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. H

    Time Invariant Signal: Help Me See Why My Solution is Wrong

    Can anyone help my to see why my solution is wrong because the in the solutions it says that it is not time invariant
  2. B

    Electromagnetic wave equation not invariant under galilean trans.

    Homework Statement Prove that the electromagnetic wave equation:  (d^2ψ)/(dx^2) + (d^2ψ)/dy^2) + (d^2ψ)/(dz^2) − (1/c^2) * [(d^2ψ)/(dt^2)]= 0 is NOT invariant under Galilean transformation. (i.e., the equation does NOT have the same form for a moving observer moving at speed of...
  3. S

    Second order pde - on invariant?

    second order pde -- on invariant? What the meaning for a second order pde is rotation invariant? Is all second order pde are rotation invariant? or only laplacian?
  4. A

    A question about invariant factors

    A Theorem in our textbook says... If R is a PID, then every finitely generated torision R-module M is a direct sum of cyclic modules M= R/(c_1) \bigoplus R/(c_2) \bigoplus ... \bigoplus R/(c_t) where t \geq 1 and c_1 | c_2 | ... | c_t . There is an example from our textbook that I...
  5. H

    Invariant Tensors and Lorentz Transformation

    It is often stated that the Kronecker delta and the Levi-Civita epsilon are the only (irreducible) invariant tensors under the Lorentz transformation. While it is fairly easy to prove that the two tensors are indeed invariant wrt Lorentz transformation, I have not seen a proof that there aren't...
  6. popbatman

    Left and right invariant metric on SU(2)

    Homework Statement I nedd some help to write a left-invariant and right invariant metric on SU(2) Homework Equations The Attempt at a Solution
  7. D

    Invariant Mass/Inelastic Collision Problem

    Homework Statement Suppose two "small" particles of equal mass m collide, annihilate each other, and create another particle of mass M > 2m . (Note that the final state is just that one "big" particle, nothing else.) If one of the small particles is initially at rest, what must be the minimum...
  8. D

    If X is a left invariant vector field, then L_x o x_t = x_t o L_x

    If X is a left invariant vector field, then L_x \circ x_t = x_t \circ L_x , where xt is the flow of X and Lx is the left translation map of the lie group G. In order to show this, I am trying to show that x_t = L_x \circ x_t \circ L_x^{-1} by showing that L_x \circ x_t \circ L_x^{-1}...
  9. G

    Invariant spaces and eigenvector problem

    Homework Statement Let W be a 1-dimensional subspace of V that is A-invariant. Show that every non zero vector in W is a eigenvector of A. [A element of Mn(F)] The Attempt at a Solution We know W is A-invariant therefore for all w in W A.w is in W. W is one dimensional which implies to...
  10. S

    Loop Invariant in Analysis of Algorithm

    I can't generally map Loop Invariant method for proving the correctness of an Algorithm. Take the case of an Insertion Sort http://csnx.groups.allonline.in/pool/Introduction%20to%20Algorithms-Cormen%20Solution.pdf in the above link, at the page 2-3, they have proved the correctness of...
  11. tom.stoer

    Invariant mass and spin in deSitter cosmology

    A friend of mine was reading Penrose's new book on CCC; I do not want to discuss this story here but a rather interesting detail which could be relevant w/o the whole CCC stuff. SR and GR rely on (global and local) Lorentz invariance. From these symmetries one can derive invariant mass M² and...
  12. phosgene

    Show that c^2(t^2) - x^2 - y^2 - z^2 is invariant under a change of frame

    Homework Statement Show that the quantity T = c^2(Δt)^2 - (Δx)^2 - (Δy)^2 - (Δz)^2 is invariant under a change of frame Homework Equations Lorentz transformations Δx' = \gamma(Δx - vΔt) Δt' = \gamma(Δt - vΔx/c^2) Δy' = Δy Δz' = Δz The Attempt at a Solution I know that the way to do...
  13. C

    Check to see whether it is time invariant or not?

    Hi the problem that I am dealing with is to check whether y[n]=T{x[n]}=x[kn] time invariant or not? My solution is I said z(n)=T{x[n-A]}=x[k(n-A)] and y[n-A]=x[k(n-A)] and because y[n-A]=z(n) so it is time invariant but solution is saying that it is time varying because...
  14. D

    MHB Diffusion Equation Invariant to Linear Temp. Transform

    Show the diffusion equation is invariant to a linear transformation in the temperature field $$ \overline{T} = \alpha T + \beta $$ Since $\overline{T} = \alpha T + \beta$, the partial derivatives are \begin{alignat*}{3} \overline{T}_t & = & \alpha T_t\\ \overline{T}_{xx} & = & \alpha T_{xx}...
  15. E

    Source tracelessness, divergencelessness at invariant speed

    Is the local propagation of an entity at invariant speed a sufficient condition for its stress-energy tensor, independently of its explicit mathematical form, to be trace-free, or to have null covariant divergence, or both in curved space-time? In the book “An Introduction To Mechanics”, by...
  16. alemsalem

    Gravitational Wave Local Energy: Not Gauge Invariant?

    I'm reading wald page 85, and he defines a stress-energy tensor for the linearized gravitational field. he mentions that it not gauge invariant as a problem. but isn't that a general property of any tensor (except scalars). so any stress-energy tensor will not be gauge invariant (change of...
  17. H

    What is the Lorentz invariance of power?

    Power, defined as P = dE/dt is Lorentz invariant according to http://farside.ph.utexas.edu/teaching/em/lectures/node130.html, Eq. 1645 But, considering another equation for the power, P = q E v, where E and v are electric field and velocity vectors, respectively; this is obviously not the...
  18. P

    Lorentz invariant theory, irreducible representations

    "In a Lorentz invariant theory in d dimensions a state forms an irreducible representation under the subgroups of SO(1,d-1) that leaves its momentum invariant." I want to understand that statement. I don't see how I should interpret a state as representation of a group. I have learned that...
  19. D

    Solving Higgs Decay Invariant Averaged Amplitude Problem

    Homework Statement I have decay of Higgs to fermion and antifermion and I need to find out the invariant, averaged amplitude. And I wrote down the Feynman diagram, and calculated everything and I came to this part: \langle|M|^2\rangle=\frac{g_w^2}{4}\frac{m_f^2}{m_w^2}(4p_1\cdot...
  20. O

    Difficulty understanding invariant 'c' implications

    In our world where 'c' is large, most people intuitively understand Galilean addition of velocities at everyday speeds i.e. if someone stands 40m behind me and rolls a ball towards me at 10m/s (assuming we are not moving relative to each other) it takes 4s to reach me. If we repeat the...
  21. F

    Prove square of four-momentum is relativistic invariant

    Homework Statement Hi everyone, I have a physics assignment that asks: Prove that the square of relativistic four-momentum for a massive particle is a relativistic invariant under Lorentz transformations. Can anyone help me to work on the problem? I'm always lost in the class ever since my...
  22. I

    Quickly: Multlinearity of exterior derivative, and proof of invariant formula

    Hi all, I am trying to prove the invariant form for the exterior derivative http://en.wikipedia.org/wiki/Exterior_derivative#Invariant_formulations_of_grad.2C_curl.2C_div.2C_and_Laplacian by following these notes...
  23. T

    Applications of Invariant Theory to Quantum Physics

    Hey everybody, I have to give a talk in our seminar on invariant theory of Lie Groups. And I'm now looking for easy applications of invariant theory to quantum physics. I want to present them to motivate the discussion. I would be lucky if someone of you has an idea where I can found...
  24. M

    Finding Impulse Transfer Function with Impulse Invariant Method

    Homework Statement Transfer functions of the continuous compensation links are given as follows. Find the impulse transfer functions of the digital compensation links using the impulse invariant method. \frac{a}{s+a} I don't know how to solve the problem correctly :cry: Homework...
  25. F

    Understanding the strain energy function invariant term

    Hi, Dear all, Facing problem to understand strain energy function invariant terms A typical strain energy function consist of strain invariant can be defined as followed W(I1,I4)=C0+C1(I1-3)(I4-1)+C2(I1-3)^2+C3(I1-4)^2+C4(I1-3)+C5(I4-1), I1 and I4 are so called invariants of Green's strain...
  26. L

    Weinberg QFT - Inner product relations, Standard momentum, Invariant integrals

    Weinberg in his 1st book on QFT writes in the paragraph containing 2.5.12 that we may choose the states with standard momentum to be orthonormal. Isn't that just true because the states with any momentum are chosen to be orthonormal by the usual orthonormalization process of quantum mechanics...
  27. Doofy

    The Jarlskog invariant and leptogenesis?

    I've been trying to find out some info about CP violation in the lepton sector at a basic (ie. a fresh postgraduate) level. We can take the neutrino mixing matrix U in its standard parametrization: \left( \begin{array}{ccc} c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta} \\ -s_{12}c_{23}...
  28. E

    Noether current for SO(N) invariant scalar field theory

    Homework Statement I understand the premise of Noether's theorem, and I've read over it in as many online lectures as I can find as well as in An Introduction to Quantum Field Theory; Peskin, Schroeder but I can't seem to figure out how to actually calculate it. I feel like I'm missing a...
  29. L

    Finding Eigenvalues and determine if there are invariant lines

    Homework Statement Find the eigen values of the following mapping and determine if there are invariant lines. (2 -4) (-3 3) is the mapping. Homework Equations det (L-λI)=0 The Attempt at a Solution L-λI= (2-λ -4) (-3 3-λ) det(L-λI)=0=ac-bd=(3-λ)(2-λ)-12 ...
  30. V

    Parity switching wave functions for a parity invariant hamiltonian?

    Hi guys, I'm reading Shankar and he's talking about the Variational method for approximating wave functions and energy levels. At one point he's using the example V(x) = λx^4, which is obviously an even function. He says "because H is parity invariant, the states will occur with alternating...
  31. S

    Invariant vectors/eigenvectors of R(., v)v

    I'm afraid I need help again... First, these two things are shown: 1) Let v \in T_{\bar p}\mathbb{CP}^n, ||v|| = 1. Then: R(w, v)v = w \forall w \in (\mathbb Cv)^\perp 2) Let v \in T_{\bar p}\mathbb{HP}^n, ||v|| = 1. Then: R(w, v)v = w \forall w \in (v\mathbb H)^\perp Afterwards...
  32. F

    E-P invariant on relativistic mecahnics problem

    Homework Statement A particle of mass m moving at speed \frac{3}{5}c collides with an identical particle at rest, and forms a new particle of mass M which moves off at speed v. Find v.Homework Equations E-P invariant: E_1^2-p_1c^2=E_2^2-p_2^2c^2=\mathrm{const.} Momentum...
  33. R

    Find all Invariant Probability Measures for P (Markov Chain)

    Homework Statement Find all Invariant Probability Measures for P (Markov Chain) E = {1,2,3,4,5} The screenshot below has P and my attempted solution. I am wondering if it acceptable to have infinitely many answers ("all" seems to indicate that is acceptable). Basically, I had too many unknowns...
  34. A

    Why is C Invariant? FORs Explained

    Why is c invariant for all FORs? Special R took it as a postulate but it did not explain it,,,
  35. Z

    What is the loop invariant for this algorithm?

    Hi, I'm having trouble getting a loop invariant expression for this algorithm: Majority(A): c = 1 m = A[0] for i = 1 to len(A) - 1: if c == 0: m = A[i] c = 1 else if A[i] == m: c = c + 1 else: c = c - 1 return...
  36. R

    How to prove invariance of I in this system?

    Homework Statement Show that I = log(u)-u+2log(v)-v is an invariant of the following system \dot{u}=u(v-2) \dot{v}=v(1-u) Homework Equations The Attempt at a Solution The question was given on a homework assignment, but I have very little idea what it is asking for and even...
  37. E

    Gauge invariant Lagrangian: unique?

    Hi all! Long story short, my QFT class recently covered gauge equivalence in QED, and this discussion got me thinking about more general gauge theory. I spent last weak reading about nonabelian symmetries (in the context of electroweak theory), and I like to think I now have a grasp on the...
  38. M

    Global U(1) invariant of Dirac Lagrangian

    Does anybody know what interpretation the invariant corresponding to the global U(1) invariance of the Dirac Lagrangian is? I have always had it in my head that it's charge, but then I realized that uncharged free particles such as neutrinos satisfy this equation too! Any thoughts much...
  39. E

    Discrete time invariant system

    the discrete time system defined by y[n]= x[n] ^ 2 Is it time varying ? I proceeded as follows x[n] → x[n]^2 x[n+a] → x[n+a]^2 so y[n+a] = x[n+a]^2 So according to me it is time invariant Am i right ?
  40. J

    Free will and Emmy Noether's theorem of time invariant systems

    Hey all, Since first learning about Emmy Noether's proof that time invariant laws of physics imply conservation of energy, I can't shake the idea that this is the argument against the notion of free will. Here is my argument: By Noether's first theorem, whenever the laws are invariant in...
  41. N

    Why is Casimir operator to be an invariant of coresponding Lie Algebra?

    Please teach me this: Why is Casimir operator T^{a}T^{a} be an invariant of the coresponding Lie algebra? I know that Casimir operator commutes with all the group generators T^{a}. Thank you very much for your kind helping.
  42. M

    Deriving Invariant Distance in Space Time: Pythagorean Theorem

    Why in deriving invariant distance in space time we use Pythagorean Theorem with a negative sign?
  43. J

    Mini black hole not Lorentz invariant?

    Let's say that we have a particle flying through space, at a collision course with a planet. As seen from an observer on this planet, the particle has an enormous energy, and its wavelength is just slightly bigger than the Planck length. As the particle falls down the gravitational well of the...
  44. K

    Potential Invariant under translation

    When I was learning translational symmetry I saw that for translation invariance, i.e [T,H]=0 the momentum P needs to be conserved [P,H]=0. This momentum is actually the generator of small translations defined as T:x→x+ε. Now, I was solving some problems and I met one which is...
  45. N

    Four-Momentum Invariant and Conservation Laws Yielding Contradictory Results

    Hello PF community! I'm having trouble with what strikes me as an inconsistency within conservation of energy, conservation of momentum, and the four-momentum invariant equation (E2-p2c2 = m2c4). For the sake of this question, I'll be using non-relativistic mass--i.e. mass is the same in all...
  46. Rasalhague

    Degree p Invariants in Linear Tensor Products: Bishop & Goldberg, p. 86-87

    ... I think I've misunderstood their definition of an invariant. The pth power of the trace function seems to be homogeneous of degree p rather than linear: I(\lambda A) = J((\lambda A)\otimes (\lambda A)) = (\text{Tr}\, \lambda A)^2 =\lambda\lambda...
  47. K

    Lie derivative of two left invariant vector fields

    Hi all, I was following Nakahara's book and I really got my mind stuck with something. I would appreciate if anybody could help with this. The Lie derivative of a vector field Y along the flow \sigma_t of another vector field X is defined as L_X...
  48. M

    Varying Gravitational Field - Invariant Tetrahedron?

    Varying Gravitational Field - Invariant Tetrahedron?? Classical Theory of Fields, Landau Lifgarbagez, page 246: "Strictly speaking, the number of particles should be greater than four. Since we can construct a tetrahedron from any six line segments, we can always, by a suitable definition of...
  49. C

    Lagrangian invariant but Action is gauge invariant

    Homework Statement So I'm having some difficulty with my QFT assignment. I have to solve the following problem. In three spacetime dimensions (two space plus one time) an antisymmetric Lorentz tensor F^{\mu\nu} = -F^{\nu\mu} is equivalent to an axial Lorentz vector, F^{\mu\nu} =...
  50. C

    Gamma5 x d^2 Lorentz invariant?

    is \overline{\Psi} γ5 \partial2 \Psi Lorentz Invariant? How does this term transform under Lorentz transformations? Here \Psi is a Dirac field. Thanks
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