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. S

    Lorentz Invariant Volume Element

    So, the upper light cone has a Lorentz invariant volume measure dk =\frac{dk_{1}\wedge dk_{2} \wedge dk_{3}}{k_{0}} according to several sources which I have been reading. However, I've never seen this derived, and I was wondering if anyone knew how it was done, or could point me towards...
  2. D

    Relativistic Mass vs. Invariant Mass.

    Hey all, I'm quite confused on this and am curious to be put straight. Now I understand the basic principles of relativity, this one just bugs me. Now I have always been taught that the famous E=MC^2 formula was proof that mass would reach toward infinity as it neared the speed of light...
  3. J

    Simple proof that the zero-point energy spectrum is Lorentz Invariant?

    In his article on the Zero-point Energy: http://www.calphysics.org/zpe.html Bernard Haisch says: That the spectrum of zero-point radiation has a frequency-cubed dependence is of great significance. That is the only kind of spectrum that has the property of being Lorentz invariant. The...
  4. F

    What is the significance of Tim Palmer's Invariant Set Postulate?

    What do people here think about Tim Palmer's Invariant Set Postulate? http://en.wikipedia.org/wiki/Invariant_set_postulate http://nanominded.com/2011/03/23/linking-two-passions-quantum-mechanics-and-fractals/
  5. I

    Properties of Systems: Memoryless, Causal, time invariant, linear

    Hi, I need a hand with my reasoning on the following question. I have answered all the questions, but not too sure weather they are correct. Please guide me or point me on the right direction if they are not correct. Regards 1. A system defined by the following equation y(t)=[x(t-2)]^2...
  6. K

    Planck constant is Lorentz invariant?

    It is widely recognized in physics textbooks that Planck constant is a "universal constant". But I nerver see a proof. As we know, in the special theory of relativity, c is a universal constant, namely a Lorentz invariant, which is Einstein's hypothesis. But How do we know the Plack constant h...
  7. D

    T invariant subspace (intro lin alg class undergrad)

    Homework Statement V=Matrix (2x2), T(A) = (0 1 ) A , and W = {A\epsilon V: A^{}t = A (1 0) Homework Equations So T(A) transformation, multiplies a 2x2 matix with entries 0 1 1 0 by A with A on the right side The Attempt at a Solution I...
  8. P

    Divergence of left invariant vector field

    Let's assume that a compact Lie group and left invariant vector filed X are given. I wonder why the divergence (with respect to Haar measure) of this field has to be equall 0. I found such result in one paper but I don't know how to prove it. Any suggestions?
  9. M

    Why does action have to be invariant?

    In classical mechanics, isn't kinetic energy not a Galilean scalar? So the action isn't invariant under Galilean transformations, but we can still use it with Galilean transformations. So why must it be a scalar in special relativity? I think I'm missing something...
  10. M

    Understanding Vector Invariance Under Coordinate Transformation

    What does it mean for a vector to remain "invariant" under coordinate transformation? I think I already know the answer to this question in a foggy, intuitive way, but I'd like a really clear explanation, if someone has it. I know all of multivariable calculus and quite a bit of linear algebra...
  11. A

    Show EM Wave equation invariant under a Lorentz Transformation

    Homework Statement Show that the electromagnetic wave equation \frac{\partial^{2}\phi}{\partial x^{2}} + \frac{\partial^{2}\phi}{\partial y^{2}} + \frac{\partial^{2}\phi}{\partial z^{2}} - \frac{1}{c^2}\frac{\partial^{2} \phi}{\partial t^2} is invariant under a Lorentz transformation...
  12. A

    Why an Invariant Subspace Has an Eigenvector

    I am following a proof in the text "Algebras of Linear Transformations" and having problem justifying this line: ... M is an invariant subspace so it has an eigenvector. Why should an invariant subspace have an eigenvector? Thank you I have a feeling this is a very simple result, if so I am sorry
  13. R

    Derivation of Lorentz invariant

    Some time ago, I came across a nice justification (by Einstein IIRC) for the formula x'^2 + y'^2 + z'^2 - c^2t'^2 = x^2 + y^2 + z^2 - c^2t^2. The argument went something like this: (1) x'^2 + y'^2 + z'^2 - c^2t'^2 = x^2 + y^2 + z^2 - c^2t^2 = 0 for light. (2) *reasoning I forget*, therefore...
  14. N

    Lorentz invariant lagrangian density

    Hi, Would someone know where I can find a derivation of the lorentz-invariant lagrangian density? This lagrangian often pops-up in books and papers and they take it for granted, but I was actually wondering if there's a "simple" derivation somewhere... Or does it take a whole theory and...
  15. jfy4

    Is there an invariant tensor for metric under all 10 motions?

    Hi, I was wondering, and I hope this isn't a ridiculous question, for the set of motions: 4 translations, 3 rotations, and 3 boosts; is there an invariant tensor for any metric under all 10 of these motions. That is, preforming these 10 motions, is there a tensor which remains unchanged...
  16. Phrak

    Lorentz Invariant Vectors and Grammatical Errors

    A vector in special relativity is the quantity: V = V^\mu \hat{e_\mu} On a change of coordinates, the basis vectors co-vary with the coordinate derivatives: \hat{e_\mu'} = \frac{\partial x_\mu'}{\partial x_\mu} \hat{e_\mu} The vector elements are the opposite. They are said to be...
  17. ShayanJ

    Invariant Quantities in SR: Explained

    I'm really confused about invariant quantities.Could someone explain which quantities are invariant in special relativity and how are they recognised? thanks
  18. J

    QED as a gauge invariant theory

    I'm just beginning to learn about Feynman diagrams and wanted to make sure I've got the correct basic understanding of QED. This is what I believe to be true right now: QED allows us to describe the interaction between an EM field and light/matter. The QED vertex is composed of a photon and...
  19. L

    Invariant Polynomials on complexified bundles with connection

    I would like to know if the following correct. Suppose I start with a connection on a real vector bundle and extend it to the complexification of the bundle. The curvature forms of the complexification seem to be the same as curvature 2 forms of the real bundle. From this it seems that the...
  20. L

    Proof Scalar action is conformally invariant

    Hi, So if we have the Lagrange density for a massless scalar field: L=\sqrt{-g}\left(-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{(n-2)}{4(n-1)} R\phi^2\right) Then under a conformal transformation g_{\mu\nu}=\omega^{-2}\tilde{g_{\mu\nu}} , then the Ricci sclar goes to...
  21. U

    Invariant 4d volume-time element

    Homework Statement Prove that the element dt\ dx\ dy\ dz is invariant under Lorentz boost with velocity \beta along z axis. Homework Equations Convention c=1 Lorentz boost in z direction: L(z)=\left[ \begin{array}{cccc} \gamma & 0 & 0 & -\gamma\beta \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\...
  22. A

    Differential cross section with invariant matrix element

    Homework Statement Calculate the differential cross section for A+B---> C+D with an invariant matrix element Homework Equations See attachment The Attempt at a Solution I have no idea how to even begin this problem. The course I am taking is an undergraduate course in intermediate modern...
  23. T

    Invariant subspace and linear transformation

    Homework Statement Let U be a subspace of V. Suppose that U is a invariant subspace of V for every linear transformation from V to V. Show that U=V. Homework Equations no The Attempt at a Solution Assume U is not trivial: Now we only need to show that U = V. Let dimV = n: We can...
  24. A

    How can I show that trace is Invariant under the change of basis?

    How can I show that trace is Invariant under the change of basis?
  25. M

    Invariant of Motion: Understand in Plain English

    Can anyone please describe me the following theory in plain english. Your help will be much appreciated. "In certain types of systems there is an invariant of motion which is equal to the sum of the kinetic energy and the potential energy of all the particles in the system. This quantity is...
  26. marcus

    Gravity as a diffeomorphism invariant gauge theory (new Krasnov paper)

    I'm hoping there will be some comment on this new paper of Kirill Krasnov http://arxiv.org/abs/1101.4788 Gravity as a diffeomorphism invariant gauge theory Kirill Krasnov 24 pages (Submitted on 25 Jan 2011) "A general diffeomorphism invariant SU(2) gauge theory is a gravity theory with two...
  27. Z

    Solving for invariant points on trig transformations

    Homework Statement Hello. I came across a question that required me to solve for invariant points between a base trig function and the function after horizontal stretch. I can't remember the exact question right now, but I'm just wondering how I would go about solving it if I didn't know...
  28. N

    HiLets say I have a Hamiltonian which is invariant in e.g. the

    Hi Lets say I have a Hamiltonian which is invariant in e.g. the spin indices. Does this imply that spin is a conserved quantity? If yes, is there an easy way of seeing this? Niles.
  29. D

    Linear Time Invariant System

    Homework Statement Propose a Second order Model in the continuous time domain Natural damping frequency of 0.6 Steady State gain of 2 units Write down the corresponding differential equation I have no idea where to begin with this, could someone help me with the process, then i will...
  30. T

    Relativistically Invariant Tensors

    In special relativity, the metric tensor is invariant under Lorentz transformations: \Lambda^\alpha{}_\mu \Lambda^\beta{}_\nu g^{\mu \nu} = g^{\alpha \beta} Is this the unique rank 2 tensor with this property, up to a scaling factor? How would I go about proving that? I know that two...
  31. M

    Question About Invariant Interval in Special Relavitiy

    This isn't a homework question, rather I believe I could use the invariant interval to check if my answers are right sometimes, but we are skipping this part in my class. Anyway I did this problem where there were two trains, one of length L moving to the right with velocity v, and another of...
  32. bcrowell

    Massless Klein-Gordon equation not conformally invariant?

    massless Klein-Gordon equation not conformally invariant?? Wald discusses conformal transformations in appendix D. He shows that the source-free Maxwell's equations in four dimensions are conformally invariant, and this makes sense to me, since with photons all you can do is measure the...
  33. J

    Is g^-1Ng a Subgroup of G? Proving Invariance in Group Theory

    Salutations all, just stuck with the starting step, I want to see if I can take it from there. Homework Statement Let G be a group and let N be a subgroup of G. Prove that the set g^{-1}Ng is a subgroup of G. The Attempt at a Solution Well, I'm going to have to show that...
  34. L

    What are the Functions of Riemann Invariants?

    Can someone tell me what a riemann invariant is in general terms please? i can't find it anywhere! thanks, lav
  35. L

    Lagrangian Invariant Under Transformation

    Verify that the Lagrangian density L= \frac{1}{2} \partial_\mu \phi_a \partial^\mu \phi_a - \frac{1}{2} m^2 \phi_a{}^2 for a triplet of real fields \phi_a (a=1,2,3) is invariant under the infinitesimal SO(3) rotation by \theta \phi_a \rightarrow \phi_a + \theta \epsilon_{abc} n_b \phi_c...
  36. F

    Geometric interpretation of the spacetime invariant

    For a euclidean space, the interval between 2 events (one at the origin) is defined by the equation: L^2=x^2 + y^2 The graph of this equation is a circle for which all points on the circle are separated by the distance L from the origin. For space-time, the interval between 2 events is...
  37. P

    Minkowski vacuum: Poincare invariant, quasi-free state

    Minkowski vacuum is Poincare invariant and quasi-free state. I wonder if these two conditions fully define it or there are more states which fulfill these conditions (or maybe Poincare invariance alone is sufficinet). Thanks for answers.
  38. T

    Modulo Z2 Invariant: What is it?

    what is modulo associated with z2 invariant?
  39. R

    Is Minkowski space the only Poincare invariant space?

    Hi everyone, I was wondering: if a space is invariant under Poincare transformations, does that mean it has to be Minkowski space? Or could it have some further isometries? By the same token, if a space is invariant under the orthogonal transformations, does it have to be Euclidean? I...
  40. S

    Is quantum field theory really lorentz invariant?

    Hi guys, Before responding to my post, please note that I am only familiar with the mathematics of nonrelativistic quantum mechanics, and don't know any quantum field theory. All I have is this vague idea that quantum field theory is the union of special relativity and quantum mechanics...
  41. R

    How to prove that rank is a similarity invariant?

    1. Prove that the rank of a matrix is invariant under similarity.Notes so far: Let A, B, P be nxn matrices, and let A and B be similar. That is, there exists an invertible matrix P such that B = P-1AP. I know the following relations so far: rank(P)=rank(P-1)=n ; rank(A) = rank(AT); rank(A) +...
  42. S

    Invariant Subspace: Understanding Definitions

    So I'm trying to get an idea of what an invariant subspace is and so please let me know if my understanding is correct. Given that you have some vector subspace being a collection of a particular number of vectors with the the space denoted as |\gamma>. If you have some other collection of...
  43. P

    Invariant Lagrangian or action

    "invariant" Lagrangian or action Hello everyone, I tried to describe my question but it seems getting too complicated and confusing to write down my thoughts in detail, so I am trying to start with the following question... Are invariance of the Lagrangian under a transformation and...
  44. E

    Proving Hopf Invariant and Degree of g for f⋅g

    Can someone give me some hints on how to prove the following statement: if f: S^3 \to S^2, g: S^3 \to S^3 then H(f\circ g) = deg~g H(f) where H is the Hopf invariant and deg g is the degree of g. I'm pretty clueless on how to start and I don't see how to get the deg to come in since that has...
  45. B

    Why is energy-momentum tensor Lorentz invariant?

    I'm studying General Relativity and facing several problems. We know that energy-momentum must be Lorentz invariant in locally inertial coordinates. I am not sure I understand this point clearly. What is the physics behind?
  46. V

    How Do You Calculate Invariant Mass in Particle Physics?

    Homework Statement so I'm doing some proof-of-concept data analysis this summer and I've never taken a relativistic mechanics class and I'm a bit stuck. i need to figure out if there was a rho0 decay to pi+/pi- in some hypothetical 900GeV collision data. If there is, there should be a spike...
  47. M

    Cross product invariant under SO(3)-matrices?

    Hi there!I'm trying to prove the following obvious statement, but am somehow stuck :( Let \vec a,\ \vec b\in\mathbb{R^3} , let M be in SO(3) and x be the cross productprove: M(\vec a\times\vec b)=M\vec a\times M\vec bI tried using the epsilon tensor, as in physics, but it doesn't really...
  48. Z

    Quadratic invariant in Chern-Simons Th. w/ ISO(2,1) group.

    Hi dudes. I'm studying the paper of Witten: 2+1 Dimensional gravity as an exactly soluble system. Before eq (2.8) the author justifies as a way to find the inner product the fact that in this theory we have the casimir \epsilon_{abc}P^aJ^b. Then he introduce the invariant quadratic form...
  49. S

    M. Frame: Invariant Mass Confinement < c Motion

    Hi, hopefully this isn't a dumb question. I've read essentially that in the center of mass/momentum frame an object has invariant mass, and that the system's total mass will be composed of the constituent particles' masses and any other kinetic and potential energies within the system. I also...
  50. H

    Prove: Invariant Subspaces are g(T)-Invariant

    Homework Statement Let T be a linear operator on a vector space V and let W be a T-Invariant subspace of V. Prove that W is g(T)-invariant for any polynomial g(t). Homework Equations Cayley-Hamilton Theorem? The Attempt at a Solution Im not sure how to begin. Ok so g(t) is the...
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