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. S

    How Does Translational Invariance Influence Variable Definitions in Physics?

    Homework Statement Consider a system of objects labeled by the index ##I##, each object located at position ##x_{I}##. (For simplicity, we can consider one spatial dimension, or just ignore an index labeling the different directions.) Because of translational invariance ##x'_{I}=x_{I}+\delta...
  2. ShayanJ

    How Does the Transformation Rule for A^\mu Ensure EM Lagrangian Invariance?

    Homework Statement [/B] Show that in order for the free Lagrangian to be invariant when ## A^\mu ## is transformed by a transformation U, it has to transform as below: ## A'^{\mu}=\frac i g (\partial^\mu U) U^{-1}+U A^\mu U^{-1} ## Homework Equations [/B] The wording of the problem is a bit...
  3. A

    I Is a Riemannian Metric Invariant Under Any Coordinate Transformation?

    Q1: How do we prove that a Riemannian metric G (ex. on RxR) is invariant with respect to a change of coordinate, if all we have is G, and no coordinate transform? G = ( x2 -x1 ) ( -x1 x2 ) Q2: Since the distance ds has to be invariant, I understand that it has to be proved...
  4. S

    A Lorentz invariance of the Heaviside function

    Consider the Heaviside function ##\Theta(k^{0})##. This function is Lorentz invariant if ##\text{sign}\ (k^{0})## is invariant under a Lorentz transformation. I have been told that only orthochronous Lorentz transformations preserve ##\text{sign}\ (k^{0})## under the condition that ##k## is a...
  5. ShayanJ

    Lagrangian invariance under infinitesimal transformations

    This is my second term in my master's and one of the courses I've taken is QFT1 which is basically only QED. In the last class, the professor said the Klein-Gordon Lagrangian has a global symmetry under elements of U(1). Then he assumed the transformation parameter is infinitesimal and , under...
  6. D

    Lorentz invariance of the Minkowski metric

    I understand that in order to preserve the inner product of two four vectors under a change of coordinates x^{\mu}\rightarrow x^{\mu^{'}}=\Lambda^{\mu^{'}}_{\,\, \nu}x^{\nu} the Minkowski metric must transform as \eta_{\mu^{'}\nu^{'}}=\Lambda^{\alpha}_{\,\...
  7. LarryS

    Gauge Invariance for field of *Uncharged* particles?

    A complex classical field Φ of particles is, by itself, invariant under global phase changes but not under local phase changes. It is made gauge invariant by coupling it with the EM potential, A, by substituting the covariant derivative for the normal partial derivative in the Lagrangian. But...
  8. kroni

    Is Lorentz invariance is true in curved spacetime?

    Hello, I am re-reading a book about quantum physics and general relativity. To introduce representation of the lorentz group, they explain the definition of lorentz group as the group of transformation that let x² + y² ... -t² unchanged. But in cuved space the distance is not the same as in...
  9. Xezlec

    Electromagnetic gauge invariance with boundary conditions

    Hello. I'm trying to wrap my head around how Lagrangians work in classical field theory. I have a book that is talking about the gauge invariance of the Lagrangian: \mathscr{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-J^\mu A_\mu. It shows that we can replace A^\mu with A^\mu+\partial^\mu\chi for...
  10. J

    Principle of relativity and invariance of c

    First postulate (principle of relativity) and 2. Second postulate (invariance of c), affect not we continue investigating new things?
  11. E

    Why must the Higgs' gauge symmetry be broken?

    The part I understand: I understand that the spontaneous symmetry breaking of the Higgs produces the 'Mexican hat' potential, with two non-zero stable equilibria. I understand that as the Higgs is a complex field, there exists a phase component of the field. Under gauge transformations of...
  12. B3NR4Y

    Invariance: What Does it Mean?

    In my GR book they discuss things that are invariant, and I know from my math classes that invariant things are very useful. However, my intuition with invariance is that when a coordinate transformation is applied, the object is the same. Scalars are the same scalar in one frame as another...
  13. S

    Invariance of quadratic form for unitary matrices

    Homework Statement Show that all ##n \times n## unitary matrices ##U## leave invariant the quadratic form ##|x_{1}|^{2} + |x_{2}|^{2} + \cdots + |x_{n}|^{2}##, that is, that if ##x'=Ux##, then ##|x'|^{2}=|x|^{2}##. Homework Equations The Attempt at a Solution ##|x'|^{2} = (x')^{\dagger}(x')...
  14. S

    Invariance of quadratic form for orthogonal matrices

    Homework Statement Show that all ##n \times n## (real) orthogonal matrices ##O## leave invariant the quadratic form ##x_{1}^{2} + x_{2}^{2}+ \cdots + x_{n}^{2}##, that is, that if ##x'=Ox##, then ##x'^{2}=x^{2}##. Homework Equations The Attempt at a Solution ##x'^{2} = (x')^{T}(x') =...
  15. K

    Please explain gauge invariance un-mathmatically

    please explain what gauge symmetry is, gauge transformation is, gauge invariance is, and also how gauge invariance deletes the timelike polarization of a massless vector boson. without fancy math and formulas.
  16. D

    Invariance of integration measure under shifts in field

    I've been trying to teach myself the path integral formulation of quantum field theory and there's a point that's really bugging me: why is the integration measure ##\mathcal{D}\phi(x)## invariant under shifts in the field of the form $$\phi(x)\rightarrow\tilde{\phi}(x)=\phi(x)+\int...
  17. nomadreid

    Understanding Invariance of Spacetime Intervals

    Category of simple questions Obviously I am misunderstanding how an interval of space- time can be invariant under coordinate transformations. The following elementary (but obviously incorrect) calculation will illustrate my difficulty. Alice is leaving her two boyfriends, Bob and Charlie. Bob...
  18. 0

    Invariance of Acceleration in Inertial Reference Frames

    Claim: The acceleration (both direction and magnitude) for any object is the same in any inertial reference frame. Is this claim true? I think it is, but someone mentioned to me that time may be an issue as it's not agreed upon in all inertial reference frames. I'd appreciate any references...
  19. K

    Spacetime scaling invariance and quantum gravity

    Neil Turok, Director of the Perimeter Institute of Theoretical Physics in Ontario, Canada suggests scaling invariance is a fundamental property of nature, including spacetime. that nature does not recognize any kind of scale, including Planck scale. if true how would this affect the leading...
  20. H

    How is the center of mass determined independently of the coordinate system?

    Hi! I have been reading about the position of the center of mass in the Marion's Classical Dynamics book, in some point of the section he states that: "The location of center of mass of a body in uniquely defined, but the position vector R(of the center of mass ofcourse) depends on the...
  21. ShayanJ

    Invariance of the determinant under spin rotations

    Homework Statement Show that the determinant of a ##2 \times 2 ## matrix ## \vec\sigma \cdot \vec a ## is invariant under ## \vec \sigma\cdot \vec a \rightarrow \vec \sigma\cdot \vec a' \equiv \exp(\frac{i\vec \sigma \cdot \hat n \phi}{2})\vec \sigma\cdot \vec a \exp(\frac{-i\vec \sigma \cdot...
  22. M

    Conservation of angular momentum invariance

    Homework Statement Given a reference frame O' moving at a constant speed $\vec{V}$ in relation to another reference frame O, I want to prove that ##\vec{r_{1B}} \times m_1\vec{v_{1B}} + \vec{r_{2B}} \times m_2\vec{v_{2B}} = \vec{r_{1F}} \times m_1\vec{v_{1F}} + \vec{r_{2F}} \times...
  23. Gvido_Anselmi

    Vacuum state Lorentz invariance

    Hi everybody! Why we don't have to prove Lorentz invariance of the Vacuum state in QFT? This fact is quite obvious in QED and follows from Lorentz invariance of electric charges. But in general case? I don't know, but it seems to me this fact is not so obvious as it treated.
  24. A

    Can we determine motion within an enclosed object using energy and mass?

    Is it a fact of invariance that a person moving in an enclosed object cannot tell if he/she is moving at constant velocity or standing still (for case when he/she is not being accelerated nor in a gravitational field)? If so, would it be possible to perform an experiment within the closed object...
  25. Safinaz

    Couplings of new Higgs scalars and CP- invariance

    Hi all, In A. Djouadi's review for Higgs, volume II, " arXiv:hep-ph/0503173v2 ", Sec. 1.2.3, it discuss the couplings of SUSY new scalars with gauge bosons, there are some points I don't understand: - CP–invariance forbids WWA, ZZA and W ZH ± couplings - For the couplings between two Higgs...
  26. C

    Intuition Behind Scale Invariance Power Spectrum

    In the book "Statistical physics for cosmic structures" at p. 171 a read a definition of scale invariance (leading to the so called scale invariant power spectrum) given as the requirement that ##\sigma^2_M(R=R_H(t)) = constant##, where ##R_H(t)## is the horizon, i.e. the maximal distance that...
  27. C

    CMB , Spherical Harmonics and Rotational Invariance

    In Dodelson's "Modern Cosmology" on p.241 he states that the ##a_{lm}##-s -- for a given ##l##-- corresponding to a spherical harmonic expansion of the photon-temperature fluctuations, are drawn from the same probability distribution regardless of the value of ##m##. Dodelson does not explain...
  28. T

    Gauge invariance is not normal invariance?

    I recently learned that with (local) gauge invariance, functional quantization needs to factor out volume factor(Faddeev-Popov procedure). Why does this has to be done?Just to remove infinity? As far as I am concerned, ##\phi^4## theory contains invariance(for example ##\phi\to\phi\cdot e^{i...
  29. VintageGuy

    Tensor indices (proving Lorentz covariance)

    Homework Statement [/B] So, I need to show Lorentz covariance of a Proca field E-L equation, conceptually I have no problems with this, I just have to make one final step that I cannot really justify. Homework Equations "Proca" (quotation marks because of the minus next to the mass part, I...
  30. B

    What is the concept of scale invariance in quantum field theory?

    Hey guys! I was reading the following paper http://arxiv.org/abs/hep-ph/0703260 for Georgi and I have a conceptual question about it. Howard Georgi was talking about this Unparticle Physics theory and at the base of his analysis is the principle of scale invariance. So Georgi is saying what if...
  31. Breo

    CPT Invariance of Hermitian & Lorentz Lagrangians

    Are all the hermitian and lorentz invariant lagrangians, invariant under the combination of CPT? If yes, how can it be proved?
  32. pastoreerrante

    Scalar triple product invariance under circular shift proof

    Homework Statement Prove that for any three vectors ##\hat a, \hat b ## and ## \hat c##, ##\hat a \cdot (\hat b \times \hat c)## = ##(\hat a \times \hat b) \cdot \hat c ## Homework Equations [/B] ## \hat i \cdot \hat i = \hat j \cdot \hat j = \hat k \cdot \hat k = (1)(1)\cos(0) = 1 ## ##...
  33. Z

    How Does the Invariance Principle Apply to Limits in Engel's Problem?

    Homework Statement Hi Guys, This is the first exampe from Engel's problem solving book. After a long period of no math I am self studying. I do not know where my knowledge deficits lie, and was recommended this site for help. "E1. Starting with a point S (a, b) of the plane with 0 < b < a...
  34. S

    Almost sure invariance principle

    I would like to understand the Almost Sure Invariance Principle: "We say that the functions f_i: [a,b] →ℝ satisfy the Almost Sure Invariance Principle with error exponent γ < 1/2 if there are a probability space supporting a Brownian motion B and a sequence ξ_i, i ≥ 1, such that (1) {f_i}_{i≥1}...
  35. Coffee_

    Lagrangian invariance, short question

    Consider a Lagrangian: ##L(x,x',t)## Define now: ##L'(x,x',t) = L + x ## We have seen that Lagrangians can differ up to a total time derivative of some function ##F(x,t)## in such cases and give the same equation. When checking explicitly these two give different equations. Why would it be...
  36. TrickyDicky

    QED vacuum and Lorentz invariance

    The measured energy density of the vacuum has a disturbing discrepance with the one theorized by imposig Poincare invariance in QFT, usually referred to as the "vacuum catastrophe". On the other hand the Heisenberg indeterminacy principle leads to a nonzero vacuum expectation value for the...
  37. D

    Gauge Invariance of Weak Gravity Approximation

    Hey guys, So I have a question about the gauge invariance of the weak field approximation. So if I write the approximation as \Box h^{\mu\nu} -\partial_{\alpha}(\partial^{\mu}h^{\nu\alpha}+\partial^{\nu}h^{\mu\alpha})+\partial^{\mu}\partial^{\nu}h=0 then this is invariant under the gauge...
  38. P

    Gauge Invariance (QED): How Does the Statement Hold?

    My book says that in this case $$e^+e^- \rightarrow \gamma \gamma $$ gauge invariance requires that $$k_{1\nu}(A^{\mu\nu} + \tilde{A}^{\mu\nu})=0=k_{2\mu}(A^{\mu\nu} + \tilde{A}^{\mu\nu})$$ Please see attachment. My question is how does this statement hold?
  39. itssilva

    What is the physics behind GR's diffeomorphism invariance?

    Hi. This is my first post here in PF ( :) ). I've been reading some threads on "passive" versus "active" diffeomorphisms, and I wondered: what is the physical motivation for having GR be diffeomorphic invariant? Sure, this allows us to have solutions to Einstein's equations (EFE) up to...
  40. Coffee_

    Generating function and Lagrangian invariance

    To make my explanation easier open the ''Generating function approach'' section on this wiki article: http://en.wikipedia.org/wiki/Canonical_transformation The function ##\frac{dG}{dt}## represents the function that always can be added to the Lagrangian without changing the mechanical...
  41. lalo_u

    Gauge invariance of electroweak Lagrangian

    I was trying to prove all those little things you spend long as the local invariance in the free Lagrangian of electroweak interaction. Taking into account the appropriate SU(2) transformations (without covariant derivatives), came to the following expression \mathcal{L}_{\text{ferm.}} =...
  42. A

    Proving Newton's third law invariant with Galilean tranfrom

    Homework Statement Consider Newton’s force law for two particles interact through a central force F12(r1',r2',u1,u2), where by Newton’s third law F12 = -F21. m1(d^2r1/dt^2) = F12(r1,r2,u1,u2) m2(d^2r2/dt^2) = F21(r1,r2,u1,u2) A. Show that Newtonian mechanics is form invariant with respect to...
  43. Einj

    Fixed point and scale invariance

    Hello everyone. I'm studying the fixed point of theory in the context of QFT. First of all, let me say what I think I understood about fixed points and then I'll state my question. Suppose we have a theory with a certain running coupling ##\lambda(\mu)##. If we have, for example, an UV fixed...
  44. P

    Invariance of Hamiltonian of strong interaction under SU(6)

    Dear all, this is my first thread in the forum. I am trying to solve the following problem. it was given during a written exam at my university (many years ago) and I really would appreciate if someone will help me to solve it 1. Homework Statement Show that if the hamiltonian of the strong...
  45. WannabeNewton

    Gauge invariance/Lorentz invariance of regulator in QED

    See the passage attached below. Consider the 1-loop vertex correction (c.f. p.2 of http://bolvan.ph.utexas.edu/~vadim/classes/2012f/vertex.pdf) and vacuum polarization diagrams in QED. A very simple UV regulator that makes the integrals for the amplitude very simple is the prescription that we...
  46. D

    Invariance of the speed of light

    SR section 1.7. Einstein states if a train and light beam are moving in the same direction, the speed of the light as seen from the train is c-v. ( c being the speed of light and v the speed of the train ). c-v being smaller than c is resolved by time dilation or length contraction. My...
  47. A

    What is the derivation and meaning behind the SO(2,1) invariance algebra?

    Hello, Please excuse me about my ignorance. I would like to know how SO(2,1) Lie algebra, is derived from operators and commutators. I have some notes, that the Lie algebra of SO(2,1) is derived from: [D,H]=-iH [K,D]=-iK [H,K]=2iD where D, H, and K are the "generators". I have no clue what does...
  48. moriheru

    String Theory & Poincaré Invariance: Why & What?

    My question concerns poincare invariance (I have left out the accent) in bosonic string theory. As far as I know, action of a 1-d String is described by poincare invariance. So my question is: why poincare invariance? And here comes the more ambarassing question: What is poincare...
  49. Enoy

    Exploring Lorentz Violations in Vacuum: A Deep Space Investigation

    Tests of lorentz violation in space outside earth, planets, our sun, other stars, galaxies and galaxygroups have shown no violations. That is fine. But do you ladies and gentlemen, know if anyone have tested (experimental and/or theorethical) if there may happen lorentz violations (or if the...
  50. ellipsis

    Is distance between particles relative? Poincare invariance?

    If you shift the universe five meters to the left, there is no observational change. If you rotate the entire universe, the inertial frame is also rotated, and there is no observable change. If you freeze time in the universe for one billion years, then resume it, there is no observable...
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