Invariant Definition and 387 Threads

In classical mechanics, isn't kinetic energy not a Galilean scalar? So the action isn't invariant under Galilean transformations, but we can still use it with Galilean transformations. So why must it be a scalar in special relativity? I think I'm missing something...

What does it mean for a vector to remain "invariant" under coordinate transformation? I think I already know the answer to this question in a foggy, intuitive way, but I'd like a really clear explanation, if someone has it. I know all of multivariable calculus and quite a bit of linear algebra...

Homework Statement Show that the electromagnetic wave equation \frac{\partial^{2}\phi}{\partial x^{2}} + \frac{\partial^{2}\phi}{\partial y^{2}} + \frac{\partial^{2}\phi}{\partial z^{2}} - \frac{1}{c^2}\frac{\partial^{2} \phi}{\partial t^2} is invariant under a Lorentz transformation...

I am following a proof in the text "Algebras of Linear Transformations" and having problem justifying this line: ... M is an invariant subspace so it has an eigenvector. Why should an invariant subspace have an eigenvector? Thank you I have a feeling this is a very simple result, if so I am sorry

Some time ago, I came across a nice justification (by Einstein IIRC) for the formula x'^2 + y'^2 + z'^2 - c^2t'^2 = x^2 + y^2 + z^2 - c^2t^2. The argument went something like this: (1) x'^2 + y'^2 + z'^2 - c^2t'^2 = x^2 + y^2 + z^2 - c^2t^2 = 0 for light. (2) *reasoning I forget*, therefore...

Hi, Would someone know where I can find a derivation of the lorentz-invariant lagrangian density? This lagrangian often pops-up in books and papers and they take it for granted, but I was actually wondering if there's a "simple" derivation somewhere... Or does it take a whole theory and...

Hi, I was wondering, and I hope this isn't a ridiculous question, for the set of motions: 4 translations, 3 rotations, and 3 boosts; is there an invariant tensor for any metric under all 10 of these motions. That is, preforming these 10 motions, is there a tensor which remains unchanged...

A vector in special relativity is the quantity: V = V^\mu \hat{e_\mu} On a change of coordinates, the basis vectors co-vary with the coordinate derivatives: \hat{e_\mu'} = \frac{\partial x_\mu'}{\partial x_\mu} \hat{e_\mu} The vector elements are the opposite. They are said to be...

I'm really confused about invariant quantities.Could someone explain which quantities are invariant in special relativity and how are they recognised? thanks

I'm just beginning to learn about Feynman diagrams and wanted to make sure I've got the correct basic understanding of QED. This is what I believe to be true right now: QED allows us to describe the interaction between an EM field and light/matter. The QED vertex is composed of a photon and...

I would like to know if the following correct. Suppose I start with a connection on a real vector bundle and extend it to the complexification of the bundle. The curvature forms of the complexification seem to be the same as curvature 2 forms of the real bundle. From this it seems that the...

Hi, So if we have the Lagrange density for a massless scalar field: L=\sqrt{-g}\left(-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{(n-2)}{4(n-1)} R\phi^2\right) Then under a conformal transformation g_{\mu\nu}=\omega^{-2}\tilde{g_{\mu\nu}} , then the Ricci sclar goes to...

Homework Statement Prove that the element dt\ dx\ dy\ dz is invariant under Lorentz boost with velocity \beta along z axis. Homework Equations Convention c=1 Lorentz boost in z direction: L(z)=\left[ \begin{array}{cccc} \gamma & 0 & 0 & -\gamma\beta \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\...

Homework Statement Calculate the differential cross section for A+B---> C+D with an invariant matrix element Homework Equations See attachment The Attempt at a Solution I have no idea how to even begin this problem. The course I am taking is an undergraduate course in intermediate modern...

Homework Statement Let U be a subspace of V. Suppose that U is a invariant subspace of V for every linear transformation from V to V. Show that U=V. Homework Equations no The Attempt at a Solution Assume U is not trivial: Now we only need to show that U = V. Let dimV = n: We can...

How can I show that trace is Invariant under the change of basis?

Can anyone please describe me the following theory in plain english. Your help will be much appreciated. "In certain types of systems there is an invariant of motion which is equal to the sum of the kinetic energy and the potential energy of all the particles in the system. This quantity is...

I'm hoping there will be some comment on this new paper of Kirill Krasnov http://arxiv.org/abs/1101.4788 Gravity as a diffeomorphism invariant gauge theory Kirill Krasnov 24 pages (Submitted on 25 Jan 2011) "A general diffeomorphism invariant SU(2) gauge theory is a gravity theory with two...

Homework Statement Hello. I came across a question that required me to solve for invariant points between a base trig function and the function after horizontal stretch. I can't remember the exact question right now, but I'm just wondering how I would go about solving it if I didn't know...

Hi Lets say I have a Hamiltonian which is invariant in e.g. the spin indices. Does this imply that spin is a conserved quantity? If yes, is there an easy way of seeing this? Niles.

Homework Statement Propose a Second order Model in the continuous time domain Natural damping frequency of 0.6 Steady State gain of 2 units Write down the corresponding differential equation I have no idea where to begin with this, could someone help me with the process, then i will...

In special relativity, the metric tensor is invariant under Lorentz transformations: \Lambda^\alpha{}_\mu \Lambda^\beta{}_\nu g^{\mu \nu} = g^{\alpha \beta} Is this the unique rank 2 tensor with this property, up to a scaling factor? How would I go about proving that? I know that two...

massless Klein-Gordon equation not conformally invariant?? Wald discusses conformal transformations in appendix D. He shows that the source-free Maxwell's equations in four dimensions are conformally invariant, and this makes sense to me, since with photons all you can do is measure the...

Salutations all, just stuck with the starting step, I want to see if I can take it from there. Homework Statement Let G be a group and let N be a subgroup of G. Prove that the set g^{-1}Ng is a subgroup of G. The Attempt at a Solution Well, I'm going to have to show that...

Can someone tell me what a riemann invariant is in general terms please? i can't find it anywhere! thanks, lav

Verify that the Lagrangian density L= \frac{1}{2} \partial_\mu \phi_a \partial^\mu \phi_a - \frac{1}{2} m^2 \phi_a{}^2 for a triplet of real fields \phi_a (a=1,2,3) is invariant under the infinitesimal SO(3) rotation by \theta \phi_a \rightarrow \phi_a + \theta \epsilon_{abc} n_b \phi_c...

For a euclidean space, the interval between 2 events (one at the origin) is defined by the equation: L^2=x^2 + y^2 The graph of this equation is a circle for which all points on the circle are separated by the distance L from the origin. For space-time, the interval between 2 events is...

Minkowski vacuum is Poincare invariant and quasi-free state. I wonder if these two conditions fully define it or there are more states which fulfill these conditions (or maybe Poincare invariance alone is sufficinet). Thanks for answers.

what is modulo associated with z2 invariant?

Hi everyone, I was wondering: if a space is invariant under Poincare transformations, does that mean it has to be Minkowski space? Or could it have some further isometries? By the same token, if a space is invariant under the orthogonal transformations, does it have to be Euclidean? I...

Hi guys, Before responding to my post, please note that I am only familiar with the mathematics of nonrelativistic quantum mechanics, and don't know any quantum field theory. All I have is this vague idea that quantum field theory is the union of special relativity and quantum mechanics...

1. Prove that the rank of a matrix is invariant under similarity.Notes so far: Let A, B, P be nxn matrices, and let A and B be similar. That is, there exists an invertible matrix P such that B = P-1AP. I know the following relations so far: rank(P)=rank(P-1)=n ; rank(A) = rank(AT); rank(A) +...

So I'm trying to get an idea of what an invariant subspace is and so please let me know if my understanding is correct. Given that you have some vector subspace being a collection of a particular number of vectors with the the space denoted as |\gamma>. If you have some other collection of...

"invariant" Lagrangian or action Hello everyone, I tried to describe my question but it seems getting too complicated and confusing to write down my thoughts in detail, so I am trying to start with the following question... Are invariance of the Lagrangian under a transformation and...

Can someone give me some hints on how to prove the following statement: if f: S^3 \to S^2, g: S^3 \to S^3 then H(f\circ g) = deg~g H(f) where H is the Hopf invariant and deg g is the degree of g. I'm pretty clueless on how to start and I don't see how to get the deg to come in since that has...

I'm studying General Relativity and facing several problems. We know that energy-momentum must be Lorentz invariant in locally inertial coordinates. I am not sure I understand this point clearly. What is the physics behind?

Homework Statement so I'm doing some proof-of-concept data analysis this summer and I've never taken a relativistic mechanics class and I'm a bit stuck. i need to figure out if there was a rho0 decay to pi+/pi- in some hypothetical 900GeV collision data. If there is, there should be a spike...

Hi there!I'm trying to prove the following obvious statement, but am somehow stuck :( Let \vec a,\ \vec b\in\mathbb{R^3} , let M be in SO(3) and x be the cross productprove: M(\vec a\times\vec b)=M\vec a\times M\vec bI tried using the epsilon tensor, as in physics, but it doesn't really...

Hi, hopefully this isn't a dumb question. I've read essentially that in the center of mass/momentum frame an object has invariant mass, and that the system's total mass will be composed of the constituent particles' masses and any other kinetic and potential energies within the system. I also...

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...

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,[SIZE="3"][SIZE="5"] 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...