Expansion of the Universe?

https://www.sciencedaily.com/releases/2017/11/171122113013.htm A University of Geneva researcher has recently shown that the accelerating expansion of the universe and the movement of the stars in the galaxies can be explained without drawing on the concepts of dark matter and dark energy…...
4. ### A Scale invariant inverse square potential

Yesterday, I was thinking about a problem I had encountered many years before, the central force problem with a ##V(r) \propto r^{-2}## potential... If we have a Hamiltonian operator ##H = -\frac{\hbar^2}{2m}\nabla^2 - \frac{A}{r^2}## and do a coordinate transformation ##\mathbf{r}...
5. ### I Scale invariance in the power spectrum

I understand the inflation predicts a nearly scale invariant power spectrum but some have claimed this was predicted before inflation (by Harrison and Zeldovitch?) My understanding is that perfectly scale invariance would predict ns=1 but inflation predicts ns =.96. So did the prior prediction...
6. ### B Understanding Vacuum Energy: A Key to Unifying QFT and GR?

I often wonder about how little I understand vacuum, and only recently I've been paying attention to this "vacuum energy" hypothetical. I see it being associated with things as small as spontaneous emissions to things as large as the expansion of the universe. This is a huge range of length...
7. ### Intuition Behind Scale Invariance Power Spectrum

In the book "Statistical physics for cosmic structures" at p. 171 a read a definition of scale invariance (leading to the so called scale invariant power spectrum) given as the requirement that ##\sigma^2_M(R=R_H(t)) = constant##, where ##R_H(t)## is the horizon, i.e. the maximal distance that...
8. ### What is the concept of scale invariance in quantum field theory?

Hey guys! I was reading the following paper http://arxiv.org/abs/hep-ph/0703260 for Georgi and I have a conceptual question about it. Howard Georgi was talking about this Unparticle Physics theory and at the base of his analysis is the principle of scale invariance. So Georgi is saying what if...
9. ### Density Perturbations: Relation to c/H & Scale Invariance

Why have the density perturbations of all lengths a relation to the constant radius c/H? I suppose this is the origin for scale invariance and gaussianity.
10. ### Fixed point and scale invariance

Hello everyone. I'm studying the fixed point of theory in the context of QFT. First of all, let me say what I think I understood about fixed points and then I'll state my question. Suppose we have a theory with a certain running coupling ##\lambda(\mu)##. If we have, for example, an UV fixed...
11. ### Einstein field equations and scale invariance

Hi, Are Einstein's field equations without the cosmological constant scale invariant? If so does the addition of the cosmological constant break the scale invariance? John
12. ### Are Random Walks with Different Step Sizes Identical in Brownian Motion Limit?

Consider a random walk (in any dimension) with N steps and a step size of 1. Take a real number \alpha > 0 and consider another random walk which takes \alpha^2 N steps but wil step size \frac{1}{\alpha}. I immediately noticed that the mean deviation after the full walk in both cases is the...
13. ### Why doesn't Diffeomorphism Invariance lead to Scale Invariance?

One of the foundations of General Relativity is diffeomorphism invariance - the fact that the laws of physics are invariant under smooth coordinate transformations, and thus the laws must involve tensors. My question is, why doesn't this imply scale invariance; after all, isn't a change of...
14. ### Scale invariance and bubble universes

Max Tegmark has provided a four part taxonomy of multiverse theories (http://arxiv.org/abs/astro-ph/0302131). The first type can be labeled the "bubble universe" multiverse, in which universes like ours are scattered throughout an infinite space in every direction. Going the other direction...
15. ### Scale Invariance in Global Terrorism

Clauset and Young, in their new paper http://www.arxiv.org/PS_cache/physics/pdf/0502/0502014.pdf , apply standard modern statistics to a database of terrorist strikes ordered by the number of people killed or injured. They find that extreme cases, such as 9/11, are not outliers, but find their...