What is Invariance: Definition and 475 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.

View More On Wikipedia.org
  1. cianfa72

    I Time dilation vs Differential aging vs Redshift

    Hi, I would like to ask for a clarification about the terms time dilation vs differential aging vs gravitational redshit. As far as I can tell, time dilation is nothing but the rate of change of an object's proper time ##\tau## w.r.t. the coordinate time ##t## of a given coordinate chart (aka...
  2. Thytanium

    I Invariance of a tensor of order 2

    Good morning friends of the Forum. For me it is difficult to geometrically imagine a tensor of order 2 and maybe that is why it is difficult for me to know, what remains invariant when making a change of coordinates of this tensor. The only thing I can think of it, is that since a tensor of...
  3. M

    I Alternative Ways to Realize Invariance: Lorentz Transformation

    The Lorentz transformation ensures different inertial observers measure the same speed of light. Are there other transformations, or other ways to setup a "space-time" that also have this property of invariance? Is the Lorentz transformation the unique solution?
  4. G

    A Virial theorem and translational invariance

    According to the virial theorem, $$\left\langle T\right\rangle =-{\frac {1}{2}}\,\sum _{k=1}^{N}{\bigl \langle }\mathbf {F} _{k}\cdot \mathbf {r} _{k}{\bigr \rangle }$$ where ##N## is the number of particles in the system and ##T## is the total kinetic energy. It is often claimed that this...
  5. Vanilla Gorilla

    B Solving for the Nth divergence in any coordinate system

    Preface We know that, in Cartesian Coordinates, $$\nabla f= \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}$$ and $$\nabla^2 f= \frac{\partial^2 f}{\partial^2 x} + \frac{\partial^2 f}{\partial^2 y} + \frac{\partial^2 f}{\partial^2 z}$$ Generalizing...
  6. S

    I Solutions that break the Lorentz invariance...?

    I was reading a discussion where some physicists participated* where the topic of Lorentz invariance violations occurring in cosmology is mentioned. There, they mention that we can imagine a Lorentz-violating solution to the cosmological equations. What do they mean by that? Can anyone specify...
  7. Z

    I The invariance of the speed of light is not only a hypothesis?

    hello Einstein assumed the invariance of the speed of light as an hyphotesis, while I was told that : "The speed of light need not have been postulated as an invariant." in other words the invariance of the speed of light could have been proven even regardless of the special relativity is it...
  8. E

    B Problem about postulate of the invariance of the speed of light

    Hello, I have a problem with the postulate of the invariance of the speed of light. When we move away from a light source it is redshift, it is the sign that the relative velocity between us and the light source has changed. If a stationary observer observes the phenomenon, he will measure that...
  9. Rafaelmado

    A Symmetry & Invariance of Pions: π+, π0, π− and Other Mesons Explained

    Pions are particles with spin 0 and they form an isospin triplet: π+, π0, π− (with the superscript indicating the electric charge). Their intrinsic parity is −1 and they are pseudoscalar mesons. In nature we also find other kind of mesons, like the ρ mesons, ρ+, ρ0 and ρ−. As pions, they also...
  10. P

    MHB Invariance of Asymmetry under Orthogonal Transformation

    Show that the property of asymmetry is invariant under orthogonal similarity transformation
  11. Tertius

    A Local phase invariance of complex scalar field in curved spacetime

    I am stuck deriving the gauge field produced in curved spacetime for a complex scalar field. If the underlying spacetime changes, I would assume it would change the normal Lagrangian and the gauge field in the same way, so at first guess I would say the gauge field remains unchanged. If there...
  12. phyz2

    I Klein Gordon Invariance in General Relativity

    Hello! I'm starting to study curved QFT and am slightly confused about the invariance of the Klein Gordon Lagrangian under a linear diffeomorphism. This is $$L=\sqrt{-g}\left(g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi-\frac{m^2}{2}\phi^2\right),$$ I don't see how ##g^{\mu\nu}\to...
  13. ergospherical

    I Lorentz Invariance of Q in Weinberg: Justifying Transformation

    If ##\partial_{\alpha} J^{\alpha}(x) = 0## then ##Q \equiv \displaystyle{\int} d^3 x J^t(x)## is time-invariant. To show that if ##J^{\alpha}(x)## is a four-vector then ##Q## is also Lorentz-invariant, he re-writes it as \begin{align*} Q = \int d^4 x J^{\alpha}(x) \partial_{\alpha} H(n_{\beta}...
  14. K

    Find the Conserved Quantity of a Lagrangian Using Noether's Theorem

    So Noether's Theorem states that for any invarience that there is an associated conserved quantity being: $$ \frac {\partial L}{\partial \dot{Q}} \frac {\partial Q}{\partial s}$$ Let $$ X \to sx $$ $$\frac {\partial Q}{\partial s} = \frac {\partial X}{\partial s} = \frac {\partial...
  15. F

    Charge invariance with Heaviside's function

    I followed a demonstration in one of my electromagnetism books, but it is not clear to me. My problem is at the starting point. The book begins by considering the office defined in the following way: $$Q=\int d^4xJ^\alpha(x)\partial_\alpha\theta(\eta_\beta x^\beta)$$ where...
  16. U

    I Why is Scalar Massless Wave Equation Conformally Invariant?

    It can be shown mathematically that the scalar massless wave equation is conformally invariant. However, doing so is rather tedious and muted in terms of physical understanding. As such, is there a physically intuitive explanation as to why the scalar massless wave equation is conformally invariant?
  17. Demystifier

    A Gauge invariance confusions: symmetry vs redundancy, active vs passive

    Symmetry transformations in physics can be either passive or active. Symmetries in field theory can be either global or local. But only the local ones, the so called gauge symmetries, are fundamental. Except that local transformations cannot be active (despite the fact that diffeomorphisms are...
  18. haushofer

    I Diffeomorphism Invariance, Passive/Active Interpretations GR Insight

    Dear all, in my current week of holidays, where all the Corona-dust settles down a bit, I came across some personal notes I made a while ago about the meaning of diffeomorphism invariance, the difference between passive and active coordinate transformations, and the notion of background...
  19. G

    B Light Speed Invariance: Experiments, Difficulties & Clarification

    Let me clarify my question, is there any experiment directly proved the invariance of light speed to observers? Let's not get to the argument of equivalence between source and observer. SR was based on the postulate that the light speed is constant and independent of both the motions of source...
  20. agnimusayoti

    Invariance of Energy Momentum Relativistic

    I try to use relativistic energy equation: $$E'=\gamma mc^2$$ But, I use $$\gamma=\frac{1}{\sqrt{(1-(\frac{v'}{c})^2}}$$ then I use Lorentz velocity transformation. $$v'=\frac{v-u}{1-\frac{uv}{c^2}}$$ At the end, I end up with messy equation for E' but still have light speed c in the terms. How...
  21. docnet

    Prove the rotational invariance of the Laplace operator

    Hello, please lend me your wisdom. ##\Delta u=\partial_{x1}^2u+\partial_{x2}^2u+...+\partial_{xn}^2u## ##Rx=\left<r_{11}x_1+...r_{1n}x_n+...+r_{n1}x_1+...+r_{nn}x_n\right>## ##(\Delta u)(Rx)=(\partial_{x1}^2u+\partial_{x2}^2u+...+\partial_{xn}^2u)\left<r_{11}x_1+...r_{1n}x_n...

    Lorentz and Gauge invariance of EM

    I have been reading the book of Chris Quigg, Gauge theories, Chapter 3, sec 3.3 in which he explains how local rotations transform wave function and variations in Schrodinger equation forces us to introduce the electromagnetic interaction between the particles. I need a bit deep concept of the...
  23. mcas

    Check invariance under rotation group in spacetime

    I started by inserting ##ds=\sqrt{dx'^{\mu} dx'_{\mu}}## and ##p'^{\mu}=mc \frac{dx'^{\mu}}{ds}##. So we have: $$\frac{dp'^{\mu}}{ds}=mc \frac{d}{dx'^{\mu}} \frac{d}{dx'_{\mu}} (x'^{\mu})$$ Now I know that ##dx'^{\mu}=C_\beta \ ^\mu dx^\beta## and ##dx'_{\mu}=C^\gamma \ _\mu dx_\gamma## where...
  24. J

    I Spacetime invariance algebraic proof

    In Phillip Harris' (U. Sussex) post on special relativity he includes on p. 45 an algebraic proof of invariance of spacetime intervals. He starts with the definition S^2 =c^t^2 - x^2 -y^2 -z^2, he inserts the Lorentz transform expressions fot t and x, and he does some algebra to show that one...
  25. JD_PM

    Showing Feynman-amplitudes' gauge invariance (for Compton Scattering)

    Show that the Feynman amplitude for Compton scattering ##\mathcal{M} = \mathcal{M}_a + \mathcal{M}_b## is gauge invariant while the individual contributions ##\mathcal{M}_a## and ##\mathcal{M}_b## are not, by considering the gauge transformations $$\varepsilon^{\mu} (\vec k_i) \rightarrow...
  26. LucaC

    A Invariance of ##SO(3)## Lie group when expressed via Euler angles

    I am trying to understand the properties of the ##SO(3)## Lie Group but when expressed via Euler angles instead of rotation matrix or quaternions. I am building an Invariant Extended Kalman Filter (IEKF), which exploits the invariance property of ##SO(3)## dynamics ##\mathbf{\dot{R}} =...
  27. F

    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...
  28. Filip Larsen

    I Rotational invariance of cross product matrix operator

    Given that the normal vector cross product is rotational invariant, that is $$\mathbf R(a\times b) = (\mathbf R a)\times(\mathbf R b),$$ where ##a, b \in \mathbb{R}^3## are two arbitrary (column) vectors and ##\mathbf R## is a 3x3 rotation matrix, and given the cross product matrix operator...
  29. cianfa72

    I Does Lorentz invariance imply Einstein's synchronization convention?

    Hi, I've read a number of posts here on PF about Einstein's clock synchronization convention. In the context of SR we know the transformation law between inertial frame's coordinates is actually the Lorentz one. The invariant speed for Lorentz transformation is c (actually it coincides with...
  30. P

    A Gauge Invariance of Transverse Traceless Perturbation in Linearized Gravity

    In linearized gravity we define the spatial traceless part of our perturbation ##h^{TT}_{ij}##. For some reason this part of the perturbation should be gauge invariant under the transformation $$h^{TT}_{ij} \rightarrow h^{TT}_{ij} - \partial_{i}\xi_{j} - \partial_{j}\xi_{i}$$ Which means that...
  31. S

    I A query regarding Rotational Invariance

    We know that Bell States follow the Rotational Invariance property i.e. the probability of results on measurement of two entangled particles do not change if the initial measurement basis (say ##u##) is rotated by an angle θ to a new basis (to say ##v##). Lets take the Bell State ##\psi = \frac...
  32. Q

    A Invariance of discrete Spectrum with respect a Darboux transformation

    According to this this the Darboux transformation preserves the discrete spectrum of the Haniltonian in quantum mechanics. Is there a proof for this? My best guess is that it has to do with the fact that $$Q^{\pm}$$ are ladder operators but I'm not sure.
  33. Math Amateur

    I Translation Invariance of Outer Measure .... Axler, Result 2.7 ....

    I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ... I need help with the proof of Result 2.7 ... Result 2.7 and its proof read as follows: In the above proof by Axler we read the following: " ... ... Thus ... ##\mid t +...
  34. Math Amateur

    MHB Translation Invariance of Outer Measure .... Axler, Result 2.7 ....

    I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ... I need help with the proof of Result 2.7 ... Result 2.7 and its proof read as follows: In the above proof by Axler we read the following: " ... ... Thus ... $\mid t + A...
  35. S

    I Lorentz Invariance Violation for Manifolds

    I was looking at this video , and I was wondering if a (Riemannian)manifold violates the "lorentz invariance" would it become a discrete manifold?
  36. sophiatev

    I Invariance of a system under symmetry operations

    I'm trying to understand the precise reason we claim that a value being conserved means that the system in question is invariant under the corresponding symmetry transformation. Take parity for example. If the parity operator satisfies the commutation relation ##[P, H] = 0## for a given...
  37. hyksos

    A What Fundamental Invariance Principle is Violated by FSC Anisotropy?

    If the Fine-structure constant was measured to deviate in the sky , and this deviation was directional, which fundamental invariance principle would be violated? Quasar survey by VLT has observed deviations in the FSC that appear to be locked against directions of distant galaxies...
  38. Luke Tan

    I Invariance of the Poisson Bracket

    I've recently been starting to get really confused with the meaning of equality in multivariable calculus in general. When we say that the poisson bracket is invariant under a canonical transformation ##q, p \rightarrow Q,P##, what does it actually mean? If the poisson bracket ##[u,v]_{q,p}##...
  39. G

    Checking Parity Invariance of the QED Lagrangian

    Hi, I'm trying to check that the QED Lagrangian $$\mathscr{L}=\bar{\psi}\left(i\!\!\not{\!\partial}-m\right)\psi - \frac{1}{4}F^{\mu\nu}F_{\mu\nu} - J^\mu A_\mu$$ is parity invariant, I'm using the general transformations under parity given by $$\psi(x) \rightarrow...
  40. sophiatev

    I Invariance of Diff. Line Elems. in Hartle's Gravity: Intro to GR

    In Hartle's book Gravity: An Introduction to Einstein's General Relativity he spends chapter 2 discussing some basic aspects of differential geometry. For example, he derives the expression for a differential line element in 2D Euclidean space: dS^2 = (dx)^2 + (dy)^2 in Cartesian coordinates...
  41. JD_PM

    Show the invariance of the complex-scalar-field Lagrangian

    a) Alright, I think that the trick here is to consider ##\phi^{\dagger}## and ##\phi## as independent scalar fields. I've read that the unitary matrices read as follows $$U = e^{i \epsilon}$$ Thus here we have to consider two separate transformations $$\phi \rightarrow \phi' = e^{i...
  42. dRic2

    Invariance of ##\epsilon^{\mu \nu \alpha \beta}##

    Hi, I'm reading some introductory notes about SR and I'm completely stuck at this problem. I imagine I should consider a transformation ##L## such that $$ \hat \epsilon^{\mu \nu \alpha \beta} = L^{\mu}_{\delta}L^{\nu}_{\gamma}L^{\alpha}_{\theta}L^{\beta}_{\psi} \hat \epsilon^{\delta \gamma...
  43. G

    A Lorentz Invariance of Lagrangian: Proof & Explanation

    Last day in class, a professor told us that, for a Lagrangian to be Lorentz Invariant, the Lagrangian density cannot have second or higher derivatives. Is this true? Because one can write the KG lagrangian as $$\mathscr{L}=\phi(\square + m^2)\phi,$$ which have second derivatives. And, where...
  44. M

    Conserved quantity due to invariance under temporal rotations

    Hi, I was looking at the so(3,3) Lie algebra which has 3 temporal rotation generators as well as the normal 3 spatial rotation generators. When I try to use Noether's Theorem to determine what the conserved quantity is, due to invariance under temporal rotations, I seem to get an integral where...
  45. S

    Invariance of a spin singlet under rotation

    I have tried doing the obvious thing and multiplied the vectors and matrices, but I don't see a way to rearrange my result to resemble the initial state again: ##(\mathcal{D_{1y}(\alpha)} \otimes \mathcal{D_{2y}(\alpha)} )|\text{singlet}\rangle = \frac{1}{\sqrt{2}}\left[ \begin{pmatrix}...
  46. M

    B How does the Gallilean Invariance Puzzle challenge our understanding of motion?

    Gallilean Invariance states that the laws of motion are the same in all inertial frames. One experiment involved being on a ship below deck with no frame of motion reference. Supposedly, there is no experiment which could show whether the ship is moving or in what direction or speed. I was...
  47. T

    Invariance of the Lorentz transform

    of course y and z terms are invariant but for the x and t terms I am getting an additional factor of 1/1-v^2/c^2
  48. Diracobama2181

    A Gauge Invariance of the Schrodinger Equation

    Given the schrodinger equation of the form $$-i\hbar\frac{\partial \psi}{\partial t}=-\frac{1}{2m}(-i\hbar \nabla -\frac{q}{c}A)^2+q\phi$$ I can plug in the transformations $$A'=A-\nabla \lambda$$ , $$\phi'=\phi-\frac{\partial \lambda}{\partial t}$$, $$\psi'=e^{-\frac{iq\lambda}{\hbar c}}\psi$$...