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. muscaria

    Gauge invariance of the vector potential

    The vector potential can be expressed in the following way: ∇^2 Ay-∂/∂y (∇∙A)=-μJy (Here only taking y components) Vector A is not determined uniquely. We may add derivatives of an arbitrary function (gradient) to the components of A, and the magnetic field does not change (curl of...
  2. B

    Phase Invariance: Is It an Invariant?

    phase invariance? I read papers claiming that the phase of a plane wave (acoustic and electromagnetic) is not an invariant. Taking into account that invoking its invariance, we could derive formulas that account for well tested experimental facts (Doppler shift, aberratiion of light, wave...
  3. J

    (global) Gauge invariance and field theory

    Hi everyone, This is my first post and I hope to get some better understanding of something that has been bugging me. I understand (global) gauge invariance in the sense that |\psi\rangle denotes the same (physical) state as e^{i\varphi}|\psi\rangle, or more generally, the physical state...
  4. C

    Time Reversal Invariance Of Hamiltonian

    Homework Statement Suppose that the Hamiltonian is invariant under time reversal: [H,T] = 0. Show that, nevertheless, an eigenvalue of T is not a conserved quantity. Homework Equations The Attempt at a Solution Using Kramer's Theorem. Consider the energy eigenvalue...
  5. C

    Gauge Invariance: Classical vs Quantum

    In classical e&m, for gauge invariance you can choose div[A]=0 or div[A]=dV/dt, where A is vector potential and V is the scalar potential; however, in qft you multiply your wavefunction by a phase factor that is dependent on space time. My question is that is there any parallel that can be drawn...
  6. J

    Simple math question regarding Lorentz Invariance

    Let us restrict ourselves to SR for the moment at least. So we have a flat spacetime. Now consider a proper force of the form: \frac{dp^\mu}{d\tau} = a v^\mu where a is a scalar. It seems to be coordinate system independent due to the definition being in tensor notation. But it seems to...
  7. E

    Invariance - Normal Linear Transformation

    Homework Statement Let W be a complex finite dimensional vector space with a hermitian scalar product and let T: W -> W be linear and normal. Prove that U is a T-invariant subspace of W if and only if V is a T*-invariant subspace, where V is the orthogonal complement of U. The attempt at a...
  8. A

    Proof of invariance of dp1*dp2*dp3/E

    If we introduce four-dimensional coordinate sistem with component of four-momentum on axis, then dp1*dp2*dp3 can be considered as zeroth component of an element of hypersurface given by p^2=m^2*c^2. Element of this hypersurface is parallel to 4-vector of momentum so we have that dp1*dp2*dp3/E is...
  9. I

    Proving Hamiltonian Invariance with Goldstein Problems

    Homework Statement I'm solving Goldstein's problems. I have proved by direct substitution that Lagrange equations of motion are not effected by gauge transformation of the Lagrangian: L' = L + \frac{dF(q_i,t)}{dt} Now I'm trying to prove that Hamilton equations of motion are not affected...
  10. e2m2a

    Invariance of scalar dot product across inertial and non-inertial frames

    I have a question concerning scalar invariance with respect to an accelerating and an inertial reference frame. Here is the problem. Suppose we have a rotating spherical object, which we denote as the rotator, attached to a near-massless wire. The other end of the wire slips loosely over a...
  11. WCOLtd

    Bends in Space-Time vs Invariance

    Imagine a beam of light being turned on at the surface of a massive body such as a neutron star, the beam of light travels along a geodesic path towards a mirror located at radius r from the planet's surface, when the light beam hits the mirror, the light bounces back to the observer located at...
  12. L

    QFT and local gauge invariance

    Why is local gauge invariance needed in qft? I read that is allows interactions whereas global gauge invariance does not but was given no reason.
  13. L

    What is the significance of singlets and gauge invariance in particle physics?

    Hi folks! Another stupid question: Consider a Yukawa coupling \lambda \bar{\psi}_1 \psi_2 \phi where \phi is a scalar field in the (2,-\frac{1}{2}) representation and \psi_1 and \psi_2 are lh. Weyl fields in the (2,-\frac{1}{2}) and (1,1) representation of \mathrm{SU}(2) \times \mathrm{U}(1)...
  14. L

    Weak Interaction Invariance Under CPT Symmetry

    Is weak interaction invaraint under CPT Symmetry? Why?
  15. F

    Quantum Field Theory: Field Operators and Lorentz invariance

    [SOLVED] Quantum Field Theory: Field Operators and Lorentz invariance Hi there, I am currently working my way through a book an QFT (Aitchison/Hey) and am a bit stuck on an important step in the derivation of the Feynman Propagator. My problem is obviously that I am not a hard core expert...
  16. Jim Kata

    Time translation invariance and the vacuum state

    I kind of get the connection, but could someone elaborate the necessity for time translation invariance for the existence of a unique vacuum state.
  17. K

    Lorentz invariance and General Coordinate transformations

    Sorry to bring up again a question that I asked before but I am still confused about this. In SR we have Lorentz invariance. Now we go to GR and one says that the theory is invariant under general coordinate transformations (GCTs). But, as far as I understand, this is simply stating that...
  18. P

    Gauge invariance and it's relation to gauge bosons

    Hello, I'm currently doing a project that is concerned with the hopeful discovery of the Higgs Boson at LHC. I'll be running some code that my supervisor has produced, but before that he wanted me to understand more of the physics that is behind the Higgs mechanism. He has proposed a...
  19. K

    Meaning of diffeomorphism invariance?

    I initially posted this question in the Beyond the Standard Model forum since diffeomorphism invariance is a key ingredient of loop quantum gravity but it was suggested that I post the question here. Why is Einstein's theory diffeomorphism invariant? A diffeomorphism is basically a map of...
  20. K

    Meaning of diffeomorphism invariance

    I was watching one of Smolin's online lecture (the link was provided by Marcus in the thread "What's new that's happening in quantum gravity" or something to that effect) and Smolin makes a big deal on the difference between diffeomorphism invariance and invariance under general coordinate...
  21. B

    Show Lorentz invariance for Euler-Lagrange's equations- how?

    Hello, I need help showing that the Euler-Lagrange equations are Lorentz invariant (if Einstein's extended energy concept is used). Is there an easy way to show this? Any help would be very much appreciated.
  22. J

    Invariance of maxwell's equations under Gauge transformation

    [SOLVED] invariance of maxwell's equations under Gauge transformation Homework Statement Show that the source-free Maxwell equations \partial_{\mu} F^{\mu\nu}=0 are left invariant under the local gauge transformation A_{\mu}(x^{\nu})\rightarrow...
  23. Y

    Proof of invariance of gas pressure

    Proof that pressure is invariant on the front and back faces of the moving box. (We are already agreed in another thread that the pressure is invariant on the other faces) Pressure in the rest frame of the box: (Frame S) When a gas particle of mass M and velocity U collides with a face...
  24. B

    Phase invariance of e.m. waves

    Many textbooks derive the formulas which account for the Doppler shift and for aberration of light from the invariance of the phase of an electromagnetic wave. Do you know an explanation for the invariance?
  25. D

    Linearity, Time Invariance, Causality, ETC.

    Homework Statement Is the following input/output (x is input, y is output) system linear, time invariant, causal, and memoryless? Answer yes or no for each one.Homework Equations y(t)=2x(t)+3The Attempt at a Solution My instinct tells me it's linear, but for some reason I have trouble showing...
  26. B

    Lorentz contraction from space-time interval invariance

    Please tell me if it is possible to derive the formula which accounts for the Lorentz contraction from the invariance of the space-time interval. Thanks
  27. D

    Lorentz Invariance and Non-Galilean Invariance of Maxwell's Equations

    I am having trouble going about proving the Lorentz invariance and non-Galilean invariance of Maxwell's equations. Can someone help me find a simple way to do it? I've looked online and in textbooks, but they hardly give any explicit examples.
  28. J

    Understanding Lorentz invariance

    On p. 32 of Quantum field theory in a nutshell, Zee tries to derive the propagator for a spin 1 field: D_{\nu\lambda} = \frac{-g_{\nu\lambda} + k_\nu k_\lambda /m^2}{k^2 - m^2} using the Lorentz invariance of the equation k^\mu \varepsilon_\mu^{(a)}=0 where \varepsilon_\mu^{(a)} denotes the...
  29. B

    Invariance of combinations of physical quantities

    Please tell me if there are motivated physical reasons to consider that combinations of dimensionless physical quantities that appear at the exponent of e in distribution functions have the same magnitude in all inertial reference frames in relative motion, Thanks in advance
  30. L

    How to Prove Lorentz Invariance of Volume Element in Momentum Space?

    Simple question.. How do you prove the volume element of momentum space (d3k/Ek) is Lorentz Invariant? I tried making it proportional to "velocity volume element" derived from the Lorentz transformations but didn't seem to get very far.
  31. M

    Is 2EN^2 dx^3 a Relativistic Invariant?

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  32. C

    Tensors: Lorentz vs Galilean invariance

    What is the physical significance of tensors? Occasionally, motivating statements are made roughly along the lines of "if an equation can be expressed purely in terms of tensors, then it is true for all observers". But that doesn't seem quite complete because different tensor-users would have...
  33. I

    Matrix Invariance: T:X->Y Explained

    I just wanted to know say a matrix X is invariant under some transformation T. So: T:X->Y is invariant... does that mean all the elements of X are the same as the elements of Y? Or is the elements in Y a scalar multiple of the elements in X?
  34. WolfOfTheSteps

    Proving Time Invariance: How to Analyze and Confirm Time Invariance in a System

    Homework Statement Show that y(t) = \frac{d}{dt}\left[e^{-t}x(t)\right] is time invariant. 2. Relevant Information I don't think this is TI! I'm told it is TI, but I think I proved that it is not TI! My proof is below. Am I wrong or is the question wrong in assuming that the...
  35. T

    CPT Invariance & Neutrinos: Exploring the Violation

    On page 3 of the following pdf: http://pdg.lbl.gov/2005/reviews/numixrpp.pdf the paragraph after equation 13.12, there's a line that says: "Thus, if U is not real, the neutrino... When CPT holds, any difference between these probabilities indicates a violation of CP invariance." I don't...
  36. samalkhaiat

    Conformal group, Conformal algebra and Conformal invariance in field theory

    I have noticed that questions about this subject get either ignored or receive some confusing answers. So I decided to write a "brief" but self-contained introduction to the subject. I'm sure you will find it useful. It is going to take about 13 or 14 post to complete the work. Be patient with...
  37. G

    Can Adding a Constant to the D'Alembertian Maintain Relativistic Invariance?

    In Feynman lectures in physics v2 28-6, Feynman points out that we can add a constant times \phi to D'Alembertian without distrupting the relativistic invariance. How and why?? Can someone work out a mathematical proof? Thanks in advance.
  38. T

    Checking for time invariance

    Homework Statement Consider the following input-output relationship: y(t) = \int_0^\infty e^{-\sigma}x(t-\sigma) d\sigma A) Is the system time-invariant? B) Find the output y(t) when the input to the system is x(t) = \mid t \mid , -\infty < t < \infty Homework Equations These are the...
  39. G

    Proving invariance of scalar product

    Hi everyone, How would I go about proving that the scalar product of two four-vectors (A,B) is invariant under a Lorentz transformation?
  40. L

    How can the invariance of the wave equation be shown without using tensors?

    The problem is, rather briefly: Show that the wave equation is INVARIANT The equation is given as: [the Laplacian of phi] - 1/(c^2)*[dee^2(phi)/dee(t^2)] dee being the partial derivative.. phi is a scalar of (x, y, z, t) Now, i want, and think i should be able, to solve this problem...
  41. B

    Invariance of uxw (velocityxphase velocity)

    Please inform me if you know places where the invariance of the product velocityxphase velocity=cc is discussed (derived?). Thanks
  42. A

    And another one on Lorentz invariance

    It is clear that a conserved current \partial_{\mu} J^\mu = 0 implies the existence of a conserved charge Q= \int d^3x J^0 . Now I want to go the other way round: Suppose we have a basis of momentum eigenstates, such that these states are also eigenstates of the charge. Then clearly the charge...
  43. A

    Another one on Lorentz Invariance

    I recently read an author making the following argument in QFT: if <m|A^0(t,0)|n>=B then <m|A^mu(t,0)|n>=(B/p^0)*p^mu by Lorentz invariance. Can anybody tell me under which circumstances this holds and how it comes about? I understand that <m|A^mu(t,0)|n> had to transform as a 4-vector but why...
  44. A

    Exploring Supersymmetry Breaking & Poincare Invariance

    I recently studied supersymmetry breaking and read there that for Supersymmetry breaking we have the energy of the vacuum state >0. However what I do not really see is why such a vacuum would not break Poincare invariance as well as the energy is part of the momentum 4-vector and so transforms...
  45. R

    Proving the Invariance of the Spacetime Interval: Importance and Applications

    How do we prove that the spacetime interval is invariant? Also why is it so important?
  46. B

    Exploring the Relativistic Invariance of Proper Time

    do you kinow a simple and convincing argument for the fact that proper time is a relativistic invariant?
  47. dextercioby

    Remark on electric charge Lorentz invariance

    Since there's another thread on the same subject in the GR forum, but on this forum about 8 months ago an interesting discussion on the same subject took place, https://www.physicsforums.com/showthread.php?t=114620, i want to draw everyone's attention on the post \#24 in that thread in which the...
  48. quasar987

    Charge Invariance: Argument Explained

    Today my professor gave us an argument in favor of why charge was frame invariant. It went like "if it were not, then moving neutral object would become charged, which is not something that we observe". It's not exactly that and it made sense at the time, but I'm missing a piece in the argument...
  49. K

    Understanding Time Invariance in Signals

    i usually have such a hard time determining whether a signal is time invariant or not ... for example, why would x[-n] not be time-invariant? please don't just tell me why x[-n] would not be time invariant ... tell me techniques that I can apply to other signals too
  50. CarlB

    Poincare Invariance from General QFT

    Derivation of Poincare Invariance from general quantum field theory C.D. Froggatt, H.B. Nielsen Annalen der Physik, Volume 14, Issue 1-3 , Pages 115 - 147 (2005) Special Issue commemorating Albert Einstein Starting from a very general quantum field theory we seek to derive Poincare...
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