Invariant Definition and 387 Threads

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...

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...

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...

How i can see the right lorentz invariance in lqg?

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...

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...

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...

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...

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 ?

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.

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...

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

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...

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...

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...

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?

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...

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...