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.

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  1. C

    Invariance of energy under change of origin

    Suppose you have a particle in one dimension in an energy eigenstate, i.e. Hψ(x)=Eψ(x) for some E. For an observer B in a coordinate frame with the origin translated some distance K to the right, the wavefunction of the particle looks like ψ'(x) = ψ(x+K). Surely, we expect the energy that B...
  2. E

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  3. E

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  4. F

    Does SR = invariance of dτ

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  5. D

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  6. C

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  7. R

    Space-Time Invariance, Weird Names and Some Questions

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  8. P

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  9. S

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  10. C

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  11. jegues

    Is the System y(t) = (1/2)(x(t) - x(-t)) Time Invariant?

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  12. lalo_u

    Why must a scalar field have a constant vacuum expectation value?

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  13. jegues

    Determining Time Invariance in Signal Statements: Examples and Solutions

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  14. bcrowell

    Gauge invariance requires gauge bosons, why not for neutral fermions?

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  15. L

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  16. L

    Lorentz invariance of [itex] E_p \delta ({\bf p}- {\bf q}) [/itex]

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  17. S

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  18. F

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  19. E

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  20. T

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  21. G

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  22. E

    Lorentz invariance of chirality

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  23. M

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  24. W

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  25. E

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  26. I

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  27. D

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  28. D

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  29. H

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  30. M

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  31. Rasalhague

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  32. alemsalem

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  33. S

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  34. P

    Relativity of Simultaneity & C Invariance Doubts

    Hi all, I'm trying to understand relativity for the first time. First of all I'm sorry for my bad English, I'll try to be as clear as possible. My doubt is the following: In my books and on many web sites I have red the thought experiment of the train and the two bolts of lightning to explain...
  35. L

    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...
  36. D

    Help me with this equation from Invariance of interval

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  37. A

    Guage symmetry - invariance under arbitrary phase change

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  38. L

    Invariance of the Fisher matrix

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  39. P

    Conformal invariance of gluon amplitudes

    Hi, I'm very ashamed to not understand how even the simplest gluon amplitudes are conformally invariant. See eg http://arxiv.org/abs/hep-th/0312171 pages 11-12. M(1^-,2^-,3^+)=\delta(\sum_i \lambda_i\tilde{\lambda}_i)\frac{\langle12\rangle^4}{\langle12\rangle \langle...
  40. N

    Maxwells Equations and Time Invariance

    Hi In my book it says that if the dielectric function ε is time invariant, we can write a solution to Maxwells equations of the form E(r, t) = E(r)exp(jωt). I agree that the ME are separable, but I don't see how they know that the time-dependence is harmonic? What is so special about exp(jωt)...
  41. S

    Relevance of Local Lorentz Invariance Violations

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  42. J

    Proving the Lorentz invariance of an integration measure? QFT related?

    So, first off, I'm thinking Lorentz invariant quantities are the same in any inertial frames S and S' regardless of their relative velocity. I'm thinking I need to show that \frac{d^3k}{(2\pi)^32E(\vec{k})} = \frac{d^3k'}{(2\pi)^32E'(\vec{k'})} where the primed & unprimed quantities denote...
  43. L

    Fibre bundles for describing gauge invariance

    Hello all ! My question: Does fibre bundles are necessary for describing gauge invariance in electromagnetic case? Or fibre bundles uses only for describing gauge invariance in cases of weak, electroweak and strong interactions? Thanks
  44. M

    Proving Invariance of Physical Laws Under All Transformations

    Hi. So if you have \frac{d p_{\alpha}}{ds} = \frac{q}{c} F^{\alpha \beta} u_{\beta} how could you possibly go on proving this its form is invariant under all coordinate transformations? Or any physical law of any form, really? I guess my point is how do you represent "all possible...
  45. S

    Gauge invariance of QED if the photon has a mass

    Hi, excuse the funny title :). In his book on quantum field theory Zee says (pag 245, fouth line) that QED gauge symmetry follows from the conservation of the current j=ψ γ^μ ψ (with the bar on the first spinor). I'm confused because that current is the noether current resulting from the...
  46. TrickyDicky

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  47. P

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  48. A

    Determine if the following system is time invariant: y(t) = x(t - 2) + x(2 - t)

    Homework Statement Determine if the following system is time invariant: y(t) = x(t - 2) + x(2 - t) 2. The attempt at a solution I know from the solutions that the system is NOT time invariant, yet whenever I try to solve it I get the opposite result. Here's what I'm doing: y1(t)...
  49. S

    Proving System is Time Invariant (T.I.) or Not

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  50. tom.stoer

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    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...
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