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. E

    Gauge invariance of interaction lagrangian

    Anyone can help me how to argue that interaction lagrangian is invariant under gauge transformation?
  2. A

    Density matrix formalism and Poincaré invariance

    The postulates of quantum theory can be given in the density matrix formalism. States correspond to positive trace class operators with trace 1 on a Hilbert space ##\mathcal{H}##. Composition is defined through the tensor product and reduction through partial trace. Operations on the system are...
  3. ShayanJ

    Lorentz invariance of Klein-Gordon Lagrangian

    I want to prove the invariance of the Klein-Gordon Lagrangian \mathcal{L}=\frac 1 2 \partial^\mu \phi \partial_\mu \phi-\frac 1 2 m^2 \phi^2 under a general Lorentz transformation \Lambda^\alpha_\beta but I don't know what should I do. I don't know how to handle it. How should I do it? Thanks
  4. maverick280857

    Diffeomorphism invariance of the Polyakov action

    [SOLVED] Diffeomorphism invariance of the Polyakov action Hi, I'm struggling with something that is quite elementary. I know that the Polyakov action is diffeomorphism invariant and Weyl invariant. Denoting the world-sheet coordinates \sigma^0 = \sigma and \sigma^1 = t and the independent...
  5. S

    Question about Lorentz Invariance and Gamma Matrices

    This is a pretty basic question, but I haven't seen it dealt with in the texts that I have used. In the proof where it is shown that the product of a spinor and its Dirac conjugate is Lorentz invariant, it is assumed that the gamma matrix \gamma^0 is invariant under a Lorentz transformation. I...
  6. M

    Asymptotic safety and local gauge invariance

    Hi folks -- does anyone know of a good survey article on the topic of whether local gauge invariance is a requirement of a fundamental theory within QFT -- hence of an asymptotically safe theory? I only have a few scattered remarks to this effect (by F. Wilczek mostly), so any good...
  7. L

    Question related to the Lorentz Invariance

    I have a question related to the Lorentz invariance. (on the book of Mark Srednicki Quantum Field Theory, page 35 prob. 2.9 c) There are representations of \Lambda and S. In order to show that result of problem, I use number of two ways. 1. I expanded \Lambda to infinitesimal form using...
  8. Greg Bernhardt

    What is the Concept of Gauge Invariance in Physics?

    Definition/Summary Gauge invariance is a form of symmetry. An experiment here today will work the same way over there tomorrow and with the apparatus pointing in a different direction. This is called "global invariance" … the laws of physics are invariant under translations, both in...
  9. carllacan

    Why does rotational invariance have to do with spin?

    Hi. According to Griffiths the conmutation relations for the angular momentum and spin operators conmutation relations can be deduced from the rotational invariance, as in Ballentine 3.3. For the angular momentum seems logical that it is so, but how is it that rotational invariance leads to...
  10. W

    Is Mean Curvature Invariant Under Coordinate Changes?

    Hi All: I am curious about the definition of mean curvature and its apparent lack of invariance under changes of coordinates: AFAIK, mean curvature is defined as the trace of the second fundamental form II(a,b). II(a,b) is a quadratic/bilinear form, and I do not see how its trace is invariant...
  11. B

    Lorentz invariance of Rarita-Schwinger action

    The Rarita-Schwinger action is \int \sqrt{g} \overline{\psi}_a \gamma^{abc} D_b \psi_c Here ##g = \det(g_{\mu \nu})##, and the indices ##a, b \dots ## are 'internal' indices that transform under e.g. ##\mathrm{SO} (3,1) ## in ##3+1## dimensions. ##\gamma^{abc} = \gamma^{[a} \gamma^{b}...
  12. B

    Exploring Bianchi IX Models for Metric Invariance

    In a Bianchi IX universe the metric must be invariant under the SO(3) group acting on the 3-sphere. Hence, the metric must be translation invariant in the spatial parts, where t=constant. This implies that the metric must take the form such that: ds^2 = dt^2 - g_ij(t)(x^i)(x^j), where g is a...
  13. ChrisVer

    What type of field can have a Lorentz invariant VEV?

    Suppose I have a field \hat{X}... What kind of operator should it be in order to develop a vev which doesn't break the Poincare invariance? I am sure that a scalar field doesn't break the poincare invariance, because it doesn't transform. However I don't know how to write it down mathematically...
  14. gn0m0n

    A manifesto on gauge invariance - how am I wrong?

    If gauge symmetries are really just redundancies in our description accounting for nonphysical degrees of freedom, then how does one explain the deep and powerful fact that if one begins with, say, just fermions and no gauge field in one's theory (and no interactions & essentially no dynamics)...
  15. P

    Spacetime symmetries vs. diffeomorphism invariance

    This is a very basic question, but I cannot get my head around the following: Any physical system should be invariant under changes of coordinates, because these are just a way of parametrizing the manifold/space in which my physical system is embedded. Now, let us consider a system that...
  16. M

    Use of chain rule in showing invariance.

    Hi, I’m a bit confused. I am familiar with the chain rule: if y=f(g(t,x),h(t,x)) then dy/dt=dy/dg*dg/dt+dy/dh*dh/dt To show that an equation is invariant under a galiliean transform, it’s partially necessary to show that the equation takes the same form both for x and for x’=x-v(T). So if you...
  17. B

    Why is the invariance of light a problem?

    Surely I am missing something, can you explain what? If we shoot a gun while traveling on a train, the speed of the bullet is its usual speed plus the speed of the train vt, because the bullet inside is already traveling at vt. If we produce an EMR on the train (or on the Earth in the case...
  18. michael879

    Gauge invariance and conserved current in SU(N)

    Hi, so I'm trying to derive the charge conservation law for a general SU(N) gauge field theory by using gauge invariance. For U(1) this is trivial, but for the more general SU(N) I seem to be stuck... So if anyone sees any flaws in my logic below please help! Starting with the Lagrangian...
  19. R

    CPT and Diffeomorphism Invariance

    I suspect this is somewhat off the beaten track here, but there may be some few that could give it a go. Einstein called his concept of coordinate independent physical theory General Covariant. The mathematicians call coordinate independent differential topology, diffeomorphism invariant...
  20. T

    Is This System Time Invariant?

    Homework Statement y(n)=x(4n+1). Is this system T.I or NOT T.I The professor marked this question wrong for my homework. He says it's NOT time invariant. I proved it is time invariant. Homework Equations System is time invariant if a shift in time in input results in the same shift in time...
  21. B

    Black holes and time invariance.

    Einstein's field equations are time invariant. So is it conceivable that a reverse black hole can exist i.e a "white hole"? Or would the second law of thermodynamics prevent such a thing?
  22. E

    Lorentz Invariance of Propagator for Complex Scalar Field

    Homework Statement Show that [\hat{\phi}(x_1),\hat{\phi}^\dagger(x_2)] = 0 for (x_1 - x_2)^2 < 0 where \phi is a complex scalar field Homework Equations \hat{\phi}=\int\frac{d^3 \mathbf{k}}{(2\pi)^3 \sqrt{2\omega}}[\hat{a}(k)e^{-ik\cdot x} + b^\dagger(k)e^{ik\cdot x}]...
  23. J

    Einstein field equations and scale invariance

    Hi, Are Einstein's field equations without the cosmological constant scale invariant? If so does the addition of the cosmological constant break the scale invariance? John
  24. fluidistic

    Wave equation invariance under Lorentz transform

    Homework Statement I must show that the one dimensional wave equation ##\frac{1}{c^2} \frac{\partial u}{\partial t^2}-\frac{\partial ^2 u}{\partial x^2}=0## is invariant under the Lorentz transformation ##t'=\gamma \left ( t-\frac{xv}{c^2} \right )## , ##x'=\gamma (x-vt)##Homework Equations...
  25. L

    Invariance of the speed of light

    Hello! Consider the law of addition of velocities for a particle moving in the x-y plane: u_x=\frac{u'_x+v}{1+u'_xv/c^2},\, u_y=\frac{u'_y}{\gamma(1+u'_xv/c^2)} In the book by Szekeres on mathematical physics on p.238 it is said that if u'=c, then it follows from the above formulae that...
  26. P

    Maxwell Equations Lorentz Invariance - Notation

    [This is mostly about notation] I was working on a problem where I had to prove that div(B) remains invariant under lorentz transformations. That was not too hard, so I came up with div(B) = \partial_{\mu} B^{\mu} must equal div(B) = \partial'_{\mu} B'^{\mu} so I did a...
  27. lonewolf219

    Is H^DaggerH invariant under rotations and translations?

    Hi, Since H^DaggerH is invariant under SU(2) X U(1), does this mean that H^DaggerH is invariant under rotations and translations? Thanks
  28. O

    Galilean invariance and kinetic energy

    I tried to look this up on the internet. I know there is a book about it but I forgot its title. I know that you can prove that the kinetic energy should be proportional to velocity squared by saying that this is the only Galilean invariant definition of kinetic energy. Can someone help me...
  29. L

    General Gauge Invariance Problem

    Hi! I have to prove that the amplitude of the process \gamma \gamma \to W^+ W^- does not depend on the gauge we will choose, R_{\xi}. So I use the most general expressions for the propagators and vertices. I find 5 diagrams. One that involves only the 4 fields and a vertex, 1 t and...
  30. U

    Four-momentum invariance between frames

    Homework Statement Homework Equations The Attempt at a Solution E2 - p2c2 = E02 I know that this is true. But how do i relate p1 to p1'? and same for energy as well. I expanded the LHS= E0,12 + E0,22 + 2E1E2 - 2(p1c)(p2c) for the RHS = E0,12 + E0,22 + 2E'1E'2 -...
  31. F

    Lorentz invariance of an equation (metric)

  32. C

    RG equation and invariance of the vertex function under scaling(Ryder)

    Hi. I have trouble understanding an argument in Lewis H. Ryder's QFT (second edition) at page 325 where he wants to write down an equation similar to the renormalization group equation which expresses the invariance of the vertex function \Gamma^{(n)} under the change of scale. The relevant...
  33. X

    Prove rotational invariance leads to conservation of ang. momentum

    Homework Statement Noether's theorem asserts a connection between invariance principles and conservation laws. In section 7.8 we saw that translational invariance of the Lagrangian implies conservation of total linear momentum. Here you will prove that rotational invariance of L implies...
  34. T

    Coulomb Gauge invariance, properties of Lambda

    Homework Statement A gauge transformation is defined so as to leave the fields invariant. The gauge transformations are such that \vec{A}=\vec{A'}+\nabla\Lambda and \Phi=\Phi'-\frac{\partial\Lambda}{\partial t}. Consider the Coulomb Gauge \nabla\cdot\vec{A}=0. Find out what properties the...
  35. C

    Vacuum expectation value and lorenz (trans) invariance

    Hi! I've seen it stated that because of Lorenz and translational invariance \langle 0| \phi(x) |0 \rangle has to be a constant and I wondered how to formally verify this?
  36. L

    Rotational invariance and degeneracy (quantum mechanics)

    Homework Statement Show that if a Hamiltonian H is invariant under all rotations, then the eigenstates of H are also eigenstates of L^{2} and they have a degeneracy of 2l+1. Homework Equations The professor told us to recall that J: \vec{L}=(L_x,L_y,L_z)...
  37. C

    How is C speed invariance - hows it suppose to work mechanically?

    Are photons partially non-local? Warping time-space to achieve this? Seems a bit confusing. I get that c is always supposed to be the same for all observers according to special relativity, but i am trying to picture what actually is supposedly happening there? Forgive me if...
  38. F

    Diffeomorphic Invariance implies Poincare Invariance?

    I have been quite puzzled for some time with the concept of Diffeomorphic Invariance. Here is what I think about it, 1) Diffeomorphic Invariance is the invariance of the theory under general coordinate transformations. For instance the Einstein-Hilbert action is diffeomorphic invariant...
  39. S

    How to maintain CPT invariance in Kaon oscillations

    Hey, I'm trying to get my head around neutral Kaon oscillations. As far as I understand it neutral Kaons can change between K^0 and \overline{K^0} as they propagate. Going through the quantum mechanics of this implies that this oscillation must be facilitated by a mass difference between the...
  40. S

    Minimal Subsitution from Lorentz Invariance

    Hello, My question is on coupling the photons to our Dirac field for electrons, we have the Dirac equation: (i\not{\partial -m })\psi=0 By Lorentz invariance we can change our space-time measure by: \partial ^\mu \rightarrow \partial ^\mu+ieA^\mu\equiv D^\mu Though I cannot see...
  41. ElijahRockers

    Is the Discrete Time System x[n] → y[n] = x[-n] Time Invariant?

    Homework Statement I am supposed to determine wether or not the discrete time system x[n] \rightarrow y[n] = x[-n] is time invariant or not. The Attempt at a Solution Let x_d[n] = x[n-n_0] y_d[n] = x_d[-n] = x[-(n-n_0)] = x[-n+n_0] y[n-n_0] = x[-(n-n_0)] = x[-n+n_0] Since y_d[n] =...
  42. H

    Invariance of the y coordinate for a boost along the x axis

    Homework Statement I've been reading through Spacetime Physics by Taylor & Wheeler, but this argument about the invariance of the y coordinate for inertial frames, one moving relative to the other on the x axis, is tripping me up. I'll just write the text word for word: I'm just not...
  43. strangerep

    Maximal invariance group for constant acceleration?

    Recently, over in the relativity forum, Micromass contributed a post: https://www.physicsforums.com/showpost.php?p=4168973&postcount=89 giving a proof that the most general coordinate transformation preserving the property of zero acceleration (i.e., maps straight lines to straight lines) is...
  44. samalkhaiat

    Non-abelian Local Gauge Invariance in Field Theories

    These are notes I made when I was studying the subject 20 years ago. They seem fine considering that I was student then. I believe they can be useful for those who are studying Yang-Mills and other related material. Sam
  45. D

    Linear Sigma Model Invariance Under O(N)

    In addition to my Faddeev-Popov Trick thread, I'm still tying up a few other loose ends before going into Part III of Peskin and Schroeder. I was able to show that the other Lagrangians introduced thus far are indeed invariant under the transformations given. But, I am hung up on what I think...
  46. A

    Physical Interpretation of point transformation invariance of the Lagrangian

    Homework Statement The problem asked us to show that the Euler-Lagrange's equations are invariant under a point transformation q_{i}=q_{i}(s_{1},...,s_{n},t), i=1...n. Give a physical interpretation. Homework Equations \frac{d}{dt}(\frac{\partial L}{\partial \dot{s_{j}}})=\frac{\partial...
  47. F

    Invariance of a Lagrangian under Transformation

    Homework Statement Show that the Lagrangian \mathcal{L}=\frac{m}{2}\vec{\dot{r}}^2 \, \frac{1}{(1+g \vec{r}^2)^2} is invariant under the Transformation \vec{r} \rightarrow \tilde{r}=\vec{r}+\vec{a}(1-g\vec{r}^2)+2g\vec{r}(\vec{r} \cdot \vec{a}) where b is a constant and \vec{a} are...
  48. jk22

    Does SR preserve the direction of light?

    does SR change the direction of light and if yes is it then possible to find a transformation keeping the velocity of light invariant and not only its speed ?
  49. jk22

    Invariance of schroedinger equation

    im trying to prove the galileo invariance of s.e. But i get stuck with an extra term prop.to v*d/dx in fact i get invariance only for scaling x' equ. ax and t' equ. at. Where does the mistake hide ?
  50. ShayanJ

    Physical significance of gauge invariance

    I've read that gauge invariance leads to a fundamental phenomenon.What is that? Thanks
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