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. T

    Gauge invariance of stress-energy tensor for EM field

    For free EM field: L=-\frac{1}{4}FabFab Then the stress-energy tensor is given by: Tmn=-Fml∂vAl+\frac{1}{4}gmnFabFab The author then redefines Tmn - he adds ∂lΩlmn to it, where Ωlmn=-Ωmln. The redefined tensor is: Tmn=-FmlFvl+gmv\frac{1}{4}FabFab It is gauge invariant and still satisfies...
  2. 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...
  3. E

    Is the Invariance of Physical Laws Linked to a Symmetry Group Structure?

    Homework Statement The invariance of physical laws to a coordinate change suggests a symmetry group structure. Can the operations of cordinate transformation be written as group operations? What is the group? Homework Equations The Attempt at a Solution At the moment I do not...
  4. E

    Invariance of vectors due to changes in coordinate systems

    Homework Statement How do I know that vector is invariant to changes of coordinate systems if i only have the components of the vector and not the basis vectors? Homework Equations let the vector in reference frame 1 be ds and the same vector in the reference frame 2 be ds1 The...
  5. F

    Is dτ Invariant Under Transformations Beyond Lorentz in Special Relativity?

    Can the metric of special relativity be derived from requiring the infinitesimal line segment, dτ, to be invariant in space and time? If we parameterize a line segment by the variable τ marked off along the line (that exists in space and time dimensions) is the length in τ of that line segment...
  6. D

    In a nutshell: getting some perspective on invariance

    My understanding of the S&G relativity is that one theory deals with reference frames at speeds near the speed of light while the other deals with reference frames that are approaching the speed of light. There are variances in observation between the two reference frames arising from their...
  7. C

    Heisenberg picture manifests Lorentz invariance?

    In several textbooks of QM I have read that Lorentz invariance is manifest in Heisnberg picture. How can we deduce that?
  8. R

    Space-Time Invariance, Weird Names and Some Questions

    Hi, so I was going over my lectures notes and I was looking at the Invariance, S2 for space time. I was just wondering why they call it time-like for S2<0 and space-like for S2>0 because, S2>0 says that there is an inertial frame where events occur at the same time (this has to do with...
  9. P

    Action invariance under galilean boost

    Hello, I've been spending a lot of time trying to solve this problem but I can't figure out a good solution. I have to show that the action of a non-relativistic particle ( Schrodinger density Lagrangian ) is invariant under Galilean boost with the form...
  10. S

    Are Random Walks with Different Step Sizes Identical in Brownian Motion Limit?

    Consider a random walk (in any dimension) with N steps and a step size of 1. Take a real number \alpha > 0 and consider another random walk which takes \alpha^2 N steps but wil step size \frac{1}{\alpha}. I immediately noticed that the mean deviation after the full walk in both cases is the...
  11. C

    Derivation of mass invariance using Lorentz transformation

    Homework Statement As the title suggests, I need help finding resources that clearly shows the step by step process of the derivation of the rest or invariant mass using the Lorentz transformation. Homework Equations Energy-momentum relation The Attempt at a Solution Not looking...
  12. jegues

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

    Homework Statement Prove whether or not, y(t) = \frac{1}{2}\left( x(t) - x(-t) \right) Is time invariant or not Homework Equations The Attempt at a Solution Shifting the output by -T results in, y(t-T) = \frac{1}{2}\left( x(t-T) - x(-(t-T)) \right) y(t-T) = \frac{1}{2}\left( x(t-T) -...
  13. lalo_u

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

    I was reading Mandle QFT book, and it says: "If we require the vacuum states to be invariant under Lorentz transformations and under translations, then this field must be a scalar field, $\phi(x)$, and its vacuum expectation value must be constant". Could anybody explain to me why is that?
  14. jegues

    Determining Time Invariance in Signal Statements: Examples and Solutions

    Homework Statement I just have a general question about what one of my professors had written on the board today in class. He was writing down examples where we had to determine whether the given statement was time invariant or not. One example was written as follows, x(-t) = y(t)...
  15. bcrowell

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

    My understanding is that for electrons, there is a standard argument that the electromagnetic interaction between them is required, not optional. Since they're identical particles, we should be able to take the wavefunction of two electrons and mix up their identities by any amount we like, and...
  16. L

    Lorentz invariance of four-volume element [itex] d^4x [/itex]

    I am slightly confused with the invariance of four-volume element. The orthodox way to show it is to prove that Jacobian is one, that I did, however in many textbooks I find a reasoning that because we have Lorentz contraction on one hand and time dilation on the other hand, the product is...
  17. L

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

    Can anybody help me with the proof that E_p \delta ({\bf p}- {\bf q}) is a Lorentz invariant object? I did a boost along z axes and used the formula \delta (f(x)) = \frac{\delta(x-x_0)}{|f'(x_0)|} and the factor in front of the delta function indeed is invariant but within the function I...
  18. S

    Invariance of U_harm of crystal to rotation

    Homework Statement Show that because a pure rotational displacement field u(r) has no effect, that the energy of a crystal only depends on the symmetric strain tensor epsilon_ij. Homework Equations As in Ashcroft & Mermin (22.72), the energy of a crystal as a function of displacement field...
  19. F

    Conformal Invariance Klein Gordon Action

    Wald Appendix D talks on why g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi is not conformally invariant when n is not equal to 2. I want to prove that the Klein Gordon Action (V=0) is not conformally invariant. However the term that I have in the action is just...
  20. E

    Understanding the Invariance of dl in Biot-Savart Law

    In the expression of Biot-Savart law B = (µo/4π) ∫ (I dl x r^)/r2 why dl does not depend on the coordinate systems ? in books they are using del X dl = 0
  21. T

    How can one prove P(τ)S(τ) = S(τ)P(0)?

    Assume u:R\rightarrow C^n and define shift operator S(\tau) with S(\tau)u(t)=u(t-\tau) and truncation operator P(\tau) with P(\tau)u(t)=u(t) for t\leq\tau and 0 for t>\tau Then P(\tau)S(\tau)=S(\tau)P(0) for every \tau>=0. Can someone please prove last statement..
  22. G

    Gauge invariance & mass shell amplitudes & pdfs

    What is means that unintegrated parton distributions and matrix elements are supposed to be gauge invariant??
  23. E

    Lorentz invariance of chirality

    Hi folks, I've been reading into the concepts of chirality & helicity and often I find a statement that chirality is Lorentz invariant in contrast to helicity (which of course depends on the frame). BUT I don't see in which way chirality IS Lorentz invariant. For massless particles things...
  24. M

    Gauge invariance Vs. Gauge covariance

    I know what gauge invariance is, but I'm not sure what gauge covariance is. Is it that a given field has a gauge covariant derivative? And under which circumstances do we get a field that is gauge invariant but not gauge covariant? And I would appreciate an example (other than the one...
  25. W

    How to prove Galilean invariance

    I'm having problems showing that Newton's second law of motion stays invariant (has the same form) under a Galilean transformation. If we write the general Galilean transformation as t=t'+t_{t} \bar{x}=R\bar{x}'+\bar{u}t'+\bar{t}_{\bar{x}} where R an orthogonal transformation, then velocity...
  26. E

    Is the Charge Q Lorentz Invariant in Quantum Field Theory?

    Hi all, I'm studying quantum field theory and I'm watching video lectures on Harward University website (Professor Colemann's lectures). Now, in lesson number six at 1h-6 minute a student asks why after trasforming field by a Lorentz transformation he doesn't transform also integration...
  27. I

    Proving the Jacobi identity from invariance

    "Proving" the Jacobi identity from invariance Hi all, In an informal and heuristic manner, I have heard that the "change" in something is the commutator with it, i.e. \delta A =[J,A] for an operator A where the change is due to the Lorentz transformation U = \exp{\epsilon J} = 1 + \epsilon J...
  28. D

    Axial anomaly and broken Lorentz invariance

    I had a look at Jackiws article on axial anomaly in scholarpedia: http://www.scholarpedia.org/article/Axial_anomaly Apparently, axial anomaly also breaks Lorentz invariance. Even if this effect would be very weak, doesn't this pull the plug on relativity?
  29. D

    Constant c and not its invariance.

    hi, value of c is experimental however c itself is theoretically lorentz invariant. So value of c depends upon fabric of cosmos right? Thanks.
  30. H

    What Are the Limits of {xn} and {yn} in the Given Sequence?

    Given x0 and y0 such that x0 > y0 > 0. Define, for n = 0,1,2,, xn+1 =xn +yn , yn+1 = 2xnyn .Find the limits of {xn} and {yn}. why is the answer lim{xn} = lim{yn} = sqrt(x0y0)?
  31. M

    Galilean Invariance and the Special Principle of Relativity

    To what extent is the PoR an extension of the galilean PoI? A stated consequence of the Galilean PoI is that inertial observers cannot determine by experiment if they are "in motion" or "at rest", with a similar consequence being mentioned for the PoR - to what extent to these differ, does...
  32. Rasalhague

    Invariance of Domain: Showing U Open in Differential Geometry | Spivak Ch.1

    In the first volume of Differential Geometry, Ch. 1, Spivak states that if U \subset \mathbb{R}^n is homeomorphic to \mathbb{R}^n, then U is open. This seems obvious: \mathbb{R}^n is open in \mathbb{R}^n, so its pre-image under a homeomorphism f:U \rightarrow \mathbb{R}^n is open. The pre-image...
  33. alemsalem

    Right handed particles and SU(2)L gauge invariance

    If only left handed fields couple in the weak force, and we can boost to a frame that changes left handed fields to right handed ones, how can that theory be relativistically invariant? thanks for the help!
  34. S

    Confused by proof of Lorentz properties from invariance of interval

    I've seen a few short proofs that if that some transformation \Lambda preserves the spacetime interval, then \Lambda^\top g \Lambda = g where g is the spacetime metric. They have all relied on an argument using some simple algebra to show that (\Lambda^\top g \Lambda) x \cdot x = g x...
  35. 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...
  36. 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...
  37. D

    Help me with this equation from Invariance of interval

    Here is an equation from proof of invariance of interval: This equation is from bernard schutz's first course in GR: I could not understand what M stands for. Can someone help me with this? I don't have advanced knowledge. I am a beginner UG.
  38. A

    Guage symmetry - invariance under arbitrary phase change

    As I understand it the heart of gauge symmetry is that I can change the phase at different points different amounts and the Lagrangian/action is unchanged. What I am not clear on is whether the changes I can make are completely arbitrary - I can make any change I want at any point - or whether...
  39. L

    Invariance of the Fisher matrix

    Fisher matrix=(minus the) average of the second derivative of the log-likelihood with respect to the parameters It seems to me the Fisher matrix for Gaussian data is invariant with respect to any (non-singular) linear transformation of the data; if correct this is a very useful property...
  40. 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...
  41. 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)...
  42. S

    Relevance of Local Lorentz Invariance Violations

    What is the relevance of Local Lorentz Invariance Violations if they would be detected in any future experiments? Does it mean there is absolute space and time in the microscopic sector below where current experiments can't probe or other absolute parameters since there would be preferred frame...
  43. 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...
  44. 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
  45. 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...
  46. 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...
  47. TrickyDicky

    Point-like particles, Lorentz invariance and QM/QFT

    As we know nonrelativistic quantum mechanics doesn't have the Lorentz invariance property and yet it makes a number of powerful predictions and gives rise to all the fundamental quantum properties (HUP, tunnelling effec, harmonic oscillator, superposition, wave-particle duality etc). What is...
  48. P

    Conformal invariance of null geodesics

    Hi, folks. I hope this is the right forum for this question. I'm not actually taking any classes, but I am doing self-study using D'Inverno's Introducing Einstein's Relativity. I have a solution, and I want someone to check it for me. Homework Statement Prove that the null geodesics of two...
  49. 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)...
  50. S

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

    Homework Statement Prove that the system is either T.I. or is not T.I. Homework Equations y(n) = x(n)*h(n) x(n) is the input signal y(n) is the output signal h(n) is the system The Attempt at a Solution Inputing x(n-n0) into the system I get out: as the output x(n-n0)*h(n)...
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