Poincare Definition and 139 Threads

Consider the following 1-form ##\omega## defined on ##U = \mathbb R^2 \, \backslash \{ 0 \}##: $$\omega = \frac {y} {x^2 + y^2} dx + \frac {-x} {x^2 + y^2} dy$$ It is closed on ##U## since ##\partial \left (\frac {y} {x^2 + y^2} \right) / \partial y = \partial \left (\frac {-x} {x^2 + y^2}...

Hi there I am trying to get into topology I am looking at the poincare conjecture if a line cannot be included as it has two fixed endpoints by the same token isn't a circle a line with two points? that has just be joined together so by the same token the circle is not allowed? Can i get a...

In the far future there will be most likely a point where a maximal state of entropy will be reached in the universe and after the last black hole evaporates there could be no more structures and no more work could be done. According to the Poincaré recurrence theorem for a closed universe...

Does Perelman’s proof of the Poincaré conjecture imply that the universe is the surface of a 3 sphere?

Namely, what does Perelman’s proof of it imply?

Hi, I was attempting a question on the dynamical systems topic of Poincare maps, and was struggling to understand a certain part of it. Knowledge from prior parts of the questions: There was a system which we converted to polar coordinates to get: (## a ## is an arbitrary real constant)...

The full Lorentz group includes discontinuous transformations, i.e., time inversion and space inversion, which characterize the non-orthochronous and improper Lorentz groups, respectively. However, these groups are excluded from the Poincare group, in which only the proper, orthochronous...

Hi. I am working through some notes on the above 2 theorems. Liouville's Theorem states that the volume of a region of phase space is constant along Hamiltonian flows so i assume this means dV/dt = 0 In the notes on the Poincare Recurrence Theorem it states that if V(t) is the volume of phase...

Is this a coincidence?

Using differential expressions for the generator, verify the commutator expression for ##[J_{\mu\nu},P_{\rho}]=i(\eta_{\mu\rho}P_{\nu}-\eta_{\nu\rho}P_{\mu})## in Poincare group Generator of translation: ##P_{\rho}=-i\partial_{\rho}## Generator of rotation...

I see that the first four equations are definitions. The problem is about the dimensions of the quotient. Why does the set Kx forms a six dimensional Lie algebra?

I'm trying to 'see' what the generators of the Poincare Group are. From what I understand, it has 10 generators. 6 are the Lorentz generators for rotations/boosts, and 4 correspond to translations in ℝ1,3 since PoincareGroup = ℝ1,3 ⋊ SO(1,3). The 6 Lorentz generators are easy enough to find in...

Is there a straight-forward, motivated, derivation of AdS Poincare coordinates, e.g. as given here: https://en.wikipedia.org/wiki/Anti-de_Sitter_space#Poincar.C3.A9_coordinates starting from global coordinates, as given here...

I created the attached file in geogebra but I don't know how to come up with less than 60 degrees. There always seems to be at least one angle that is large, no matter how I manipulate the curves. Quadrilateral is points US(1)V(1)W. The attached does not include angle measures but I assume the...

Weinberg QFT book aside, what are good sources for the representation of the Poincare group used in physics?

I'm generating poincare sections of a double pendulum, and they mostly look okay, but some of them have weird discontinuities that seem wrong. The condition for these sections is the standard ##\theta_1 = 0## and ##\dot{\theta}_1 > 0##. Looking at one of the maps, we see that most of the...

Homework Statement Show that for $$W^\mu = -\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}M^{\nu\rho}P^{\sigma},$$ where ##M^{\mu\nu}## satisfies the commutation relations of the Lorentz group and ##\Psi## is a bispinor that transforms according to the ##(\frac{1}{2},0)\oplus(0,\frac{1}{2})##...

I'll have to think a bit about what you've written, but just to note this is not true due to Haag's theorem.

Homework Statement Homework Equations [/B] I believe that ##\frac{\partial x^u}{\partial x^p} =\delta ^u_p ## (1) ##\implies ## (if ##\delta^a_b ## is a tensor, I'm not sure it is?) : ##\frac{\partial x_u}{\partial x^p} = g_{au} \delta ^a_p ## (2) The Attempt at a Solution [/B] sol...

When Grisha Perelman submitted his proof of the Poincare conjecture, he may have been reasonably sure that it contained no mistakes. But he could not have been 100% sure as he is, after all, human. Each time it was checked, say by the referee of an academic journal, the probability that it...

I read that in 1911 Poincare asked Einstein a rather deceptive question. What is the mechanical basis of SR. Einstein replied none, and promtly left, while Poincare was left in shock. I will not say what I think (yet), but was Poincare right to be shocked Einstein dismissed such a question...

Hi all, I'm writing a fortran 90 program to simulate the Duffing oscillator. I have it working and have produced a cos wave which I expected for D (the driving force) being set to 0. Here's the code: program rungekutta implicit none integer, parameter :: dp = selected_real_kind(15,300)...

What is more cool... to be a member of the Poincare Group or Lorentz Group? What name would you choose for a school science team and why?

Hello! :smile: On page 51 where he want to invert $$\Lambda^{\mu}_{\nu} = \tfrac{1}{2} \text{tr}( \bar{\sigma}^{\mu}A \sigma_{\nu} A^{\dagger})$$ the person says we may use $$\sigma_{\nu} A^{\dagger} \bar{\sigma}^{\nu} = 2 \text{tr}(A^{\dagger})I.$$ to do that ... how do you prove this formula...

Context The following is from the book "Ideas and methods in supersymmetry and supergravity" by I.L. Buchbinder and S.M Kuzenko, pg 56-60. It is about realizing the irreducible massive representations of the Poincare group as spin tensor fields which transform under certain representations of...

Homework Statement Does ##x_p\partial_v\partial_u-x_v\partial_p\partial_u=0## Homework Equations I need this to be true to show a poincare algebra commutator. We have just shown that ##[P_u, P_v] =0 ##, i.e. simply because partial derivatives commute. Where ##P_u=\partial_u## The Attempt...

The Poincare algebra is given by isl(2, R) ~ sl(2,R) + R^3. What exactly does the i stand for? Thanks a lot in advance!

We know that Poincare matrix which is 0 Kx Ky Kz ( -Kx 0 Jz -Jy ) describes the boost and rotation is very similar to -Ky -Jz 0 Jx...

Hi all, is Fock space Poincaré invariant? As far as I can see, the scalar product in Fock space involves the scalar products in its N-particle subspaces, which, in turn, are the integrals of the properly (anti-)symmetrized wave functions over space. This works well in a Galilei-invariant...

I was reading about entropy, Poincare recurrence theorem and the arrow of time yesterday and I got some ideas/questions I'd like to share here... Let's think a about a system that is a classical ideal gas made of point particles, confined in a cubic box. Suppose that at time ##t=0## all the...

This is a continuation of a side issue from another thread.

My question concerns both quantum theory and relativity. But since I came up with this while studying QFT from Weinberg, I post my question in this sub-forum. As I gather, we first work out the representation of Poincare group (say ##\mathscr{P}##) in ##\mathbb{R}^4## by demanding the Minkowski...

I need to study in detail the rappresentations of the Poincare Group, i am interessed in the idea that particles can be wieved as irriducible representations of it. Do you have some references about it?

I want to learn how to write down a particle state in some inertial coordinate frame starting from the state ##| j m \rangle ##, in which the particle is in a rest frame. I know how to rotate this state in the rest frame, but how does one write down a Lorentz boost for it? Note that I am not...

Is Poincare' Recurrence Theorem (PCRT) considered a possible explanation for the "low entropy" initial conditions of the universe? Is the following a roughly correct paraphrasing of it? For a phase space obeying Liouville's theorem (closed, non-compress-able, non-decompress-able), the...

Keep in mind I am a complete layman when it comes to physics. Is the Poincare recurrence time the time it will take for the universe to be exactly in the state again as it is now or the time before the universe will even begin to be able to starting producing basic patterns from chaos? What is...

This is one of those "existential doubts" that most likely have a trivial solution which I can't see. Veltman says in the Diagrammatica book: Although the reasoning makes perfect sense for a Hilbert space spanned by momentum states, intuitively it doesn't make sense to me, because a...

My question concerns poincare invariance (I have left out the accent) in bosonic string theory. As far as I know, action of a 1-d String is described by poincare invariance. So my question is: why poincare invariance? And here comes the more ambarassing question: What is poincare...

Well if I have a worldline given by x^{\mu}(\tau) And I want to make a Poincare transformation: x^{\mu} (\tau) \rightarrow \Lambda^{\mu}_{\nu} x^{\nu}(\tau) + a^{\mu}. I have one question,why can't a, \Lambda explicitly depend on \tau? that is to have: x^{\mu}(\tau) \rightarrow...

If you shift the universe five meters to the left, there is no observational change. If you rotate the entire universe, the inertial frame is also rotated, and there is no observable change. If you freeze time in the universe for one billion years, then resume it, there is no observable...

The postulates of quantum theory can be given in the density matrix formalism. States correspond to positive trace class operators with trace 1 on a Hilbert space ##\mathcal{H}##. Composition is defined through the tensor product and reduction through partial trace. Operations on the system are...

Hello, I am facing some problem with Poincare disc. (1) How to visualize a Poincare disc? (2) The arc which runs at the end cannot be reached and runs till infinity. How does it happen?

I need to plot a Poincaré map for a double pendulum where the string connecting one mass to the other is elastic with elasticity constant k and rest length s. The equations of motions are: \dot{\theta}_1= \frac{p_{\theta_1}}{m_1 r_1^2} - \frac{p_{\theta_2}}{m_1 r_1 r_2} \cos{(\theta _1-...

Hey guys, as this is a basic QFT question, I wasn't sure to put it in the relativity or quantum section. Since this question specifically is about manipulating tensor expressions, i figured here would be appropriate. My question is about equating coefficients in tensor expressions...

Suppose I have a field \hat{X}... What kind of operator should it be in order to develop a vev which doesn't break the Poincare invariance? I am sure that a scalar field doesn't break the poincare invariance, because it doesn't transform. However I don't know how to write it down mathematically...

According to the experts, Poincaré made huge leaps in his proofs, often leaving lesser mathematicians scratching their heads. I'm wondering if some of those leaps later turned out to be false and if so how often.

ΩHomework Statement Given the metric ds^2 = \frac{dx^2 + dy^2}{y^2} find a set of coordinates which yield local flat space. i.e. ( g_{\mu\nu} = \delta_{\mu\nu} + second order terms). My text outlined a process to go through to find the flat space coordinates, but the actual execution is...

Homework Statement The problem statement is to prove the following identity (the following is the solution provided on the worksheet): Homework Equations The definitions of L_{\mu \nu} and P_{\rho} are apparent from the first line of the solution. The Attempt at a Solution I get to the...

I want to discuss this because I afraid that the answer is no. In SR we stuck with the transformations from Poincare Group because this transformations leave invariant the exact form of the Lorentz Metric tensor. Any other transformation will change the components of the Lorentz Metric Tensor...

Dear all, Poincare transformation construct a group, better to say noncompact Lie group. I want to prove this fact but I don't know how...; I mean the general characteristics- associativity, closure, identity element and inversion element. I would appreciate it if anyone could help me or...