What is Invariant: Definition and 405 Discussions

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. For example, if we consider the action of the special linear group SLn on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SLn.

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  1. Z

    Maxwell eqs invariant under other transforms

    Has anyone ever seen a proof that lorentz transforms are the only transforms for maxwells equations to remain invariant between two reference frames moving at a uniform velocity with respect to each other?
  2. J

    Calculating the Invariant Interval for Two Events

    Homework Statement Consider two events ct_{1}\; =\; 3\; m,\; x_{1}\; =\; 2\; m,\; ct_{2}\; =\; 5\; m,\; x_{2}\; =\; 6m What is the time difference between the two events? Find a reference frame for which the time difference is the negative of the time difference in the original frame...
  3. L

    Adiabatic Invariant: Modifying Entropy Derivation by Laura

    I modified an incorrect derivation of entropy from phase space volume of a gas in the Wikipedia entry http://en.wikipedia.org/wiki/Adiabatic_invariant "Adiabatic expansion of an ideal gas" and I'd like to know if my modified derivation is also incorrect somehow. I realize it doesn't include...
  4. Grimble

    Is Proper Time in Relativity an Invariant Quantity?

    A very basic question, perhaps, but I am starting from basics and checking all my understanding. In Relativity is τ (tau), the proper time experienced by an observer adjacent to a clock in an inertial frame of reference, an invariant quantity? And if not, in what way can it vary?
  5. V

    Showing that commutator is invariant under orthchronous LTs

    I'm having difficulty deciphering my notes which 'proove' that the commutor of two real free fields φ(x) and φ(y) (lets call it i∆) ie. i∆=[φ(x),φ(y)] are Lorentz invariant under an orthocronous Lorentz transformation. Not sure if it helps but φ(x)=∫d3k[α(k)e-ikx+α+(k)eikx]. Now, apparently I...
  6. Pengwuino

    Invariant stress tensor = Invariant force?

    So we know that \frac{d}{dt}(P_{mech} + P_{field}) = \oint_S {T_{\alpha \beta } n_\beta da} that is, the time rate of change of the momentum of a system plus the momentum of the electromagnetic fields is equal to the surface integral of the term with the Maxwell Stress Tensor where there is...
  7. W

    Calculate Invariant Mass for Kaon & Pion - Help from W.

    We have a collision involving a Kaon plus and proton initially resulting in the same plus a neutral pion (ie. Kp to Kp(pi)). The question asks to calculate the invariant mass of just the outgoing kaon and pion, given the outgoing momenta of the particles, the angle between them and their masses...
  8. R

    Lorentz Invariant Majorana Neutrinos

    I have a two component Weyl spinor transforming as \psi \rightarrow M \psi where M is an SL(2) matrix which represents a Lorentz transformation. Suppose another spinor \chi also transforms the same way \chi \rightarrow M \chi. I can write a Lorentz invariant term \psi^T (-i\sigma^2) \chi where...
  9. F

    QFT: calculating decay rates from invariant matrix element M

    Hi! I am currently taking a first course in QFT with Peskin & Schroeder's book. I've got stuck with the equation that relates the differential decay rate of a particle A at rest into a set of final particles with the invariant matrix element M of the process. M can be found from the Feynman...
  10. snoopies622

    How Do Invariant Hyperbolae Relate to Spacetime Distances?

    Consider the upper half of the hyperbola (ct)^2 - x^2 = a^2 where a^2 is a positive constant. The spacetime distance between any point on this curve and the origin is the positive number a. A thought experiment helps give this some physical meaning to me: If I'm at x=0 with a...
  11. N

    Convolution in a Continous Linear Time Invariant System

    Dear Experts, For convolution to work any input signal we should be able to represent the input signal in terms of appropriately scaled and shifted unit impulses. This one holds good for discrete time system in which the input signal can be represented as sum of scaled shifted...
  12. J

    Is/are there any invariant OBJECTS in relativistic? Is there a substratum?

    Hi, I'm currently writing a paper on Relativity, which mostly uses original papers of Einstein. For this reason, I have little idea what the ultimate fallout of all his upheaval is. I am aware that electromagnetic fields become "shadows" of the complex mathematical entity called the...
  13. W

    The ground state of a time-reversal invariant system must has zero momentum?

    if the ground state is non-degenearate, this is easily understood But what if the ground state is non-degenerate?
  14. N

    Calculate the invariant problem

    Homework Statement Calculate the invariant E^{\alpha \beta} E_{\alpha \beta} Homework Equations The Attempt at a Solution we apply the metric in this case, E^{\alpha \beta} E_{\alpha \beta} = g_{\alpha n} g_{\beta m} E_{n m} E^{n m} is that even correct?
  15. B

    Physical interpretation of Lorentz invariant fermion field product?

    Hey all! Just a very short question: May I interpret the Lorenz invariant quantity \bar\psi\psi as being the probability density of a fermion field? Thanks! Blue2script
  16. M

    Is Simplifying Lorentz Invariant Measures by Coordinate Change Valid?

    Hi I have a question about Lorentz invariant measures, consider an integral of the form: \int d\mu(p) f(\Lambda^{-1}p) where d\mu(p) = d^3{\bf p}/(2\pi)^3(2p_0)^3 is the Lorentz invariant measure. Now to simplify this I can make a change of coordinates \int d\mu(\Lambda q) f(q)...
  17. K

    Why is invariant interval invariant?

    The invariant interval is defined to be \Delta {s^2} = \Delta {x^2} + \Delta {y^2} + \Delta {z^2} - {c^2}\Delta {t^2} and despite which inertial frame we are in, \Delta s for two particular events would be the same. If I use Lorentz transformation, this can be proved easily. But is there any...
  18. B

    Invariant moments of 2d images

    Hi there I'm thinking about using the rotation invarient moments by Hu/Flusser (http://en.wikipedia.org/wiki/Image_moment#Rotation_invariant_moments). I'm a physicist by trade, with exceptionally poor math..! I'm not comfortable with exactly what these invariants are. The wikipedia link...
  19. MathematicalPhysicist

    Hamiltonian which is invariant under time reversal question.

    Homework Statement Assuming that the Hamiltonian is invariant under time reversal, prove that the wave function for a spinless nondegnerate system at any given instant of time can always be chosen to be real. Homework Equations \psi(x,t)=<x|e^{-iHt/\hbar}|\psi_0> The Time-Reversal...
  20. V

    Invariant Tensors in GR and SR

    Hello all, this is my first post on this forum, though I have been perusing it for a while. I am currently re-reading through Carroll's text on SR and there is a curious comment on p24 that intrigues me. Carroll says that the *only* tensors in SR which are invariant are the Kronecker delta...
  21. T

    Particle physics - invariant mass question

    Homework Statement At HERA 30 GeV electrons collided head on with 820 GeV protons. Calculate the invariant mass of ep collisions. (masses: e=0.0005GeV, p=0.938GeV) Homework Equations M^2 = (E1 + E2)^2 - (p1 + p2)^2 ? The Attempt at a Solution I know the numerical answer to...
  22. e2m2a

    Is the Change in Rotational Kinetic Energy Frame Invariant?

    Is the Change in Rotational Kinetic Energy Frame Invariant? -------------------------------------------------------------------------------- I know the translational kinetic energy of an object is frame dependent. That is, in the center of mass frame of the object, the kinetic energy is...
  23. F

    Time invariant Green's function (inpulse response)

    Hello Forum, given a input=delta located at time t=0, the system will respond generating a function h(t). If the delta is instead located at t=t0 (delayed by tau), the system will respond with a function g(t)=h(t-tau), just a shifted version of the response for the delta a t=0... If...
  24. maverick280857

    No (Lorentz) Invariant tensor of rank 3?

    Hi everyone, (This isn't a homework problem). How does one show that there is no Lorentz invariant tensor of rank 3 and the only Lorentz invariant tensor of rank 4 is the 4D Levi Civita tensor? Thanks in advance.
  25. E

    Proving that the cartesian metric is rotation invariant

    I'm trying to prove that the cartesian metric g_{mn}=\delta_{mn} doesn't change under a transformation of coordinates to another cartesian coordinate set with different orientation. As a starting point I am using ds^2=\delta_{mn}(x)dx^m dx^n=\frac{\partial x^m}{\partial y^r}\frac{\partial...
  26. B

    Prove that the third invariant is equal to the determinant

    Homework Statement This is all in summation notation. Given a 3x3 matrix A_{ij}, show that det[A]=1/6(A_{ii}A_{jj}A_{kk}+2A_{ij}A_{jk}A_{ki}-3A_{ij}A_{ji}A_{kk}) Homework Equations I've been told that we're supposed to begin with det[A]=1/6\epsilon_{ijk}\epsilon_{pqr}A_{ip}A_{jq}A_{kr}...
  27. E

    Invariant functions of the four-momenta

    Why is the following statement true? The only functions of p^mu that are left invariant under proper proper, orthochronous Lorentz transformations are p^2 = p_mu p^mu and for p^2<=0 also the sign of p^0. I can see that they are invariant, but why are these the only invariants?
  28. R

    How to check if Lagrangian is parity invariant?

    The Lagrangian \mathcal L =\psi^{\dagger}\gamma^0 \gamma^\mu (1-\gamma^5)\partial_\mu \psi should violate parity, but I'm getting that it doesn't. \psi(x) changes to \gamma^0 \psi( Px) where Px=(t,-x) and x=(t,x). \gamma^j goes to - \gamma^j , while \gamma^0 stays the same...
  29. X

    Lorentz or Poincare invariant?

    Generally we say GR is local Lorentz invariant. Does it mean the action or field equation? Why not Poincare invariant? Thanks!
  30. A

    Is c invariant in Accelerating frames?

    Simple question. I would like to know if there is a definitive answer , consensus in the field, on the question of the measurement of light in an accelerating system. Whether one way measurements from the front to the back and vice versa would result in (c +v) = (c-v) = c as usual...
  31. pellman

    Why should the action be Lorentz invariant?

    Why should the action be Lorentz invariant? Every time I come across this it is assumed by the author without qualification. As too obvious to explain maybe? Ain't obvious to me.
  32. Rasalhague

    Why there can be only one invariant speed

    I've just come across the following argument as to why there can be only one invariant speed for massless particles. It's from Applications of Classical Physics by Roger Blandford and Kip Thorne. But I don't understand. Obviously, it's a contradiction to say that the hypothetical speed c_0 is...
  33. G

    Find Invariant Lines for Matrix Transformations | y=mx Form | Solutions

    Homework Statement Find all invariant lines, of the form y=mx for the matrix transformation. a) \left( \begin{array}{cc} 5 & 15 \\ -2 & 8 \end{array} \right) b) \left( \begin{array}{cc} 3 & -5 \\ -4 & 2 \end{array} \right) The Attempt at a Solution \left(...
  34. G

    Find Invariant Lines of Matrix Transformation y=mx+c

    Homework Statement find in the form y= mx+c, the invariant lines of the tranformation with matrix \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)\left( \begin{array}{c} x \\ \text{mx}+c...
  35. E

    When are linear transformations not invariant?

    I am studying invariance, and I came across this dilemma. Suppose we have a subspace with the basis <v1, v2> of the subspace (lets say U2) and we were to map v=c1v1+c2v2 and we let c2=0. Now c1T(v1)+c2T(v2)=k1c1v1+0*T(v2)= k1c1v1. I am doing a proof and need to know what the question means by...
  36. A

    Why are the primordial fluctuations called scale invariant?

    The slope of the primordial power spectrum (the power spectrum of density fluctuations produced by inflation in the very early Universe before it had been modified by gravitational/hydro dynamics) is often written, P(k) = A * k and then in the same line referred to as scale invariant or the...
  37. Spinnor

    Invariant of a helicoid, like an electron but not quite.

    Invariant of a helicoid, like an electron but not quite. Consider the surface of a helicoid whose axis extends to infinity, see for example: http://images.google.com/images?hl=en&q=helicoid&btnG=Search+Images&gbv=2 This surface has an interesting geometrical invariant. Consider a...
  38. T

    Proving d'Alembertian Invariant under Lorentz Transformations

    Homework Statement Show that (D'Alembertian)^2 is invariant under Lorentz Transformation. Homework Equations The book (E/M Griffiths) describes the D'Alembertian as: \square^2=\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial t^2} The Attempt at a Solution I don't really...
  39. N

    Cohomology = invariant forms

    Prove the following result: let G be a compact Lie group, H its closed subgroup and X = G/H. Let T(X) denote the space of G-invariant differential forms on X (e.g. \omega \in T(X) \Leftrightarrow \forall g \in G g^{*}\omega = \omega). Then T(X) is isomorphic to H^{*}(X), de Rham cohomology...
  40. H

    Proof that the E.M Field is invariant under guage transformation.

    To prove: F \overline{} \mu\nu = \nabla \overline{} \muA \overline{} \nu - \nabla \overline{} \nuA \overline{} \mu is invariant under the gauge transformation: A \overline{} \mu \rightarrow A \overline{} \mu + \nabla \overline{} \mu\LambdaI end up with: F \overline{} \mu\nu = F \overline{}...
  41. S

    Weyl invariant scalar field theory

    I'm not sure if this is the right place for this question, so feel free to move it. Anyway, my question is, is there any good reason why the following field theory should be Weyl invariant in an arbitrary dimension d>1: S = \int d^d x \sqrt{g} \left( g^{\mu \nu} \partial_\mu \phi \partial_\nu...
  42. Z

    Proving g-Orbit of z is Invariant Under g

    Suppose g\in Isom C, z\in C: Prove that the g-orbit of z is invariant under g. I just need some clarification on what this is asking for: 1.) Are we assuming that g is a group of the isometries of C under composition? 2.) To show invariance, would I only have to show that the g-orbit...
  43. R

    Krockner delta is an invariant symbol

    A representation of SU(2) is "pseudo-real". Can one form the product \phi^{\dagger i}\rho_{i} , where \phi_i and \rho_i transform in the fundamental representation? If a representation is complex, Krockner delta is an invariant symbol, so you can form such a product. SU(2) is not...
  44. S

    Finding invariant subspaces

    Homework Statement Let V be a finite dimensional, nonzero complex vector space. Let T be be a linear map on V. Show that V contains invariant subspaces of dimension j for j=1, ..., dim V. Homework Equations Since V is complex, V contains an invariant subspace of dimension 1. The...
  45. J

    Electromagnetic energy is not Gauge invariant?

    I assume I am making a mistake here. Can you please help me learn how to fix them? In electrodynamics, the gauge transformations are: \vec{A} \rightarrow \vec{A} + \vec{\nabla}\lambda V \rightarrow V - \frac{\partial}{\partial t}\lambda These leave the electric and magnetic fields...
  46. N

    Showing px-Et is invariant using Lorentz Transformations

    1. Using the Lorentz Transformations, show that the quantity px - Et is invariant, where p and E are the momentum and energy, respectively, of an object at position x at time t. 2. px - Et 3. I needed help on starting the problem. Where should I begin?
  47. C

    Invariant Lagrangian Homework: Find Solutions

    Homework Statement http://img261.imageshack.us/img261/5923/14254560bc0.th.jpg the question is in the image exactly as i wrote it down in class. but it's basically asking what systems have potential and kinetic energies that form a Lagrangian which is invariant to some transformation...
  48. S

    Minimal Invariant Subspaces: The Role of Orthogonal Linear Transformations

    I have a question about this theorem. Let V be an n-dimensional inner product space, and let T:V-->V be an orthogonal linear transformation. Let S be a minimal invariant subspace under T. Then S is one dimensional or two dimensional. I understand what this theorem says and I follow the...
  49. P

    Electromagnetic tensor - invariant

    Homework Statement Hi, I have to calculate the invariant: \tilde{F}^{\mu \nu} \, F_{\mu \nu} where F is the electromagnetic field tensor and \tilde{F} the dual one. Homework Equations First, the contravariant components of the electromagnetic field tensor are given by...
  50. L

    Lqg is still local lorentz invariant?

    How i can see the right lorentz invariance in lqg?
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