Invariance Definition and 476 Threads

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. F

    I Diffeomorphism invariance of GR

    it is often stated in texts on general relativity that the theory is diffeomorphism invariant, i.e. if the universe is represented by a manifold ##\mathcal{M}## with metric ##g_{\mu\nu}## and matter fields ##\psi## and ##\phi:\mathcal{M}\rightarrow\mathcal{M}## is a diffeomorphism, then the sets...
  2. P

    I Time invariance of weak interaciton e-p->\nu n

    The question is rather simple, but I cannot seem to find a solid answer. I need the cross section of the following interaction: e^- + p\rightarrow n+ \nu . I need the cross section using the form factors. There are many solutions for the interactions like: n+ \nu\rightarrow e^- + p or...
  3. P

    Dirac Lagrangian invariance under chiral transformation

    Consider the Dirac Lagrangian, L =\overline{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi, where \overline{\psi}=\psi^{\dagger}\gamma^{0} , and show that, for \alpha\in\mathbb{R} and in the limit m\rightarrow0 , it is invariant under the chiral transformation...
  4. e2m2a

    Invariance of elastic potential energy

    At non-relativistic speeds is the elastic potential energy of a compressed spring frame-invariant? That is, would all reference frames agree on how much elastic potential energy is stored in the spring?
  5. A

    I String theory and Lorentz invariance - 10D vs. 4D....

    Hi all, Clarification question: I've read that string theory is manifestly Lorentz invariant - however, I'm confused about this being true in 4D spacetime or in the full 10D setting of the theory (well, one version anyway). At some point I'd also read in a paper that 4D Lorentz invariance...
  6. davidge

    B Conservation of Newtonian Force and the Invariance of Maxwell's Equations

    The electromagnetic wave equation being of the same form in all intertial frames is because Newton's force is a vector quantity? I mean, if the wave equation changes its form from a intertial frame to another one, would the electromagnetic force be different in the two frames? I know that one...
  7. Pushoam

    Galilean invariance: Newton's 2nd Law of motion

    Consider a frame S' moving with speed u along +ve x direction with respect to another frame S. Consider a body moving with speed v along +ve x direction with respect to frame S . Both frame are inertials. here,force acting in S frame on the body is $$ F\hat x=\frac {dp} {dt}\hat x,$$...
  8. S

    I Lorentz Invariance of the Lagrangian

    Hello! I started reading stuff on QFT and it seems that one of the main points is for the Lagrangian to be Lorentz invariant, so that the equations of motion remain the same in all inertial reference frames. I am not sure however i understand how do non inertial reference frames come into play...
  9. vanhees71

    I Tensor Invariance and Coordinate Variance

    <This thread is a spin-off from another discussion. Cp. https://www.physicsforums.com/threads/wedge-product.914621/#post-5762138> Also again, be warned about this sloppy notation of indizes. You should put the prime on the symbol (or in addition to the symbol). Otherwise the equations don't...
  10. F

    I What is the meaning of invariance for an equation f=0?

    Hey. When talking about invariance of a function f under some transformation T we mean that T(f)=f. But what is meant by invariance of an equation f=0? As far as I can see it makes sense to call an equation invariant when the transformed equation T(f)=T(0) is equivalent to the original equation...
  11. G

    Invariance under SU(2) in quantum mechanics

    Homework Statement Hi, I'm trying to self-study quantum mechanics, with a special interest for the group-theoretical aspect of it. I found in the internet some lecture notes from Professor Woit that I fouund interesting, so I decided to use them as my guide. Unfortunately I'm now stuck at a...
  12. Phys pilot

    Showing the invariance of Levi-Civita symbol in 4 dimensions

    Homework Statement i am showing the invariance of the Levi-Civita symbol in 4 dimensions The Attempt at a Solution $$\varepsilon_{ijkl}'=R_{im}R_{jn}R_{kp}R_{lt}\varepsilon_{mnpt}$$...
  13. S

    A Time-reparameterization invariance

    This post considers an aspect of time-reparametization invariance in classical Hamiltonian mechanics. Specifically, it concerns the use of Lagrange multipliers to rewrite the action for a classical system in a time-reparametization-invariant way...
  14. S

    A Diffeomorphism invariance and gauge invariance

    Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes (http://www.hartmanhep.net/topics2015/) on Quantum Gravity: In gravity, local diffeomorphisms are gauge symmetries. They are redundancies. This means that local correlation functions like ##\langle...
  15. S

    Can transformation coefficients be interchanged in symmetric tensors?

    Homework Statement The lecture notes states that if ##T_{ij}=T_{ji}## (symmetric tensor) in frame S, then ##T'_{ij}=T'_{ji}## in frame S'. The proof is shown as $$T'_{ij}=l_{ip}l_{jq}T_{pq}=l_{iq}l_{jp}T_{qp}=l_{jp}l_{iq}T_{pq}=T'_{ji}$$ where relabeling of p<->q was used in the second...
  16. F

    I Momentum cut-off regularisation & Lorentz invariance

    Why is it that introducing a hard cut-off ##p^{2}=\Lambda^{2}## breaks Lorentz invariance? Is it simply that it introduces an energy scale and energy is not a Lorentz invariant quantity? Sorry if this is a trivial question, but I just want to make sure I understand the reasoning as I've...
  17. M

    B GR vs quantum vacuum Lorentz invariance

    is spacetime Lorentz invariant like the quantum vacuum? They say the quantum vacuum is Lorentz invariant.. you can't locate it at any place.. but if spacetime manifold is also Lorentz invariant and you can't locate it at any place.. how come the Earth can curve the spacetime around the Earth...
  18. K

    A Adiabatic Invariance: Why Do Closed Orbits Remain Closed?

    How does the adiabatic invariance of actions J imply that closed orbits remain closed when the potential is deformed adiabatically? Is it because a closed orbit has commensurate angular frequencies $$\omega_i$$ defined by $$\omega_i = \frac{\partial H}{\partial J_i}$$ where H is the...
  19. M

    Invariance of length of curve under Euclidean Motion

    Homework Statement Show that the length of a curve γ in ℝn is invariant under euclidean motions. I.e., show that L[Aγ] = L[γ] for Ax = Rx + a Homework Equations The length of a curve is given by the arc-length formula: s(t) = ∫γ'(t)dt from t0 to tThe Attempt at a Solution I would imagine I...
  20. N

    Doesn't MWI violate Lorentz Invariance?

    Doesn't the Many Worlds Interpretation violate Lorentz symmetry when the universe splits?
  21. Kara386

    I Is Observable A Invariant Under Symmetry Generators G_i?

    How do I know if an observable is invariant, specifically under some set of transformations described via the generators ##G_i##? Which conditions would this observable have to fulfil?
  22. F

    A Is the Metric Tensor Invariant under Lorenz Transformations in M4?

    I'm stuck on an apparently obvious statement in special relativity, so I hope you can help me. Can I define Lorenz transformations as transformations that don't change the spacetime interval in M4 and from this deduct that the metric tensor is invariant under LT? I've always read that the...
  23. S

    A Gauge and Lorentz invariance for Lagrangians

    Consider the following Lagrangian: ##YHLN_{1}^{c} + Y^{c}H^{\dagger}L^{c}N_{1} + \text {h.c.},## where ##L=(N_{0}, E')## and ##L^{c} = (E^{'c}, N_{0}^{c})## are a pair of ##SU (2)## doublets and ##N_{1}## and ##N_{1}^{c}## are a pair of neutral Majorana fermions...
  24. A

    I Holographic principle, non-locality, and Lorentz invariance

    Hi all, Some recent comments from the forums here led me to do a bit of reading on the holographic principle, and to a posting on "The Reference Frame" by Lubos Moti about the (likely lack of) 'holographic noise' in the experiment by Craig Hogan at Fermilab...
  25. A

    I Vacuum energy cutoff and Lorentz invariance....

    One more question before Santa comes. There are a number of different related threads, so hopefully I'm not repeating this - however, I haven't found a crisp answer yet. If one introduces a UV cutoff in the vacuum energy (in an attempt to avoid having infinite vacuum energy), is it possible at...
  26. Narasoma

    I Lorentz invariance of quantum theory

    I read Lucien Hardy's paper whose tittle was "Quantum Mechanics, Local Realistic Theories, and Lorentz Invariant Relativistic Theories". There, he argued that lorentz invariant observables which involved locality assumption contradict quantum mechanics. I tried to follow his argument, but got...
  27. N

    I Magnetic Charges & Lorentz Invariance: Finding Papers

    How to find some papers on Lorentz invariant extensioning of standard electromagnetism that include magnetic charges
  28. ShayanJ

    A Fourier transform and translational invariance

    Can anyone explain what does the author mean by the statement below? page 27 of this paperI don't understand the relation between the Fourier transform and translational invariance. Thanks
  29. S

    I Gauge invariance of momentum of charged particle

    I know that, in the presence of a magnetic field, the momentum of a charge particle changes from ##p_{i}## to ##\pi_{i}\equiv p_{i}+eA_{i}##, where ##e## is the charge of the particle. I was wondering if this definition of momentum is gauge-invariant? How about ##\tilde{\pi}_{i}=p_{i}-eA_{i}##?
  30. TheSodesa

    A real parameter guaranteeing subspace invariance

    Homework Statement Let ##A## and ##B## be square matrices, such that ##AB = \alpha BA##. Investigate, with which value of ##\alpha \in \mathbb{R}## the subspace ##N(B)## is ##A##-invariant. Homework Equations If ##S## is a subspace and ##A \in \mathbb{C}^{n \times n}##, we define multiplying...
  31. S

    A Gauge invariance and covariant derivative

    Consider the covariant derivative ##D_{\mu}=\partial_{\mu}+ieA_{\mu}## of scalar QED. I understand that ##D_{\mu}\phi## is invariant under the simultaneous phase rotation ##\phi \rightarrow e^{i\Lambda}\phi## of the field ##\phi## and the gauge transformation ##A_{\mu}\rightarrow...
  32. S

    Invariance of direction of vorticity

    Homework Statement The vorticity vector ##\vec{\omega} = \text{curl}\ \vec{v}##, defined as usual by ##\omega^{2}=i_{{\vec{\omega}}}\text{vol}^{3}##, is ##\textit{not}## usually invariant since the flow need not conserve the volume form. The mass form, ##\rho\ \text{vol}^{3}##, however...
  33. binbagsss

    I Solving SR Invariance: Minkowski Metric, Poincare Transformation, Index Notation

    I am following some lecture notes looking at the invariance of Poincare transformation acting on flat space-time with the minkowski metric: ##x'^{u} = \Lambda ^{u}## ##_{a} x^{a} + a^{u} ## [1], where ##a^{u}## is a constant vector and ##\Lambda^{uv}## is such that it leaves the minkowski...
  34. P

    Lorentz Invariance of Plane Wavefront

    Homework Statement For a plane, monochromatic wave, define the width of a wavefront to be the distance between two points on a given wavefront at a given instant in time in some reference frame. Show that this width is the same in all frames using 4-vectors and in-variants. Homework...
  35. W

    Time Invariance of f_n for Quasi-Linear PDE Boundary Conditions

    Homework Statement let y(x, t) be a solution to the quasi-linear PDE \frac{\partial y}{\partial t} + y\frac{\partial y}{\partial x} = 0 with the boundary condition y(0, t) = y(1, t) = 0 show that f_n(t) = \int_0^1 y^n\,\mathrm{d}x is time invariant for all n = 1, 2, 3,... Homework EquationsThe...
  36. D

    Time Invariance of System with x(t) and y(t) Equations

    Okay so the question looks like this Determine whether the system with input x(t) and output y(t) defined by each of the following equations is time invariant: (c) y(t) =∫t+1t x(τ−α)dt where α is a constant; (e) y(t) = x(−t); There are more sub-questions but I was able to solve them. The reason...
  37. W

    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...
  38. B

    I What causes confusion in testing time invariance?

    I understand what time invariance means but there are a few catches that I'm completely confused about: Suppose we have $$y(t)=x(\alpha t-\beta)$$ to test time invariance we shift the input then "plug" it into the output:$$x_1(t_1)=x(t-t_o)$$ so this is when I become confused; when we plug...
  39. nmsurobert

    How do I prove Lorentz Invariance using 4-vectors?

    Homework Statement I'm asked to prove that Et - p⋅r = E't' - p'⋅r' Homework Equations t = γ (t' + ux') x = γ (x' + ut') y = y' z = z' E = γ (E' + up'x) px = γ (p'x + uE') py = p'y pz = p'z The Attempt at a Solution Im still trying to figure out 4 vectors. I get close to the solution but I...
  40. S

    I Local Gauge Invariance Explained: Physics & Math Insight

    Hello! Can someone explain to me what exactly a local gauge invariance is? I am reading my first particle physics book and it seems that putting this local gauge invariance to different lagrangians you obtain most of the standard model. The math makes sense to me, I just don't see what is the...
  41. K

    I Has Large Hadron Collider shown Higgs conformal invariance

    the higgs naturalness problems has several solutions 1- natural susy 2- technicolor 3- extra dimensions 4- conformal invariance Large Hadron Collider has to date strongly disfavored susy, technicolor, extra dimensions. it is highly unlikely susy is the answer to the higgs hierarchy problem...
  42. G

    A Fock space and Poincaré invariance

    Hi all, is Fock space Poincaré invariant? As far as I can see, the scalar product in Fock space involves the scalar products in its N-particle subspaces, which, in turn, are the integrals of the properly (anti-)symmetrized wave functions over space. This works well in a Galilei-invariant...
  43. Thor90

    A Proof of Local Lorentz Invariance of Feynman Propagator in Curved Spaces

    I am looking for a proof that the Feynman propagator is locally a lorentz invariant (at least for scalar fields) also in curved space-times if the background geometry is smooth enough. I mean, since it is of course a lorentz invariant on flat spaces, this should also be true if a choose a...
  44. G

    I Relativistic Momentum Invariance in Perpendicular Boosts

    Hi, I've found a derivation of the formula for the relativistic momentum where they considered a car crashing into a wall in the system of the car and in an inertial system that moves parallel to the wall (and therefore perpendicularly to the movement of the car). They argue that since both...
  45. powerof

    I Curl from requiring invariance under inertial coordinate changes

    While investigating about the curl I have found this interesting perspective: http://mathoverflow.net/a/21908/69479 I lack the knowledge to do the derivation on my own so I would like to ask for your help. I am an undergraduate. I do not understand what a "first order differential operator"...
  46. F

    A Invariance of Wess Zumino Action under SUSY

    Hi guys, I have a very basic question about the WZ model. I want to show that it is invariant under SUSY transformations. The action is \int{d^4 x} \partial^\mu \phi* \partial_\mu \phi +i\psi^† \bar{\sigma}^\mu \partial_\mu \psi The SUSY transformations are \delta\phi = \epsilon \psi ...
  47. F

    Prove Lorentz invariance for momentum 4-vector

    Homework Statement I am meant to show that the following equation is manifestly Lorentz invariant: $$\frac{dp^{\mu}}{d\tau}=\frac{q}{mc}F^{\mu\nu}p_{\nu}$$ Homework Equations I am given that ##F^{\mu\nu}## is a tensor of rank two. The Attempt at a Solution I was thinking about doing a Lorents...
  48. Y

    B About the Lorentz invariance of Planck constant

    Is it proved experimentally that the Planck constant is invariant in the moving system? If that experiment exists, would you show me that in detail?
  49. S

    A Proof of Lorentz invariance of Klein-Gordon equation

    I would like to prove the Lorentz invariance of the Klein-Gordon equation by proving the invariance of the action ##\mathcal{S} = \int d^{4}x\ \mathcal{L}_{KG}## under a Lorentz tranformation. I would like to do this by first proving the Lorentz invariance of the ##\mathcal{L}_{KG}## and then...
  50. S

    Lorentz invariance of Klein-Gordon eqn & Maxwell Lagrangian

    Homework Statement 1. Show directly that if ##\varphi(x)## satisfies the Klein-Gordon equation, then ##\varphi(\Lambda^{-1}x)## also satisfies this equation for any Lorentz transformation ##\Lambda##. 2. Show that ##\mathcal{L}_{Maxwell}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}## is invariant under...
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