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.

View More On Wikipedia.org
Recent contents Top users
  1. D

    Collective modes and restoration of gauge invariance in superconductivity

    After the first explanation of superconductivity by Bardeen, Cooper and Schrieffer, it was for several years a matter of concern to render the theory charge conserving and gauge invariant. I have been reading the article by Y. Nambu, Phys. Rev. Vol. 117, p. 648 (1960) who uses Ward identities to...
  2. P

    Diffeomorphism invariance of metric determinant

    Hi; I am pretty sure that sqrt(-g) is diffeomorphism-invariant. I am wondering if all powers of this are diffeo-invariant too. For example, are -g, g^2, etc. all invariants too?
  3. D

    Rotational Invariance of X and Y on Rk: Proving Independence & Distribution

    suppose that X and Y are independent and each rotationally invariant on Rk a) Let P denote any orthogonal projection with dim P = k1 determine the distribution of the correlation coefficient r= X'PY/(|PX||PY|) I think r is a special case of ∑(Xi-barX)(Yi-barY) = X'PX where P = I-n^(-1)11' but...
  4. e2m2a

    Learn Tensor Calculus: Understand Tensor Invariance

    I am trying to learn tensor calculus, but I must be confused about tensor invariance. I know the definition of a tensor is a number or function that transforms according to certain rules under a change of coordinates. The transformation leaves the number or function invariant if it is a...
  5. M

    Space and Time Invariance (Classical Wave Equation)

    Hey, I've come across a part in my notes which I can't figure out. Essentially it says: \frac{\partial^{2}y}{\partial t^{2}} = v^{2} . \frac{\partial^{2}y}{\partial x^{2}} is space and time invariant. Whereas: \frac{\partial y}{\partial t} = -v . \frac{\partial y}{\partial x} is not...
  6. P

    Problem: Prove invariance of momentum factor

    Hi, In the derivation of scattering amplitudes (e.g. page 94 in http://kcl.ac.uk/content/1/c6/06/20/94/LecturesSM2010.pdf ) does anyone have a clue as to how to prove that the momentum uncertainty element (\delta p)^3/E is Lorentz invariant? I know how to do it for the measure d^3p/E...
  7. M

    Functional determinants and gauge invariance

    Hi all, I've been studying the path-integral quantisation of gauge theories in Zee III.4. My understanding is roughly as follows: that one can think of the differential operator in the quadratic tems in the lagrangian as a linear operator between infinite dimensional spaces (morally...
  8. E

    Entropy, microscopic quantum theory of space-time, lorentz invariance

    The existence of entropy in gravity implies that there are microscopic degrees of freedom in space that carries the entropy. This implies space is discrete. Discrete space breaks lorentz invariance, which has been strongly constrained by both FERMI and thought experiments. String theory...
  9. N

    Invariance of Pauli-matrices under rotation

    I'm trying to prove that the helicity operator \pmb{\sigma}\cdot\pmb{\hat{p}} is invariant under rotations. I found in Sakurai: Modern Quantum Mechanics page 166 that the Pauli matrices are invariant under rotations. Clearly that is sufficient for the helicity operator to be invariant under...
  10. J

    Galilean Invariance: Determining Invariant Quantities in Newtonian Mechanics

    Homework Statement Explain which of the following quantities are invariant in Newtonian mechanics. Position Distance between two points Velocity Acceleration Momentum Kinetic Energy Potential Energy (I presume gravitational) Homework Equations N/A The Attempt at a Solution I understand...
  11. T

    Gauge Invariance: Finding Energy Spectrum in 1D Ring

    Homework Statement So I was doing a problem out of Merzbacher 3rd edition (end of chapter 4 problem 3); the homework set has already been turned in but I wanted to run this by you all and see what you thought. I am essentially working with a particle in a 1-d ring constrained to the x-y plane...
  12. J

    Curled up dimensions and Lorentz invariance

    If we start with minkowski spacetime in 4 dimensions and then add several curled up spatial dimensions attached at every spacetime point, then: I'll label a spacetime point as: (ct,x,y,z)[a1,a2,a3,..,an] where the bracketted coordinates are the 'curled' coordinates. - If we label the...
  13. D

    Proving x^2-c^2*t^2 invariance

    How do you prove x2-c2t2 is invariant under the lorentz transformations given that;
  14. S

    Understanding the Invariance of the Speed of Light: A Question on Relativity

    Hi, I have a question which to many may seem quite stupid but it honestly has been perplexing me for a while now. I'm actually not sure if this is the correct place to post this but the question does seem to be based on the theory of relativity so here goes. I think I'm correct in supposing...
  15. C

    New limit on lorentz invariance violation

    Seasons greetings all, I am trying to dissect a really interesting article: http://www.nature.com.libproxy.ucl.ac.uk/nature/journal/v462/n7271/full/nature08574.html but I am struggling with some of the more technical terms in it. I have shown it to some lecturers at my uni and even they...
  16. H

    Testing System Linearity and Shift Invariance

    please anyone can help me how make check the linearity and shift invarient for the system I want to determine whether the system is linear and shift invarientby steps g(m,n) = f(m,-1) + f(m,0) + f(m,1) g(x) = (integration from +infinety to - infinety) f(x,z) dz please help me...
  17. S

    Galilean invariance question

    Hi, I've been reading Vic. Stenger's book "The comprehensible Cosmos" and have a question about the example he gives for Galilean invariance. In his example, Galileo drops a weight from the tower of Pisa and to a person standing near the tower (and thus in the same inertial? frame of the tower)...
  18. H

    The presumed Invariance of the atom

    The physics building is based on the invariance of the atom. Is there any principle or law or experiment ? I think that we only presume. What if there is no foundation for our 'truth' ?
  19. M

    Galilean invariance and conserved quantities

    Hi I have a simple question what is the conserved quantity corresponding to the symmetry of galilean invariance? and Lorentz invariance? cheers M
  20. C

    Aharonov Bohm Effect (gauge invariance)

    I was reading an article about the Aharonov - Bohm effect and gauge invariance ( J. Phys. A: Math. Gen. 16 (1983) 2173-2177 ) and there is something I really don't get it. The facts are: The problem is the familiar Aharonov-Bohm one, in which we have a cylinder and inside the cylinder \rho...
  21. ZapperZ

    Lorentz invariance verified almost to Planck scale?

    So whose quantum gravity theory will crash-and-burn if this is correct? http://physicsworld.com/cws/article/news/40834 Zz.
  22. C

    Foundations: Newton's Third Law and time reversal invariance

    Let me propose a list of principles of classical dynamics, specifically designed for education, for introduction to novices: - In the absence of any force: objects in motion move along straight lines, covering equal distances in equal intervals of time - Composition of motion: position...
  23. D

    Invariance of the action under a transformation

    If the action of a theory is invariant under a transformation (i.e. a lorentz transformation or a spacetime translation), does this imply that the Lagrangian is also invariant under the transformation? L \to L + \delta L \;\;;\;\; \delta L = 0?
  24. G

    Proving lorentz invariance of Dirac bilinears

    I'm trying to work through the proof of the Lorentz invariance of the Dirac bilinears. As an example, the simplest: \bar{\psi}^\prime\psi^\prime = \psi^{\prime\dagger}\gamma_0\psi^\prime = \psi^{\dagger}S^\dagger\gamma_0 S\psi = \psi^{\dagger}\gamma_0\gamma_0S^\dagger\gamma_0 S\psi =...
  25. H

    Proving Invariance of Transformations and the Linearity of a Specific Operation

    Hey guys, I was wondering how you would go about proving that the image of a transformation T, im(T), is invariant? And following that, how would you prove T(W1 \bigcap W2) is invariant if T(W1) and T(W2) are both invariant. On an unrelated note, another questions asks to show that TX =...
  26. F

    Time-Reversal Invariance at M Point of Surface Brioullin Zone

    Hello, i have a question on the M point of a surface brioullin zone. why is that point a point with time-reversal invariance? Thanks
  27. P

    Proving Invariance of Subsets under Group Actions

    Homework Statement Let G be a group acting on a set X, and let g in G. Show that a subset Y of X is invariant under the action of the subgroup <g> of G iff gY=Y. When Y is finite, show that assuming gY is a subset of Y is enough. Homework Equations If Y is a subset of X, we write GY for...
  28. mnb96

    Scaling and Translation Invariance

    Let's say I have a vector x in \mathcal{R}^3. Let's also suppose that any vector x undergoes the transformation x' = kx (where k is a positive real). Obviously, normalizing the vector will give us a quantity which is invariant to uniform scaling. In fact, \frac{\mathbf{x'}}{|\mathbf{x'}|} =...
  29. L

    Spacetime Invariance and Lorentz Equations

    Homework Statement So, I am working on a question that requires me to prove that s^2 = s'^2 from the Lorentz equations. It seemed like it'd be trivial... and then I ended up here a few hours later, not willing to waste any more time. Homework Equations By definition: s^2 = x^2 - (ct)^2 &...
  30. A

    Proof of U(1) gauge invariance

    Homework Statement I want to show explicitly that the Lagrangian... L_\Phi = (D_\mu \Phi)^\dagger (D^\mu \Phi) - \frac{m^2}{2\phi_0 ^2} [\Phi^\dagger \Phi - \phi_0 ^2]^2 where \Phi is a complex doublet of scalar fields, and D_\mu = (\partial_u + i \frac{g_1}{2} B_\mu) is the...
  31. B

    Conformal invariance / reparametrization invariance

    Hi! I have little questions about symmetries. I begin in the field, so... First about conformal symmetry. As I studied, in 2-d, a transformation (\tau, \sigma) \to (\tau', \sigma') changing the metric by a multiplicative factor implies that the transformation (\tau, \sigma) \to (\tau'...
  32. I

    What is Gauge Invariance in QFT?

    According to Steven Weinberg ('The quantum theory of fields', vol.1), the principle of gauge invariance stems from the fact, that one cannot build the 4-vector field from the creation/annihilation operators of massless bosons with spin >= 1. This '4-vector field' ('vector potential'), if we...
  33. T

    Invariance of Schrödinger's equation

    I thought i had a basic to intermediate understanding of quantum physics and group theory, but when reading hamermesh's "group theory and it's application to physical problems" there's something in the introduction i don't understand. first of all, i know the parity (or space inversion)...
  34. D

    Time discreteness and lorentz invariance

    I understand that some do not accept LQG in particular, but any discrete spatial geometry in general, because that would be a violation of lorentz symmetry. It was explained to me as meaning that If a 2D plane were discretized into a grid or lattice, a vector would not have a continuous...
  35. S

    About the invariance of similar linear operators and their minimal polynomial

    About the invariance of similar linear operators and their minimal polynomial Notations: F denotes a field V denotes a vector space over F L(V) denotes a vector space whose members are linear operators from V to V itself and its field is F, then L(V) is an algebra where multiplication is...
  36. R

    Gauge invariance of superpotential

    The superpotential is basically a product of left chiral superfields, taking the \theta \theta component. However, under a supergauge transformation, the left chiral superfields change, and the superpotential does not seem to be supergauge invariant. In fact, under supergauge...
  37. E

    Prove Invariance of Intersection Under T

    Homework Statement Suppose T is contained in the set of linear transformations from V to V. Prove that the intersection of any collection of subspaces of V invariant under T is invariant under T. Homework Equations The Attempt at a Solution Choose a basis for V. This basis...
  38. J

    Are there other Lorentz invariant bilinear combinations of E and B?

    Hello all, this is my first post so I hope the question I have is interesting, at least to a few, and that I can learn something from the discussion. I had problem for a class not long ago in which I had to prove that: \vec{E} \cdot \vec{B} and E^2 - c^2B^2 were Lorentz invariant. I was...
  39. T

    Is Proper Time Invariance Proven?

    is proper time invariant? proof? thanks...
  40. T

    Spacetime Translational Invariance vs(?) Lorentz Covariance

    Hello, I have been reviewing some relativity notes, and I am confused over something. I apologize if this seems like a silly or obvious point, but humor me. When we are talking about Lagrangians in field theory and in regular mechanics, we are often looking at symmetries. Namely, almost...
  41. B

    Invariance of del^2 operator under rotation of axes

    Homework Statement A scalar function can be represented as a position on the x-y plane, or on the u-v plane, where u and v are axes rotated by θ from the x and y axes. Prove that the 2-dimensional \nabla^2 operator is invariant under a rotation of axes. ie, \frac{\partial^2 f}{\partial...
  42. Spinnor

    Systems with global phase invariance, 3D string?

    From: http://en.wikipedia.org/wiki/Quantum_mechanics#Mathematical_formulation ... In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac[7] and John von Neumann[8], the possible states of a quantum mechanical system are represented by unit vectors (called...
  43. H

    Difference between Invariance and Covariance

    Hi, what is the difference between Lorentz Invariance and Lorentz Covariance? From my lecture note (Group theory course) Invariance and Covariance where defined as follows: Invariance: refers to the property of objects being left unchanged by symmetry operations. Covariance: refers to...
  44. B

    Why demand gauge invariance?

    I apologise if this question has been asked before, but I coudlnt find it, so: Is there some deeper reason for demanding gauge invariance other than that it allows us to include interactions between the gauge field and the fermions? I have seen people claim that it is "in keeping with the...
  45. J

    Proving Invariance of Domain Theorem

    Hello, For spring break homework, I'm supposed to prove the Invariance of domain theorem (stating that continuous injective functions from an open set in R^n to R^n are open maps). Does anyone know of any books/sources of any kind which will help? Thanks!
  46. P

    Lorentz Invariance & Finding Lambda Expression

    Homework Statement I have two four vectors v and w with v^{2} = m^{2} > 0, v_{0} > 0 and w^{2} > m^{2}, w_{0} > 0 . Now we consider a system with w' = (w_{0}', \vec{0}) and v' = (v_{0}', \vec{v} \, ') and in addition we consider the quantity \lambda = \vert \vec{v}' \vert \, \sqrt{...
  47. L

    Diff invariance allows loop variables why?

    I don't understand why a diffeomorphism invariance allows the extention of the loops variables in the continuum limit. Can someone give me some detailed reference?
  48. K

    Stupid question about invariance of maxwell equations

    Hi, as you all know one can write the Maxwell-equations in covariant form, namely: \partial_a F^{ab} = \frac{4\pi }{c} j^{b} and \partial_a G^{ab}=0 where \textbf{G} is the dual Tensor to \textbf{F}. Now the two simple questions. I can see that they are invariant, because I...
  49. R

    Prove Lorentz Invariance of a^{\mu}b_{\mu} Equation

    Homework Statement Show that a^{\mu}b_{\mu} \equiv -a^0b^0 + \vec{a} \bullet \vec{b} is invariant under Lorentz transformations. Homework Equations \Lambda_{\nu}^{\mu} \equiv \left( \begin{array}{cccc} \gamma & -\beta \gamma & 0 & 0 \\ -\beta \gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 &...
  50. B

    Space-time interval invariance question

    Cinsider please the invariance of the space-time interval in an one space dimension approach (x-0)2-c2(t-0)2=(x'-0)2-c2(t'-0)2 My question is: does it hold for arbitrary events (x,t) in I and (x',t') in I? Does it hold only in the case when the events are genertated in I and I' by the same...
Back
Top