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. tom.stoer

    Breaking of Lorentz invariance

    Today one tries to find indications for quantum gravity indirectly via low-energy effects induced by "foamy" or "discrete" structures replacing space-time at the Planck regime. It is by no means clear whether and how such discrete structures necessarily indice Lorentz symmery breaking, neither...
  2. A

    Jordan-Brouwer Separation Theorem and Invariance of Domain

    So I'm having trouble understanding how these two are related, i.e., how one proves the other. I understand the ideas behind both of them: For J-B, you're basically taking R^n and throwing in a sphere, so the inside of the sphere is bounded and everything outside the sphere is unbounded. For...
  3. S

    Galilean invariance of Maxwell equation

    always say us Maxwell equations are not covariance under Galilean Transformation They say merely this because of constant speed of light that the result of Maxwell Equations But there arent any excitability prove for Non-Ggalilean invariance of Maxwell equation I Decided try to show...
  4. M

    Spacetime Inverval Invariance using Lorentz Transformations

    Homework Statement Prove that the spacetime interval -(ct)^{2} + x^{2} + y^{2} + z^{2} is invariant. [/itex] Homework Equations Lorentz transformations \Deltax' = \gamma(\Deltax-u\Deltat) \Deltay' = \Deltay \Deltaz' = \Deltaz \Deltat' = \gamma(\Deltat-u\Deltax/c^{2}) The...
  5. P

    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...
  6. K

    An detail in proving the homotopy invariance of homology

    I'm reading Allen Hatcher's topology book.In order to prove a theorem about homotopic maps induce the same homomorphism of homology groups,given a homotopy F:X \times I \to Y from f to g,the author construct a prism operators P:C_n (X) \to C_{n + 1} (Y) by P(\sigma ) = \sum\nolimits_i {( - 1)^i...
  7. T

    Galilean invariance verification (kinetic energy and momentum)

    Hi all, First of all, sorry for not using the template, but I think in this situation it's better to explain my problem right away: I'm studying for a physics test, but I think I don't really understand Galilean invariance. In my textbook there is an example in which they proof that if you...
  8. S

    Verifying Galilean Invariance of the KdV Equation

    Homework Statement Show that the KdV has Galilean invariance. That is ut + 6uux + uxxx = 0 is invariant under the transformation xi = x - ct, tau = t, psi = phi - c/6 Homework Equations The Attempt at a Solution Do we use the chain rule on these and plug into the KdV?
  9. T

    Proving M_{0i} = 0 in Special Relativity

    http://books.google.com/books?id=qhDFuWbLlgQC&lpg=PP1&pg=PA11#v=onepage&q&f=false" Until he arrives at eq. 1.5 I don't understand the steps, can anyone explain it? thanks
  10. A

    Time invariance of Schrodinger equation

    The Schrodinger equation is linear in time. I was wondering if that means that is not invariant under time reversal. That would be a surprise because all other microscopic laws (Maxwell's equations, Newton's equations) are time invariant. Can you please clear this doubt?
  11. U

    Modifying Lorentz Transformations for Invariant Length

    I've been reading up on the idea of an invariant length (planck length possibly) and have tried my hand at modifying the lorentz transformations of SR such that they accommodate an invariant length. Unfortunately I haven't been able to find a way to do it. Does anyone know how one would go about...
  12. S

    Invariance of scalar products on Lie algebras

    Hi folks, If I have a Lie algebra \mathfrak{g} with an invariant (under the adjoint action ad of the Lie algebra) scalar product, what are the conditions that this scalar product is also invariant under the adjoint action Ad of the group? For instance, the Killing form is invariant under...
  13. S

    Non-abelian Gauge invariance (chapter 15.1 in Peskin/Schroeder)

    I am trying to understand the derivation of the covariant derivative in Peskin/Schroeder (chapter 15.1, page 483). This is the important stuff: n^\mu\partial_\mu\psi=\lim_{\epsilon \rightarrow 0} \frac{1}{\epsilon}\left[\psi(x+\epsilon n)-\psi(x)\right] Scalar quantity: U(y,x): U(y,x)...
  14. Q

    Phase invariance of an EM wave in special relativity

    Homework Statement So I'm trying to show for a specific, given EM plane wave in vacuum that kx - \omega t = k' x' - \omega' t' but I'm running into some difficulties. I'm hoping someone can show me where I'm going wrong. Here's the setup: In the lab frame K, a plane EM wave traveling in...
  15. O

    Understanding Positive invariance

    Hello everyone, I am new to the forum and was wondering if someone can help explain something to me. I would like to understand the meaning of a positivly invariant system. I have checked the definition on wikipedia, http://en.wikipedia.org/wiki/Positive_invariance" and it is...
  16. R

    What If Lorentz Invariance Violation Were Detected?

    What if someday we would have news that Lorentz Invariance Violation was detected? Is this possible at all? But our Special Relativity is based on Lorentz Invariance and the more general General Covariance in General Relativity. Does this mean that Lorentz Invariance violation is almost...
  17. R

    Diffeomorphism Invariance in GR

    Does anyone know of any website that has animations of what this Diffeomorphism Invariance in General Relativity can do? I read a lot of articles about it but can't seem to get the essence or visualize how it actually occurs exactly. Thanks.
  18. R

    Lorentz Invariance as local limit of Bigger Manifold

    Is it possible that Lorentz invariance is just a lower limit of a larger manifold that has a priveleged frame? Even if Bell's experiments can't transmit signal faster than light. The spirit of relativity is still violated by say instantaneous correlation between 10 billion light years. As...
  19. A

    Interaction energy and gauge invariance

    Hi everybody, i have a question concerning potential energy (in all its forms, which basically means all forms of energy except the kinetic one). The kinetic energy of a system is always well defined: in the rest frame it is m² (convention c=1), in a frame moving at a relative speed v compared...
  20. H

    Lorentz invariance of wave eqn.

    Hello! Hopefully somebody could give me a push from behind on this one :) Homework Statement Show that the classical wave equation is lorentz invariant. The Attempt at a Solution I tried to exchange all derivatives by the chain rule: (c^2 \frac{d^2 }{dt^2} + \frac{d^2 }{dx^2} + \frac{d^2...
  21. A

    Conditions for translational invariance in space/time for a 2-point function?

    Hi, Assume the following action: \int d^4 x L[\phi,A]+ \int d^4 x A_{\mu} (x) J^{\mu}(x) What are the conditions on the form of action to have space/time translational invariance for a two point function: \left\langle J_{\mu}(x) J_{\nu}(y) \right\rangle = G_{\mu \nu}(x-y)...
  22. P

    Modular invariance in string theory

    Is it proved that the bosonic string and superstring partition functions are modular-invariant for arbitrarily high loop order? If not, how many loops have been analyzed?
  23. C

    Is the QED Action Invariant Under Gauge Transformation?

    Hello, I don't understand two steps in solution to the problem: I. Homework Statement Show that QED action is invariant under gauge transformation. II. Relevant equations QED action: S= \int{d^{4} x \left[\overline{\Psi}\left(i\gamma^{\mu} D_{\mu} -m \right)\Psi...
  24. TrickyDicky

    Does GR imply small Lorentz violations in practice?

    How is Lorentz invariance handled in GR? I know that there is no global Lorentz invariance in GR, instead it only holds locally, meaning that it is obeyed in the limit at infinity:when r goes to infinity by considering infinite distance or infinitely small point mathematical objects. But when...
  25. S

    Time-reversal invariance and irreversibilities

    We know basic classical mechanics is time-reversal invariant while there is a concept of irreversibility in thermodynamics. Is there a simple (by which I mean undergrad level and more preferably lower undergrad level) explanation for this apparent paradox? Someone please either explain this or...
  26. R

    Thread on Lorentz Invariance Violation

    Yesterday there was a thread here on a claimed violation of Lorentz invariance, but I can't locate it today. Was the thread moved? Can someone point me to its new location? (I don't remember the exact title of the thread, but the posts referred to a letter in the Sep 2010 issue of European...
  27. G

    Contradiction in Taylor-Wheeler's example of spacetime invariance

    In section 1.2 of Taylor and Wheeler's Spacetime Physics, a rocket moves past a laboratory (on Earth). Attached to the rocket is a pin. From that pin a spark is emitted at two locations in the lab, separated by 2 meters. The observer in the rocket measures the elapsed time between the sparks, as...
  28. andrewkirk

    Gauge invariance of Euler-Lagrange equations

    I have been trying to teach myself Lagrangian mechanics from a textbook “Lagrangian and Hamiltonian Mechanics” by MC Calkin. It has covered virtual displacements, generalised coordinates, d’Alembert’s principle, the definition of the Lagrangian, the Euler-Lagrange differential equation and how...
  29. I

    About Gauge invariance - again

    First of all, let me remind about an older thread on this topic: https://www.physicsforums.com/showthread.php?t=330517 Here I'd like to thank again to everybody, who participated in that discussion. However, I still find myself at a deadlock with some questions about Gauge Invariance (GI)...
  30. A

    Is there a connection between covariance, invariance, and dark matter?

    Covariance and Invariance We consider the equation: {\frac {{d}^{2} {x^{\alpha}}}{{d }{{\tau}^{2}}}}{=}{-}{{\Gamma}^{\alpha}}_{\beta\gamma}{\frac{{d}{x^{\beta}}}{{d}{\tau}}}{\frac{{d}{x^{\gamma}}}{{d}{\tau}}} The covariant form is preserved in all coordinate systems. But the Christoffel...
  31. M

    Lorentz invariance and General invariance

    Hi I am confused about these two related but different terms Lorentz invariance/covariance and General invariance/covariance As I understand it a Lorentz invariant is a scalar which is the same in all inertial reference frames i.e. it acts trivially under a Lorentz transformation an example...
  32. S

    Gauge Invariance and the Photon Self-Energy Correction

    Short intro.: I'm a 2nd year M.Sc. student in particle physics, with basic quantum field theory and knowledge of the SM and perhaps a bit more. I've read the forums before and tried to find questions/answers that were similar to my own until I decided, "why not just join so I can ask exactly...
  33. Q

    Linearity, time invariance, causality

    Homework Statement For each of the following systems, determine whether or not the system is linear, time-invariant, and causal. a) y[n] = x[n]cos(0.2*PI*n) b) y[n] = x[n] - x[n-1] c) y[n] = |x[n]| d) y[n] = Ax[n] + B, where A & B are constants. Homework Equations The Attempt...
  34. R

    Rotational Invariance: Bosons vs Fermions

    hi, is it correct to say that any particle or object that is invariant under rotation of 2 pi is a boson, whereas fermions need 4 pi? what is the accurate statement about this? thank you for your reply
  35. G

    The invariance of Lagrange's equations with a given time

    The invariance of Lagrange's equations with a given "time" Homework Statement What is the change in the Lagrangian in order that the Lagrangian equations of motion retain their form under the transformation to new coordinates and "time" give by: q = q(Q, \tau) t = t(Q, \tau)Homework Equations...
  36. E

    Negative squares using the space time interval invariance

    Hallo I'm new to this (wonderful) forum, and to SR too... I've a general question about the space time interval invariance. Say we have two points A and B, at rest each other, at distance AB. Now A and B simultaneously in their reference frame emit a flash of light. The space time interval...
  37. W

    Is Elastic Collision Invariance Predictable Across All Inertial Frames?

    Homework Statement A collision between two particles in which kinetic energy is conserved is described as elastic. Show, using the Galilean velocity transformation equations, that if a collision is found to be elastic in one inertial reference frame, it will also be found to be elastic in all...
  38. W

    Conservation of energy predict invariance of elastic collisions?

    If one observer in an inertial reference measures a collision to be elastic, then all observers in an inertial reference frame will measure the collision to be elastic - can this be explained with the conservation of energy? What exactly does the conservation of energy principle say in regards...
  39. C

    A question about Lorentz invariance for Klein-Gordon field

    Homework Statement Hi everyone, in Peskin & Schroeder, P36, the derivative part of KG field is transformed as eqn (3.3). But why does the partial derivative itself not transform? Homework Equations \partial_{\mu} \phi (x) \rightarrow \partial_{\mu} ( \phi ( \Lambda^{-1} x) ) = (...
  40. C

    Diffeomorphism invariance and Noether's theorem

    I've read that GR is diffeomorphism invariant, I asked a math buddy of mine and I have a VERY BASIC idea of what that means in this case - the theory is the same regardless of your choice of coordinates? Noether's theorem states that for every symmetry there's a corresponding conservation...
  41. N

    Can a theory have local Lorentz invariance but not diffeo invariance?

    This is related to the thread on the meaning of diffeomorphism invariance but is adressing a distinct point (at least I think so, but I may be proven wrong). As Rovelli discusses in his book, the action of the Standard Model coupled to gravity has three types of invariance: under the gauge...
  42. M

    Is the Lorentz force invariant under Lorentz transformations?

    I am trying to establish whether the force defined by the Lorentz equation below is invariant under the Lorentz transforms: [1] F = F_E + F_B = qE + qvB In the context of this equation, [q] is moving with velocity [v] such that it is acted on by both an electric E-force and magnetic...
  43. N

    Short question about diffeomorphism invariance

    I am posting my question in this forum because it is about a basic conceptual aspect of LQG discussed in Rovelli's book Quantum Gravity. He makes the following statement on page 67 (here, "e" refers to the vierbein): I do not understand the part in boldface. First, he means that the...
  44. 3

    Interaction scalar field invariance

    The problem: http://www.gobigbang.nl/cft/cft.jpg An attempt of the solution: http://www.gobigbang.nl/cft/DSCN2722.JPG My problem is question c. I don't have a clue how to see that the improved current is conserved... Can anyone help me? Update I managed to solve the question by using the...
  45. N

    Larmor radiation formula invariance

    Hi, I have been taking a classical electrodynamics course, in which we established the classical well-known larmor formula for the radiation of a classically accelerated point charge in vacuum. Then, since the radiated power is a Lorentz invariant, we just assumed that the correct...
  46. A

    Proof of Area Invariance of Closed Curve

    Hello! Quite some time ago I'd asked for help with a proof that proves that area of a closed curve is invariant i.e : its independent of the way it is spliced into. Say we splice a closed curve into one set of rectangles with parallel sides and we then splice an identical curve with...
  47. R

    Coulomb Gauge, Lorentz Invariance & Photon Polarization in Field Theory

    In electrodynamics, the Coulomb gauge is specified by \nabla \cdot A=0 , i.e., the 3-divergence of the 3-vector potential is zero. This condition is not Lorentz invariant, so my first question is how can something that is not Lorentz invariant be allowed in the laws of physics? My second...
  48. M

    Invariance of spacetime interval

    I've tried proving the invariance of the spacetime interval from Lorentz transformations 3 times now, but every time I end up with two extra terms that don't cancel! Could I have some help?
  49. G

    Noether currents associated with diffeomorphism invariance

    Having some generic curved spacetime, what are the Noether currents that are guaranteed to exist by diffeomorphism invariance? Is the energy-momentum tensor such a current?
  50. S

    Explain Paradox of Light Moving at c in All Frames

    My friend is moving in my inertial frame and receives light in the two directions parallel to her movement: from behind and from ahead. The photons that reach her travel at c in my frame, so I presume they will approach my friend at different speeds (as I view it): faster from ahead and slower...
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