Invariant Definition and 406 Threads

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...

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?

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...

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...

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?

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...

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...

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...

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...

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...

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...

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...

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...

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

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?

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

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)...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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}...

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?

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...

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

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...

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.

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...

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(...

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...

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...

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...

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...

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...

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...

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{}...

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...

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...

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...

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...

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...

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?

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...

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...

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...