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. M

    How can I compute translation invariance for p in a one dimension wave function?

    Hi everyone, Let \psi (x) be a one dimension wave function. We suppose \langle p \rangle =q How can I compute the new \langle p \rangle when we set \psi _1 (x) := e^{\frac{ip_0x}{\hbar}}\psi (x) I want to comute it with the formula \langle p \rangle=-i \hbar \int \psi _1^*...
  2. Breo

    Internal Symmetries (Prove Lagrangian is invariant)

    Homework Statement Note: There is an undertilde under every $$\phi$$ Imagine $$ \phi ^t M \phi $$ . M is a symmetric, real and positive matrix. Prove L is invariant: $$ \mathcal{L} = \phi ^t M \phi + \frac{1}{2} \partial_\mu \phi ^t \partial ^\mu \phi $$ Trick: Counting parameters. Homework...
  3. A

    Why is force invariant in Newtonian mechanics?

    Recently, I've been pondering deeply on relativity (both Galilean and SR) and all of a sudden I find that I don't grasp even the basic concepts of physics (or life) anymore, i.e. I can't go back to my previous, "normal" mode of thinking. Consider Newtonian mechanics, take the ground to be at...
  4. R

    Showing that Energy-momentum relation is invariant

    Homework Statement [/B] A particle of mass m is moving in the +x-direction with speed u and has momentum p and energy E in the frame S. (a) If S' is moving at speed v, find the momentum p' and energy E' in the S' frame. (b) Note that E' \neq E and p' \neq p, but show that...
  5. mnb96

    Question about invariant w.r.t. a group action

    Hello, I have a group (G,\cdot) that has a subgroup H \leq G, and I consider the action of H on G defined as follows: \varphi(h,g)=h\cdot g In other words, the action is simply given by the group operation. Now I am interested in finding a (non-trivial) invariant function w.r.t. the action of...
  6. KleZMeR

    Invariant quantities of a lagrangian?

    Given a basic Lagrangian, how would I determine invariant quantities? My hunch says it would be quantities that do not depend on position or time? Saying that, perhaps using the Lagrange equation to solve for equations of motion and along the way whatever terms disappear would be my invariant...
  7. AbhiFromXtraZ

    The Principle of Invariant Light Speed

    While reading Special Theory of Relativity from Feynman Lectures, I fell into the confusion about invariant speed of light. What I'm asking for is an explanation about this. No matter whether physical explanation or mathematical. So my question is Why the speed of light is same for a person...
  8. Greg Bernhardt

    What is Poincare's Integral Invariant

    Definition/Summary Poincare's integral invariant is the most fundamental invariant in Hamiltonian Dynamics. For any phase space set, the sum of the areas of all of its orthogonal projections onto all the non-intersection canonically conjugate planes is invariant under Hamiltonian evolution...
  9. M

    Why must the form of v_i v_j be independent of coordinate system?

    Hey PF! I am trying to understand what is meant when we say a vector is invariant, which I believe is independent of a coordinate system. I have already read a PF post here: https://www.physicsforums.com/showthread.php?t=651863. I'm looking at DH's post, and this makes a lot of sense...
  10. ChrisVer

    Calculating the Invariant Matrix

    I am having one question... If we know the form of the effective Lagrangian, let's say the form: L= g (\bar{\psi}_{e} \gamma^{\mu} P_{L} \psi_{\nu})(\bar{\psi}_{p} \gamma_{\mu} P_{L} \psi_{n}) How can someone calculate the spin averaged invariant matrix \large M? I mean I can do the...
  11. B

    Entropy of canonical ensemble - invariant wrt energy shift?

    I have a short question which I have been discussing with a fellow student and a professor. The question (which is not a homework question!), is as follows: If you shift all the energies E_i \to E_i + E_0 (thus also shifting the mean energy U \to U + E_0), does the entropy of the system remain...
  12. A

    MHB A question on ergodic theory: topological mixing and invariant measures

    Hi All, This is a question on ergodic theory - not quite analysis, but as close as you can get to it, so I decided to post it here. Suppose I have a compact metric space $X$, with $([0,1], B, \mu)$ a probability space, with $B$ a (Borel) sigma algebra, and $\mu$ the probability measure...
  13. A

    Digital signal processing, linear time invariant system,

    I really confused, I found in a book that the following system, y[n]= x[n+1]-x[n], is not causal! But from the definition of causality that the output y[n0] depends only on the input samples x[n] for n<=n0,,, So I think that this system is causal... If you agree with me please tell me that...
  14. M

    Particle number lorentz invariant?

    Hi folks -- does anyone know of a proof that particle (quanta) number in QFT is / is not Lorentz invariant? I'd be happy to hear of it -- so thanks!
  15. N

    Is Yang-Mills scale invariant?

    I remember hearing this, but not sure if it's true.
  16. C

    MHB Invariant subspaces of representations

    Let $\varrho :\mathbb{Z}\rightarrow GL_3(\mathbb{R})$ be the representation given by $\varrho (n)=A^n$ where A=$\begin{pmatrix} 2 & 5 & -1 \\ 2 & \frac{5}{2} & \frac{11}{2} \\ 6 & \frac{-2}{2} & \frac{3}{2} \\ \end{pmatrix}$ Does ρ have any 1-dimensional invariant subspaces? Do I have to...
  17. B

    MHB Invariant subspace for normal operators

    I have proved the spectral theorem for a normal operator T on an infinite dimensional hilbert space, and am now trying to deduce that T has non-trivial invariant susbspaces. Case 1: If the spectrum of T consists of a single point: My book says that if this is the case then the set of continuous...
  18. pellman

    The invariant momentum-space volume element?

    When we way that \frac{d^3p}{p_0}=\frac{d^3p}{\sqrt{m^2+\vec{p}^2}} is the invariant volume element, is that with respect to all Lorentz transformations or just proper orthochronous Lorentz transformations?
  19. C

    MHB Show all invariant subspaces are of the form

    [solved] show all invariant subspaces are of the form i don't even know how to begin (Angry) C_x is a subspace spanned by x that belongs to V C_x = {x, L(x), L^2(x),...} edit: SOLVED
  20. C

    Proving the dor product of 4-vectors is Lorentz invariant

    Homework Statement Let A and B be 4-vectors. Show that the dot product of A and B is Lorentz invariant. The Attempt at a Solution Should I be trying to show that A.B=\gamma(A.B)? Thanks
  21. L

    Projection to Invariant Functions:

    Context: T : X \rightarrow X is a measure preserving ergodic transformation of a probability measure space X. Let V_n = \{ g | g \circ T^n = g \} and E = span [ \{g | g \circ T = \lambda g, for some \lambda \} ] be the span of the eigenfunctions of the induced operator T : L^2 \rightarrow...
  22. N

    Does a symmetry need to leave the whole action invariant?

    Typically a symmetry is taken to be something that leaves the action invariant. However, on a classical level, isn't that asking way too much? To match what we conceptually mean by symmetry, we only need something that maps solutions to solutions, so something which leaves the action invariant...
  23. junfan02

    Transformations that are scalar invariant

    I am a bit confused about something! Exactly under what kind of transformations are scalars invariant in the domain of classical mechanics? The fact which is disturbing me is, say we have a moving body of certain kinetic energy in a certain inertial frame of ref, and then we choose to.observe...
  24. shounakbhatta

    Is the Lagrangian Invariant or Variant in a Coordinate System Shift?

    Hello, Is Lagrangian invariant? I am in a conversation, where one is saying that: "Shifting the coordinate system changes the value of the potential energy with respect to the same reference level, that's why the Lagrangian changes" While the other: "Shifting the coordinate...
  25. A

    Invariant Spacetime Interval for Classical Spacetime

    In special relativity we have the invariant spacetime interval ds2 = dx2 - c2dt2. If we think about classical (non-relativistic) space and time as one spacetime in which the transformation between reference frames is given by the Galilean transformation, is there a corresponding spacetime...
  26. A

    Translation Invariant: Seeing it Intuitively & Mathematically

    How do I see that when my hamiltonian is translation invariant i.e. H = H(r-r') it means that it is diagonal in the momentum basis? I can see it intuitively but not mathematically.
  27. Y

    Is temperature an invariant in Special Relativity?

    In another current thread on the possible invariance of pressure, I mentioned: What is the current consensus on how temperature transforms in relativity? Here is another simple thought experiment. Consider two very long rectangular objects A and B, that are moving relative to each other...
  28. V

    Is pressure an invariant in Special Relativity?

    HI : Consider a cylinder of length L and volume V that contains one mole of an ideal gas. The familiar ideal gas law states that: PV = RT Now, if the cylinder were to move with velocity v parallel to the length direction, special relativity requires the...
  29. Sudharaka

    MHB Invariant Orthogonal Compliment

    Hi everyone, :) Here's a question with my answer. I would be really grateful if somebody could confirm whether my answer is correct. :) Problem: Prove that the orthogonal compliment \(U^\perp\) to an invariant subspace \(U\) with respect to a Hermitian transformation is itself invariant...
  30. R

    General relativity and invariant mass

    The invariant mass of special relativity: m_0{^2} = E^2 – p^2 There doesn't seem to be any quantity with units of mass that is invariant in general relativity. Invariant mass loses significance, as other than an approximation where space-time is sufficient flat. But at the same time, mass is...
  31. E

    Proving The Hamiltonian Is Invariant Under Coordinate Transformation

    Homework Statement (a) Consider a system with one degree of freedom and Hamiltonian H = H (q,p) and a new pair of coordinates Q and P defined so that q = \sqrt{2P} \sin Q and p = \sqrt{2P} \cos Q. Prove that if \frac{\partial H}{\partial q} = - \dot{p} and \frac{\partial H}{\partial p} =...
  32. D

    Proving that the orthogonal subspace is invariant

    Hi guys, I couldn't fit it all into the title, so here's what I'm trying to do. Basically, I have a unitary representation V. There is a subspace of this, W, which is invariant if I act on it with any map D(g). How do I prove that the orthogonal subspace W^{\bot} is also an invariant subspace...
  33. P

    MHB Proof of Eigenvector Existence for Linear Maps on Finite-Dimensional Spaces

    From wikipedia I read that every linear map from T:V->V, where V is finite dimensional and dim(V) > 1 has an eigenvector. What is the proof ?
  34. B

    Is the geometric multiplicity of an eigenvalue a similar invariant?

    If two matrices similar to one another are diagonalizable, then certainly this is the case, since the algebraic multiplicity of any eigenvalue they share must be equal (since they are similar), and since they are diagonalizable, those algebraic multiplicities must equal the geometric...
  35. J

    Calculating Invariant pT in 2->2 Processes

    What's the definition of invariant pT in a 2->2 process? I know how to calculate the invariant mass in this case, but I am not sure about pT.
  36. fluidistic

    Time translation invariant equation

    Homework Statement I must show that the equation ##\frac{\partial E}{\partial t}=t\nabla \times E## is invariant under time translations and I must also find its symmetries if it has any. Where E is a function of time and position.Homework Equations Already given. The Attempt at a Solution I...
  37. D

    Are the following 2 equations lorentz invariant?

    \partialμ\phi\partialμ\phi and \partialμ\partialμ\phi with \phi(x) a scalar field
  38. P

    Conceptual question: invariant tensors, raising and lowering indices

    Homework Statement When we raise and lower indices of vectors and tensors (in representations of any groups) we always use tensors which are invariant under the corresponding transformations, e.g. we use the Minkoski metric in representations of the Lorentz group...
  39. SamRoss

    Proof Minkowski metric is invariant under Lorentz transformation

    Ok, this should be an easy one but it's driving me nuts. When we take the Lorentz transformations and apply them to x2-c2t2 we get the exact same expression in another frame. I can do this math easily by letting c=1 and have seen others do it by letting c=1 but I have never seen anyone actually...
  40. lonewolf219

    Why is this invariant under U(1) X SU(2) ?

    Could anyone explain why these are invariant under U(1) X SU(2)? H^{dagger}H (H^{dagger}H)^{2} What is the condition for invariance under U(1) and similarly, under SU(2)? Unfortunately, I am not familiar with tensor contraction or tensor products...
  41. M

    In geometry, why the invariant properties that matter?

    Dear all, i'm trying to understand geometry by studying the subject myself. i came across idea that I'm very much confuse of. it say's that 'geometry is a studies of geometric properties that is invariant under transformation' such as distance for euclidean geometry. my question is: why do...
  42. caffeinemachine

    MHB Degree of extension invariant upto isomorphism?

    Let $K$ be a field and $F_1$ and $F_2$ be subfields of $K$. Assume that $F_1$ and $F_2$ are isomorphic as fields. Further assume that $[K:F_1]$ is finite and is equal to $n$. Is it necessary that $[K:F_2]$ is finite and is equal to $n$?? ___ I have not found this question in a book so I don't...
  43. D

    Natural examples of metrics that are not Translation invariant.

    I am trying to explain metric spaces and finding it hard to come up with simple to understand, interesting examples of metrics that are not translation invariant. The audience is people who are just now studying general metric spaces.
  44. B

    Scaling Invariant, Non-Linear PDE

    Homework Statement Consider the nonlinnear diffusion problem u_t - (u_x)^2 + uu_{xx} = 0, x \in \mathbb{R} , t >0 with the constraint and boundary conditions \int_{\mathbb{R}} u(x,t)=1, u(\pm \inf, t)=0 Investigate the existence of scaling invariant solutions for the equation...
  45. ash64449

    Confused on how to calculate invariant interval.

    Hello friends, I have now began to read a new book called space-time physics by Edwin.F.Taylor. In the first chapter(parable of surveyors),First it talks about invariant interval and it says the equation is: ## \sqrt{ (ct)^2 - x^2 } ## In Wikipedia,i saw a different answer for the...
  46. H

    Exploring Invariant Matrix Norms: Understanding the Frobenius Norm and Beyond

    Hi, I don't get which of the many matrix norms is invariant through a change of basis. I get that the Frobenius norm is, because it can be expressed as a function of the eigenvalues only. Are there others of such kind of invariant norms? Thanks
  47. D

    When is something Lorentz invariant.

    Hello there, I'm having a real problem understanding when a certain 'something' (for example Eddington-Finkelstein coordinates) is Lorentz invariant or how you can 'calculate' it. Heck, I'm not even sure if a coordinate system must be lorentz invariant, or if the metric in the equations...
  48. G

    Free action invariant under galliean boosts?

    Is the non-relativistic Lagrangian: \mathcal L=\frac{1}{2}m \dot{x}^2 invariant under boosts x'=x+vt? It doesn't seem like it is. Surely something must be wrong?
  49. X

    Why isn't there such a thing as an invariant velocity for all objects?

    I don't know enough about general relativity to know why the following argument is wrong and I would love to hear why because I'm sure it will be a valuable lesson. The path of light is influenced by gravitationally lensing. Since the speed of light is invariant, the observed path of light...
  50. T

    Invariant symbol with four spinor indices

    Hi, from Srednickis QFT textbook, we know the following coupling of Lorentz group representations: (2,1)\otimes (2,1) = (1,1)_A \oplus (3,1)_S, which yields \epsilon_{a b} as an invariant symbol. Generalising, we can look at (2,1)\otimes (2,1) \otimes (2,1) \otimes (2,1) = (1,1) \oplus...
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