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. Fra

    Diff invariant (measurement theory and observables)

    The recent threads makse we want to as a simple question. How many considers the notion of a _fundamental_ "diff invariant observables" as a clear and unquestionable requirement of the future theory of QG? To me this far from clear from the conceptual point of view. It's not even clear...
  2. B

    Proof: V is an invariant subspace of Hermitian H

    Homework Statement If \vec{x} is an eigenvector of a Hermitian matrix H, let V be the set of vectors orthogonal to \vec{x} . Show that V is a subspace, and that it is an invariant subspace of H. The Attempt at a Solution The Hermitian H must act on some linear space, call it K and of...
  3. J

    Invariant subspaces under linear operators

    Homework Statement Prove or give a counterexample: If U is a subspace of V that is invariant under every operator on V, then U = {0} or U = V.Homework Equations U is invariant under a linear operator T if u in U implies T(u) is in U.The Attempt at a Solution Assume {0} does not equal U does not...
  4. F

    Invariant divergence and christoffel symbols

    Homework Statement show that the definition of the invariant divergence divA = 1/√g ∂i (√g Ai) is equivalent to the other invariant definition divA = Ai;i Ai;k = ∂Ai/∂xk + ГiklAl Гkij = gkl/2 (∂gil/∂xj+∂glj/∂xi-∂gij/∂xl) Homework Equations g is the metric tensor...
  5. R

    A question of fully invariant subgroup

    A subgroup H of a group G is fully invariant if t(H)<=H for every endomorphism t of G. Let G is finite p-group has a fully invariant subgroup of order d for every d dividing |G|. What is the structure of G ?
  6. S

    Proving that the scalar product is invariant

    Is there a general way of proving that the scalar product xuxu = (x0)2 - (x1)2 - (x2)2 - (x3)2 is invariant under a Lorentz transformation that applies no matter the explicit form of the transformation.
  7. W

    QFT: Invariant Measures & Rotational Invariance Explained

    I am reading through sidney colemans lectures on QFT and I am stuck on what seem to be a silly question: He talks about the fact that the measure used in a calculation should be invariant in order to prove unitarity and later on that operators transform properly. He uses the example of...
  8. M

    Generalized eigenspace invariant?

    Hey, Is the generalized eigenspace invariant under the operator T? Let T be finite dimensional Linear operator on C(complex numbers). My understanding of the Generalized Eigenspace for the eigenvalue y is: "All v in V such that there exists a j>=1, (T-yIdenitity)^j (v) = 0." plus 0. thanks
  9. N

    What is the difference between Lorentz invariant and Lorentz covariant?

    invariant "spacetime velocity" This is related to this thread https://www.physicsforums.com/showthread.php?t=207251". To make responding easier, I have marked questions in red. I have tried to address some concerns pre-emptively. These I have marked in silver. If you would like to...
  10. S

    Degeneracy of rotationally invariant potentials

    I can appreciate the degeneracy of an infinite cubical well, in which there are three different directions, and hence three different separation constants from Schrodinger's equation which determine three separate n's (for lack of a better word.. principal quantum numbers, i suppose. it really...
  11. Y

    Is black hole entropy invariant?

    Hawking gave the black hole entopy equation: (1) S_{BH} = {k c^3 \over 4G \hbar} A where A is the surface area of the black hole event horizon. All the other factors on the right hand side of the equation are constants. From the point of view of an observer moving relative to the black...
  12. A

    Showing null space and range are invariant

    If V is any vector space and S and T are linear operators on V such that ST=TS show that the null space and the range of T are invariant under S. I think I need to begin by taking an element of the range of T and having S act on it and show that it stays in V? Can you help get me started?
  13. S

    Linear Algebra - Invariant Subspaces/Adjoint

    Homework Statement Suppose T is in L(V) and U is a subspace of V. Prove that U is invariant under T if and only if Uperp is invariant under T*. Homework Equations V = U \oplus Uperp if v \in V, u \in U, w \in Uperp, then v = u + w. <Tv, w> = <v, T*w> The Attempt at a Solution If U is...
  14. C

    Proving T is Scalar Multiple of Identity Operator: Invariant Subspace

    Homework Statement Suppose T is a linear operator on a finite dimensional vector space V, such that every subspace of V with dimension dim V-1 is invariant under T. Prove that T is a scalar multiple of the identity operator. The Attempt at a Solution I'm thinking of starting by letting U...
  15. T

    Rotation Invariant of Sch. Equation and [Lx,H]=0] ?

    Rotation Invariant of Sch. Equation and [Lx,H]=0] ?? Here is my first post, How can I prove that the rotation invariability of the Sch. equations? How can I prove that [Lx,H]=0
  16. I

    Calculating Invariant Mass Using Momentum and Rest Mass

    Hello, I'm working on this problem and I'd like to know how to find the invariant mass using just the lab-frame momentum and rest mass. I've found a lot of equations that deal with E, and I'm not completely sure what that is either. Thanks
  17. P

    Is null(T-\lambdaI) invariant under S for every \lambda \in F?

    Suppose S, T \in L(V) are such that ST=TS. Prove that null(T-\lambdaI) is invariant under S for every \lambda \in F. I can't find anything helpful in my book, I'm not sure where to start...
  18. malawi_glenn

    Rotationally invariant hamiltonian

    [SOLVED] rotationally invariant hamiltonian Homework Statement Show that the Hamiltonian H = p^2/2m+V_0r^2 corresponding to a particle of mass m and with V_0 constant is a) rotationally invariant. Homework Equations Rotation operator: U_R(\phi ) = \exp (-i \phi \vec{J} / \hbar...
  19. I

    [Special Relativity] Energy-momentum invariant question

    I was just wondering why what I've done in a spec rel question is wrong. Homework Statement A particle of mass m is traveling at 0.8c with respect to the lab frame towards an identical particle that is stationary with respect to the lab frame. If the particles undergo an inelastic collision...
  20. B

    Linear Time Invariant System Response

    Homework Statement The Relationship between the input x(n) and the output y(n) for the discrete System A is described by the expression: \frac{x(n) - 2x(n-1) + x(n-2)}{2} What is: (i) The impulse resopnse function h(n)? (ii) The frequency response function H(f)? (iii) The aplitude...
  21. L

    Relativity problem, calculating invariant P2u*Pu2

    Hello good veiled I have a problem in relativity ,how to show the following relation disintegration is considered one particle a of mass Ma was at rest desintegre in two particles 1 and 2 of mass M1 and M2 convention: Pij: I in index and J in top P2u=Pau-P1u and...
  22. M

    Linear Algebra: Invariant Subspaces

    Homework Statement Prove or give a counterexample: If U is a subspace of V that is invariant under every operator on V, then U = {0} or U = V. Assume that V is finite dimensional. The attempt at a solution I really think that I should be able to produce a counterexample, however...
  23. K

    Invariant Spacetime Interval

    I am working on a homework problem, and because it is a homework problem, I will not tell you the specifics or ask for an answer. My question has very little to do with the problem in particular, I just wanted to make that disclaimer. Oh, and the derivation of the Lorentz transformation was NOT...
  24. B

    Invariant and covariant in special relativity

    In anglo-american literature -a physical quantity is invariant if it has the same magnitude in all inertial reference frame, -an expression relating more physical quantities is covariant if it has the same algebraic structure in all inertial reference frames (rr-cctt) ? Thanks in advance
  25. D

    Lorentz invariant mass of electromagnetic field?

    An photon has mass zero by virtue of its momentum canceling its energy in m^2c^4 = E^2-p^2c^2 But in electromagnetism a field configution only has momentum when both a magnetic field and an electric field are present, e.g. in an electromagnetic wave. Now when there is only an electric or...
  26. R

    Unitary, invariant subspaces

    Homework Statement Let U be a unitary operator on an inner product space V, and let W be a finite-dimensional U-invariant subspace of V. Prove that (a) U(W) = W (b) the orthogonal complement of W is U-invariant (for ease of writing let the orthogonal complement of W be represented by...
  27. I

    Confused about invariant mass in particle collision

    Ok, when you use positrons to shoot at stationary electrons in a collider with enough energy so that you make a pair of proton and antiproton. The total energy of the pair would be E = T + MC^2, where M is the total invariance mass of the pair, namely 2*938Mev, or I can use E^2 = (pc)^2 +...
  28. Mentz114

    Invariant quantities in the EM field

    I understand that the quantities E^2 - B^2 \vec{E} \cdot \vec{B} (the dot is vector inner product). where E and B are the electric and magnetic components of an EM wave, are invariant under Lorentz/Poincare transformations. Can someone explain the physical significance of this ? Is...
  29. F

    Light as a Relativistic invariant

    Hey guys, I know this subject is a bit of an old chesnut but i thought id ask it anyway. What logical steps did the various physicists take to realize that c was constant in all reference frames? I've sort of found some weird ways to justify that fact in my head, but its more reverse...
  30. F

    Left invariant vector fields of a lie group

    Comment: My question is more of a conceptual 'why do we do this' rather than a technical 'how do we do this.' Homework Statement Given a lie group G parameterized by x_1, ... x_n, give a basis of left-invariant vector fields. Homework Equations We have a basis for the vector fields...
  31. N

    Heisenberg's uncertainty relation not invariant ?

    I'm not sure whether this has been discussed before. If one just looks at Heisenberg's uncertainty relation (energy-time), one easily sees that this is not Lorentz invariant. Even very simple results, such as the energy of a particle in a potential well seems not to transform according to the...
  32. Q

    Are there other phenomena (besides light) whose speed is invariant?

    Are there any other phenomena (optical or not) whose speed is constant and does not depend on whether the source is moving or not?
  33. P

    Inertial mass, invariant mass and the photon?

    In some cases, inertial mass does not equal invariant mass? What is the relation between the two? So the photon can have non zero inertial mass but always 0 invariant mass?
  34. S

    Charge Invariant: Intuitive Reason?

    Why is charge an invariant quantity? My professor once said that that is an experimental fact. I believe him. But is there an "intuitive" reason for why a charge should be an invariant quanity?
  35. A

    Invariant quantities for antimatter

    Since antiparticles have reversed proper time, can I conclude that all invariants are reversed for antiparticles?
  36. M

    Conserving Lorentz Invariant Momentum in Particle Collisions

    Hi, All, First time post, and this is quite possibly a very basic question: Is there a way to describe a particle's momentum such that the momentum itself is Lorentz invariant? The reason I am asking is this: As I understand it, if for example an electron and a positron were to collide and...
  37. N

    Is the vacuum stress energy tensor Lorentz invariant ?

    In many textbooks on relativity, one finds at some point a statement that the vacuum stress energy tensor should be Lorentz invariant, from which it then follows that the vacuum pressure is minus the vacuum energy density. However, the vacuum energy density (or stress tensor) is not an...
  38. pellman

    Dirac Lagrangian not invariant under rotations?

    First, I need to be able to do equations in my post but it has been a long time since I posted here. Someone please point me to a resource that gives the how-to. If you make a infinitesimal rotation of the free-field Lagrangian for the Dirac equation, you get an extra term because the Dirac...
  39. E

    Gauge invariant incorporation of particle widths?

    Introducing particle width via the Breit-Wigner propagator can break gauge invariance. Anyone know of some "nice" way to incorporate widths while still retaining gauge invariance?
  40. L

    E.H is a Lorentz invariant, when is it different from 0 ?

    From the Lorentz-invariant Faraday tensor F(H,E) two scalar invariants can be constructed: Inv1 = H²-E² and Inv2 = E.H Thinking at waves, electrostatic fields, magnetostatic fields, I see examples where Inv2 = 0. It is however easy to arrange an electrostatic field...
  41. M

    How is adiabatic invariant proved in a simple dynamic system?

    I'm reading a book on quontum mechanics in japanese (Quontum Mechanics by Shinichiro Tomonaga) and am stuck in proving the action variable "J" is constant in a one dimensional cyclic movement. i.e. The action variable "J" created by the trajectory of H(p(t),q(t),a(t/T)) = E(t) doesn't...
  42. A

    Poincaré invariant action of a point particle

    I am an MPhys graduate currently reading Joseph Polchinski’s, String Theory, Vol. 1. Unsurprisingly I’m stuck on the first real bit of maths… :p I quote from page 10, heh: “The simplest Poincaré invariant action that does not depend on the parametrization would be proportional to the proper...
  43. A

    Poincare invariant action of a point particle

    I am an MPhys graduate currently reading Joseph Polchinski’s, String Theory, Vol. 1. Unsurprisingly I’m stuck on the first real bit of maths… :p I quote from page 10, heh: “The simplest Poincaré invariant action that does not depend on the parametrization would be proportional to the proper...
  44. P

    Lagrangian remains invariant under addition

    what is the reason that the lagrangian remains invariant under addition of an arbtrary function of time?
  45. R

    Lorentz Invariant: What & When?

    What makes an equation lorentz-covariant? When is an equation lorentz-invariant?
  46. M

    Is Quantum Mechanics Scale Invariant?

    I'm wondering what happens if the scale factor of cosmological expansion increases without limit? Will you always calculate the same quantum foam and virtual particle production rates no matter what the scale is of the universe's expansion? If particles are extended objects, then of course...
  47. F

    Help me about loop invariant of LCM

    please help me, i was stuck in here 2 days! algorithm to prove lcm: a:=m b:=n while a != b if a < b a:= a + m else b:= b + n //postcondition: a is the lcm(m,n) what's the loop invariant?I thought it is(not sure): lcm(ak, bk) = lcm(ak/m, bk/n) *lcm(m,n) I am...
  48. T

    Difference between lorentz invariant and lorentz covariant

    title says it all. I've heard these two phrases. Lorentz invariant: Equation (Lagrangian, or ...?) takes same form under lorentz transforms. Lorentz covariant: Equation is in covariant form. I'm don't think I know what I mean when I say the latter. Can someone elucidate the...
  49. G

    Does Every Linear Operator Have a Nontrivial Invariant Subspace?

    Does every linear operator have a nontrivial invariant subspace? My professor mentioned this question in class, but never actually answered it. I am curious if this is true or not and why.
  50. S

    Invariant under Lorentz transformation

    HI guys first post :cool: I need to show that B^2-E^2/C^2 is invariant under Lorentz transformation (E and B are electromagnetic fields) now: B^2-E^2/C^2=B^2_x+B^2_y+B^2_z-E^2_x/C^2-E^2_y/C^2-E^2_z/C^2) and E'_x=E_x E'_y=\gamma(E_y-\frac{v}{c}B_z)...
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