# What is Poincare: Definition and 154 Discussions

Poincaré is a French surname. Notable people with the surname include:

Henri Poincaré (1854–1912), physicist, mathematician and philosopher of science
Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
Lucien Poincaré (1862–1920), physicist, brother of Raymond and cousin of Henri
Raymond Poincaré (1860–1934), French Prime Minister or President inter alia from 1913 to 1920, cousin of Henri

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1. ### B Confusion with the basics of Topology (Poincare conjecture)

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...
2. ### I Could entropy be reversed eventually in the far future?

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...
3. ### A Question about the Poincaré conjecture

Does Perelman’s proof of the Poincaré conjecture imply that the universe is the surface of a 3 sphere?
4. ### A What is the significance of the Poincaré conjecture?

Namely, what does Perelman’s proof of it imply?
5. ### Dynamical Systems: how to find equation for Poincare map?

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)...
6. ### I Why are discontinuous Lorentz transformations excluded from the Poincare group?

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...
7. ### I Liouville's Theorem and Poincare Recurrence Theorem

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...
8. ### B 10 dimensional Poincaré group

Is this a coincidence?
9. ### I Unitary Representation of Poincaré Group: Classical Relativistic Mechanics

This thread is a shameless self-promotion of a recent work of mine: https://arxiv.org/abs/2105.13882 In the paper an operational version of classical relativistic dynamics (for massive particles) is obtained from an irreducible representation of the Poincaré group. The formalism has kets...
10. ### Proving Poincare Algebra Using Differential Expression of Generator

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...
11. ### I Poincaré algebra and quotient group

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?
12. ### I Matrix Representations of the Poincare Group

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...
13. ### Exploring the Inönü-Wigner Contraction of Poincaré $\oplus$ $\mathfrak{u}$(1)

Please see https://physics.stackexchange.com/questions/552410/inönü-wigner-contraction-of-poincaré-oplus-mathfraku1Inönü-Wigner contraction of Poincaré \oplus \mathfrak{u}(1) Metric = (-+++), complex $i$'s are ignored. _____________________________________________________________________ The...
14. ### I Deriving AdS Poincare Coordinates from Global Coordinates

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...
15. ### Construct a quadrilateral in a Poincare half plane

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...
16. ### Quantum Good sources for the representations of the Poincare group?

Weinberg QFT book aside, what are good sources for the representation of the Poincare group used in physics?
17. ### Discontinuities in a Poincare map for a double pendulum

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...
18. ### Poincare algebra and its eigenvalues for spinors

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})##...
19. ### A Haag's Theorem and the Poincare Group

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.

28. ### Massive spin-s representations of the Poincare group

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...
29. ### A Deriving the Poincare patch from global coordinates in AdS##_{3}##

I have been reading Thomas Hartman's lecture notes (http://www.hartmanhep.net/topics2015/) on Quantum Gravity and Black Holes. -------------------------------------------------------------------------------------------------------------------------------- In page 97, he derives (9.4), which is...
30. ### Poincare Algebra -- Quick Question

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...
31. ### A 3dim Poincare Algebra - isl(2,R)

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!
32. ### I Relation between Poincare matrix and electromagnetic field t

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...
33. ### A Fock space and Poincaré invariance

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...
34. ### About Entropy and Poincare Recurrence

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...
35. ### A Is Poincare symmetry the real thing?

This is a continuation of a side issue from another thread.
36. ### Locally flat coordinates on the Poincaré half plane

Homework Statement Find the locally flat coordinates on the Poincaré half plane. Problem I.6.4 by A. Zee Homework Equations [/B] Poincaré Metric: ##ds^2 = \frac{dx^2 + dy^2}{y^2}## The Attempt at a Solution First, I'm having problems with the explanation in Zee's book. He said that we can...
37. ### Poincare group representations

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...
38. ### Representations of Poincare group

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?
39. ### 2j+1 d representation for Poincaré group

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...
40. ### Confusion: Fluctuation Theorem, Poincare Recurrence Theorem

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...
41. ### Several questions about the poincare recurrence time?

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...
42. ### Hilbert space transformation under Poincaré translation

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...
43. ### String Theory & Poincaré Invariance: Why & What?

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...
44. ### Poincare Transformations: Parametrization-Independent

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...
45. ### Is distance between particles relative? Poincare invariance?

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...
46. ### Density matrix formalism and Poincaré invariance

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...
47. ### Visualizing the Poincare Disc: Understanding its Limits

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?
48. ### Poincaré Sections of the double pendulum with Mathematica

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-...
49. ### Manipulating Tensor Expressions to Derive the Poincare Algebra

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...
50. ### What type of field can have a Lorentz invariant VEV?

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...