Poincare Definition and 139 Threads

I have been quite puzzled for some time with the concept of Diffeomorphic Invariance. Here is what I think about it, 1) Diffeomorphic Invariance is the invariance of the theory under general coordinate transformations. For instance the Einstein-Hilbert action is diffeomorphic invariant...

Fluctuation theorem says that there will be fluctuations in microscopic scale that results local decreases in entropy even in isolated systems. According to the Poincaré recurrence theorem, after sufficiently long time, any finite system can turn into a state which is very close to its initial...

I heard that some physicists are trying to determine the spacial/geometric curvature of the universe by measuring the angles of distant stars (a very large triangle). Is this possible? Or is Poincare correct when he said that there is no preferred geometry and that there is no experiment...

What is the theory behind mapping of the latitude and longitude of the sphere in the Poincare Circle to the polarization of the TEM wave? That is, why: 1) Linear polarization when ε=0 deg? 2) Circular polarization when ε=+/- 45 deg? 3) Elliptical when ε is not 0 or +/- 45 deg? 4) RH rotation...

The question is here: http://physics.stackexchange.com/questions/45341/representing-a-polarization-vector-for-light-as-a-manifold-of-two-state I'd appreciante any help

Hello, it is well-known that with stereographic projection we can obtain a 1-1 correspondence between the points of the 2d Cartesian plane (plus the point at infinity), and the points on the Riemann sphere. What is the geometrical construction that corresponds to a 1-1 mapping between the...

1) Since Wigner it is well known that for massless particles of spin s the physical states are labelled by helicity h = ±s; other states are absent. So e.g. for photons the physical states are labelled by |kμ, h> with kμkμ = 0 and h = ±1 and we have two d.o.f. 2) For gauge theories with...

Homework Statement Consider the Klein-Gordan action. Show that the Noether charges of the Poincare Group generate the Poincare Algebra in the Poisson brackets. There will be 10 generators.Homework Equations \{ A,B \}=\frac{\delta A}{\delta \phi}\frac{\delta B}{\delta \pi}-\frac{\delta...

Hi all! I'm trying to study the Poincare group and I have one problem. I'm reading a book: Gross D. Lectures on Quantum Field Theory (there is section about it). So I do not understand how the second part of (3.26 and 3.27) folows from the first part i.e I do not understand how was obtained...

There exists Green's Functions such that the solutions appear to be retro-causal. The Klein-Gordon equation allows for antiparticles to propagate backwards in time. Does this mean the future can influence the past and present? Then again The Poincare Recurrence Theorem states that over a...

I am reading Thurston's book on the Geometry and Topology of 3-manifolds, and he describes the metric in the Poincare disk model of hyperbolic space as follows: ... the following formula for the hyperbolic metric ds^2 as a function of the Euclidean metric x^2: ds^2 = \frac{4}{(1-r^2)^2} dx^2...

Hi! I'm trying to give a few examples of symmetric manifolds. In the article "Introduction to Symmetric Spaces and Their Compactification" Lizhen Ji mentions the Poincaré disk as a symmetric space in the following way: D = \{z \in \mathbb C | |z| < 1\} with metric ds^2 =...

I've been doing a project on Henri Poincare and I am attempting to explain his conjecture to my Calculus class so I am using the common lower dimensional equivalent to do so. If a rubber band is wrapped around an object and becomes smaller and smaller until it is a point than that object is...

This is how Wikipedia summarizes the Poincaré Recurrence Theorem: This is wrong, isn't it? Don't you need to ensure the phase space is bounded, and isn't conservation of energy an insufficient justification for that? Like, imagine throwing two baseballs away from each other into infinite...

Homework Statement System in polar coordinates \dot{r} = 2r - r3(2 + sin(\theta)), \dot{\theta} = 3 - r2 Use a trapping region to show there is at least one periodic orbit? Homework Equations By using Poincare Bendixson's Theorem The Attempt at a Solution I am struggling to...

Hi everyone, I know that there is a result that corresponding to a closed p-form α, I can find a p-1 form β, such that dβ = α. I wanted to ask, what the tensorial analog of this would be. I mean would it be right to say that on a manifold with a lorentzian metric, If I have a vector...

Here is what I've done so far How do I draw the Phase portrait for this system? Have I done everything correct so far? Thanks

i am trying to figure out how to calculate the circumference of a circle in the Poincare Half Plane. I know that vertical lines are geodesics so using the arclength formula, the distance between 2 points (x_0, y_0) and (x_1, y_1) on a vertical line is ln(y_1/y_0) . Thus, if i have a circle...

Hey Guys! I'm having MAJOR difficulty with a problem regarding stability of a DE. The problem goes as following: Find the equation of the phase paths of x˙=1+x^2, y˙=−2xy. It is obvious from the phase diagram that y=0 is Poincaré stable. Show that for the path y=0, all paths which start in...

The words Poincare and Lorentz sound pretty elegant. I think they are French words like Loreal Or Laurent. I know Poincare has to do with spacetime translation and Lorentz with rotations symmetry. But how come one commonly heard about Lorentz symmetry in Special Relativity and General...

I read here http://www.eoht.info/page/Genius that henri poincare got a 35 on an iq test. I think that is really interesting, if its true! I've heard that people with mental deficiency score in the 60's, so 35 would be really low! I used to think iq meant something years ago, but I dropped...

I'm reading a physics book and in the section on relativity they are using the Einstein summation convention, with 4vectors and matrices. They say that the transformations take the form: x^{\prime\mu}=x^{\nu}\Lambda^{\mu}_{\nu}+C^{\mu} where it is required that \Lambda^{\mu}_{\nu} satisfy the...

Alright, so I was reading Ryder and he defines the generator corresponding to a^{\alpha} as the following X_{\alpha}=\frac{\partial x'^{\mu}}{\partial a^{\alpha}}\frac{\partial}{\partial x^{\mu}} (\alpha =1,...r) for r-parameter group of transformations Now this makes sense for...

The poincare' group is the group of isometries of Minkowski spacetime, in a nutshell. In terms of an actual physical definition it is the group of all distance preserving maps between metric-spaces in Minkowski spacetime. What is the difference between this and geodesics?

Hi, I'm sorry to have to ask this, but I can't seem to reason this one out by myself at the moment. Given the metric is the Minkowski spacetime, is the group of diffeomorphisms the poincare group, or are there diffeomorphisms for flat-space that are not metric preserving? I would really...

Given [M^{\mu \nu},M^{\rho\sigma}] = -i(\eta^{\mu\rho}M^{\nu\sigma}+\eta^{\nu\sigma}M^{\mu\rho}-\eta^{\mu\sigma}M^{\nu\rho}-\eta^{\nu\rho}M^{\mu\sigma}) and [ P^{\mu},P^{\nu}]=0 I need to show that [M^{\mu\nu},P^{\mu}] = i\eta^{\mu\rho}P^{\nu} - i\eta^{\nu\rho}P^{\mu}We've been given the...

Hi, I am working through Maggiore's QFT book and have a small problem that is really bothering me. It involves finding the representation of the Poincare algebra generators on fields. I always end up with a minus sign for my representation of a translation on fields compared to Maggiore...

I've read that it is also a Lie group. But what does that have anything to do with special relativity or different reference frames? The wiki definition is "a 10-dimensional noncompact Lie group. The abelian group of translations is a normal subgroup while the Lorentz group is a...

Reading this article: http://arxiv.org/abs/math-ph/0102011 made me wonder: 1.) So, it appears that Galilean transformations are not the most general symmetry transformations of nonrelativistic mechanics. Fine. 2.) The article states that the two additional symmetries are the nonrelativistic...

I'm new here and I have checked the FAQs. I'm not sure if this question has been posted before. This may actually be a silly question. Why do we study Lorentz and Poincare Groups? I have studied a bit of the theory but was wondering what exactly are we talking about when we study the...

Considering that the smallest particles in nature are supposed to be strings, which are donut and line shapes. And the poincare conjecture says the simplest shape in nature is a sphere. wouldn't it make sense that the true fundamental particles are sphere shaped and that if they combine to form...

Can any continuous coordinate transformation on a differential manifold be viewed as a poincare transformation locally in every tangent space of this manifold? Thx!

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.

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...

Homework Statement I have to show that the Poincare disk satisfies the incidence axiom that any line contains at least two points. Homework Equations There are two kinds of lines on the Poincare disk. I've found 2 points for the first kind, which are straight lines going through the...

Hi. I want to talk about the derivation of the form of space-time transformations T between inertial references. As it's known, this is the Poincaré group, defined as the group that leaves invariant the (+,-,-,-) metric. This derives from two things: c is constant and space-time is homogeneous...

Hi, I have a system of ODE's for which I want to compute points for a Poincaré section. I used NDSolve and EventHandler in Mathematica but EventHandler appears rather messy/limited for my condition: Suppose I want to save 4 iterations of a Poincaré map and then abort integration imediately...

Hi, I'm rather confused about the procedure in which people obtain gravity from gauged (super)-Poincare algebras. Let me outline what this procedure is. *First you gauge the Poincare algebra with generators P and M *You obtain two gauge fields: the vielbein (associated with P) and the...

*I am using the conventions of circular polarisation according to electrical engineering , not the one used in optics* Let us take a uniform plane TEM wave traveling in +z direction which is composed of two linearly polarised TEM waves , one whose electric field lies in X direction , the other...

Suppose that we write a function that changes coordinates from one inertial frame to another in the form x\mapsto\Lambda x+a where \Lambda is linear, with components \Lambda=\gamma\begin{pmatrix}1 & \alpha\\ -v & \beta\end{pmatrix} in the standard basis. (This is the most general...

Does someone know something about this theory? Is it popular, or just accepted by very few people? What is the flaw or defect of this theory? Has it been ruled out by some observations or experiments? Thx.

Einstein is commonly regarded as the primary inventor of special relativity. I'd like to trust the collective wisdom of others, but I never feel fully convinced. The argument is typically "Though people like Lorentz and Poincare have worked out most of the math it's only Einstein who realized...

Hello, in relativistic quantum field theories all particles are members of (unitary) representations of the Poincare group. For massive particles m² > 0 one gets the usual scalar / spinor / vector representations with spin J = 0, 1/2, 1, 3/2, ... and dim. rep. = 2J+1. For massless particles...

Hi, Can someone please explain the following statement on page 62 of Weinberg's Vol 1 on QFT: (I understand the part for P ~ mv, so the "quote" is slightly distorted, intentionally). Also how is ? Thanks in advance!

hello! i would like to be able to understand and appreciate the proof of the poincare conjecture. i have some idea of where to begin, and my supervisor is going to help me out (i'm starting a master's in pure math and my supervisor does geometric analysis), but i was wondering if anyone here...

It is my understanding that in string theory, loop quantum gravity, the 'asymptotic safety' approach, and in semiclassical quantum gravity, local Poincare symmetry is exact. But there are things like DSR (does the D stand for Deformed, or Doubly? I've heard people say it either way), which...

Benjamin Crowell writes here, "The discontinuous transformations of spatial reflection and time reversal are not included in the definition of the Poincaré group, although they do preserve inner products." http://www.lightandmatter.com/html_books/genrel/ch02/ch02.html So, if I've...

How do I write u''+u*e^x = 0 as a planar system?

hi there What is a Poincare' disc and why is the edges of disc represent infinity? thanks

(I posted this in Particle Physics too) We call a group G "simply connected" if every curve C(t) in G which is closed (that is, C(0) = C(1) = I) can be continuously deformed into the trivial curve C'(t) = I (where I is the unit element in G). This is formalised saying that, for each closed...