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. binbagsss

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    Hi, this is probably a stupid question, but, does rotational invariance in ##d=2+1## mean to only rotate the spatial coordinates and not the time. I mean bascially I want to show that ## \int d^3 x \epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho} ##, yes epsilon the antisymmetric tensor, is...
  2. binbagsss

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  3. C

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  4. Demystifier

    I Special relativity vs Lorentz invariance

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  5. joneall

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  6. kmm

    I Bernard Schutz Proves Invariance of Interval

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  7. L

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  8. R

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  9. G

    A How does Lorentz invariance help evaluate tensor integrals?

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  10. J

    A Question about Lorenz invariance and cluster decomposition

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  11. A

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  12. K

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  13. V

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  14. T

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  15. G

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  16. PeroK

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  17. G

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  18. fresh_42

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  20. D

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  21. D

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  22. E

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  23. I

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  24. K

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  25. G

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  31. mjda

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  32. W

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  33. A

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  34. e2m2a

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  35. G

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  36. Gene Naden

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  37. Gene Naden

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  38. J

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  39. F

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  40. D

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  41. A

    B Scale Invariance and dark matter/dark energy

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  42. wolram

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  43. T

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  44. Delta2

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  46. ohwilleke

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  47. jlmccart03

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  48. entropy1

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  49. dextercioby

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  50. 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...
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