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Sure, but keep the article called Poincaré group and merge material from Poincaré symmetry with this one.
I am currently trying to improve the articles on Lorentz group and Möbius group, and will probably have some related suggestions for this one. To name just one: why not list ten generators of the Lie algebra, in the form
Maybe there should be a kind of simple infobox template for listing generators of a Lie algebra? I'd like to eventually modify existing articles to explain at an undergraduate level why thinking of a vector field as a linear first order differential operator with nonconstant coefficients is to useful in math/physics. ---2 July 2005 04:23 (UTC)
Please don't do this. The Poincare group is the semidirect product (its Lie algebra the semidirect sum) of the homogeneous Lorentz group × the Translation group of Minkowski-space. This means, it is represented by a split short exact sequence of these two groups (the notation for the semi-direct product here is a very elegant one, although it needs the action of the Lorentz-group on Minkowski-space; but since this is the natural one it can be dropped). This is the most general definition - yours is a very special realization of its Lie algebra in terms of partial derivatives. The Lorentz group is definied as the invariance group of a Minkowski-form. Please don't speak of generators - this is a kind of unmathematical jargon. The elements of the Lie algebras are best named „elements of the Lie algebra". In this connection there should be added a remark whether the Lorentz group is exponential, that is - given by exp(element of the Poincare Lie algebra) - or whether it is only generated by such elements. Who knows a proof of this or can give a reference? Moreover, there is no need to introduce a basis of the underlying Minkowski-space. Everything can be represented in a basis-free way, including the commutation relations of the pseudo-orthogonal Lie algebras, given in this old-fashioned index notation below. Even the Killing form of these Lie algebras can be written down elegantly, only in terms of the Minkowski-form. So the only structures involved is the 4-dimensional real Minkowski-space and its Minkowski-form, say <,>. And please don't use for Minkowski-space an ℜ4, because even those of physics are not all of that type: The Pauli-matrices together with the identity, the four Dirac-matrices and the four Duffin-Kemmer-matrices are not of that type, but are Minkowski-spaces with respect to the canonical bilinear form trace(AB)-trace(A)trace(B) on square matrices. Exactly because of this, Dirac's linearization of the Klein-Gordon equation works, giving rise to a Clifford algebra on Minkowski-space. So its for physical reasons to work with a general Minkowski-space.— Preceding unsigned comment added by 130.133.155.68 ( talk) 18:28, 31 October 2012 (UTC)
Poincaré algebra links to this article, so what is a Poincaré algebra? Is it the Lie algebra of the Poincaré group? This needs to be made clear. - 72.58.19.66 03:48, 8 May 2006 (UTC)
I've taken logic up through completeness and compactness (but not group theory), and am familiar with the Poincare (and especially the Riemann) models of hyperbolic spaces. And though I know what a group is, I came here to understand the the Poincare group because it's so important in general relativity. But I still don't know what it is or how it is used, because this obscure concept is described in terms of other obscure concepts.
Before you put up your elitist force-field shields of "no stupid people need apply", remember that Einstein said "if you can't explain it to your grandmother, you don't understand it yourself". Feynman was particularly good at this, and being in lovw with him, I try to do that too.
I have not yet found a math or science topic I couldn't make understandable to non-Jedi. For example, [here's] my explanation of tensors that my grandmother could understand (if the horrible woman wasn't in hell now).
Can one of you wizards explain the poincare group in a way that Feynman would approve of? Helvitica Bold 04:00, 31 July 2011 (UTC) — Preceding unsigned comment added by Helvitica Bold ( talk • contribs)
This wording does not match to the definition in the affine group article. Either this "affine group of …" is a deeply substandard term, or we miss a dab hatnote. Incnis Mrsi ( talk) 12:55, 3 April 2012 (UTC)
This post is meant to pick up from this note at the WPM talk page. The goal is to feel out what we can adopt from these edits and what the objections are. Here's what occurred to me:
The Poincare algebra is a REAL Lie algebra, why is everybody nowadays writing the commutators with the imaginary unit? This drives me crazy! There is no $i$ in the category of real Lie algebras. So please get rid of this! — Preceding unsigned comment added by 178.12.206.11 ( talk) 12:49, 26 November 2013 (UTC)
These groups and algebras are of principal interest in physics. Physicists untilize hermitean operators whose exponentials are unitary ones with the inclusion of an i. The commutator of two hermitean operators is antihermitean, hence the i, to redress the imbalance in the Lie algebra expressions. Absorbing the i in the generators to make them antihermitean can only unleash untold grief, which has done its unwholesome damage in the past, and has been deprecated by decades-old consensus. Cuzkatzimhut ( talk) 16:13, 27 November 2013 (UTC)
User:Rgdboer added and aside on the Galilean group at the bottom of section 1, and deleted my explanation that it is a group contraction of the Poincaré as c→∞, leaving it's "comparability" to it as the only excuse of it being discussed at such a crucial section. Since I have no clue what "explanation" is required, beyond the self-evident explanation in group contraction, I terminate my involvement in this business, leaving it to somebody else to satisfy the exigeant. I have no intention of entering in an edit war, and am herewith deleting this article off my watch list, and wondering how it could ever ascend above its doomed starter status. The Galilean group is always (non-negotiably!) introduced as a group contraction of the Poincare. That is, in the language of section 2 here, mapped to section 4 of Galilean group, M ↦ L ; P ↦P ; P₀ ↦ H/a ; K ↦ a C where a is a c-number, a function of c diverging as c→∞ : Given this divergence, the commutation relations of the Poincare contract to those of the Galilean group. The contraction parameter a may be chosen to make the limit of the representations prettier.
I believe it is the article on the Galilean group that could take this "explanation", and not this one--it's got enough asides and loopy non-sequiturs to make it far less useful to the novice than it could be. I suspect this article could benefit from a minimal statement on the contractive origin of the Galilean group, if it has to be brought up at all, and that should be enough. But I'm through. Cuzkatzimhut ( talk) 00:37, 22 May 2014 (UTC)
Relating to this edit comment (isometry means invariance of distance; in this case the precise term is Lorentz invariance):
Thus, the term isometry is correct, and is the more general term. In a group-theoretic article, the use of the general term is appropriate. — Quondum 17:56, 28 February 2015 (UTC)
reflection through a plane (three degrees, the freedom in orientation of this plane);
Which "plane"? A Euclidian plane in a Euclidian space?? Does it hold when reflected through a sphere in a Spherical space?
-- 216.52.207.72 ( talk) 20:13, 13 July 2015 (UTC)
The new addition
needs a citation. My understanding is that you can take the symmetry group of spacetime to be either the Lorentz group or its cover. There is a price to pay with either choice. Projective representations in one case and the tossing of in favor of in the other (big conceptual change). Reference for this is Weinberg, vol I. YohanN7 ( talk) 13:41, 10 January 2017 (UTC)
The article does not properly distinguish between the three groups in section Poincaré_group#Poincaré_group. For example, is a double cover of the zero-connected component of , not of (where is the double cover). This section needs a cleanup, and more importantly we need to decide which group we call the Poincaré group:
See e.g. Blagoje Oblak - BMS Particles in Three Dimensions, p. 80, who introduces the former as the Poncaré group and the latter as the connected Poincaré group, but then uses "Poincaré group" for the latter as the former is not relevant for the rest of the book. EduardoW ( talk) 18:51, 12 November 2017 (UTC)
![]() | This ![]() It is of interest to the following WikiProjects: | |||||||||||||||||||||||
|
Sure, but keep the article called Poincaré group and merge material from Poincaré symmetry with this one.
I am currently trying to improve the articles on Lorentz group and Möbius group, and will probably have some related suggestions for this one. To name just one: why not list ten generators of the Lie algebra, in the form
Maybe there should be a kind of simple infobox template for listing generators of a Lie algebra? I'd like to eventually modify existing articles to explain at an undergraduate level why thinking of a vector field as a linear first order differential operator with nonconstant coefficients is to useful in math/physics. ---2 July 2005 04:23 (UTC)
Please don't do this. The Poincare group is the semidirect product (its Lie algebra the semidirect sum) of the homogeneous Lorentz group × the Translation group of Minkowski-space. This means, it is represented by a split short exact sequence of these two groups (the notation for the semi-direct product here is a very elegant one, although it needs the action of the Lorentz-group on Minkowski-space; but since this is the natural one it can be dropped). This is the most general definition - yours is a very special realization of its Lie algebra in terms of partial derivatives. The Lorentz group is definied as the invariance group of a Minkowski-form. Please don't speak of generators - this is a kind of unmathematical jargon. The elements of the Lie algebras are best named „elements of the Lie algebra". In this connection there should be added a remark whether the Lorentz group is exponential, that is - given by exp(element of the Poincare Lie algebra) - or whether it is only generated by such elements. Who knows a proof of this or can give a reference? Moreover, there is no need to introduce a basis of the underlying Minkowski-space. Everything can be represented in a basis-free way, including the commutation relations of the pseudo-orthogonal Lie algebras, given in this old-fashioned index notation below. Even the Killing form of these Lie algebras can be written down elegantly, only in terms of the Minkowski-form. So the only structures involved is the 4-dimensional real Minkowski-space and its Minkowski-form, say <,>. And please don't use for Minkowski-space an ℜ4, because even those of physics are not all of that type: The Pauli-matrices together with the identity, the four Dirac-matrices and the four Duffin-Kemmer-matrices are not of that type, but are Minkowski-spaces with respect to the canonical bilinear form trace(AB)-trace(A)trace(B) on square matrices. Exactly because of this, Dirac's linearization of the Klein-Gordon equation works, giving rise to a Clifford algebra on Minkowski-space. So its for physical reasons to work with a general Minkowski-space.— Preceding unsigned comment added by 130.133.155.68 ( talk) 18:28, 31 October 2012 (UTC)
Poincaré algebra links to this article, so what is a Poincaré algebra? Is it the Lie algebra of the Poincaré group? This needs to be made clear. - 72.58.19.66 03:48, 8 May 2006 (UTC)
I've taken logic up through completeness and compactness (but not group theory), and am familiar with the Poincare (and especially the Riemann) models of hyperbolic spaces. And though I know what a group is, I came here to understand the the Poincare group because it's so important in general relativity. But I still don't know what it is or how it is used, because this obscure concept is described in terms of other obscure concepts.
Before you put up your elitist force-field shields of "no stupid people need apply", remember that Einstein said "if you can't explain it to your grandmother, you don't understand it yourself". Feynman was particularly good at this, and being in lovw with him, I try to do that too.
I have not yet found a math or science topic I couldn't make understandable to non-Jedi. For example, [here's] my explanation of tensors that my grandmother could understand (if the horrible woman wasn't in hell now).
Can one of you wizards explain the poincare group in a way that Feynman would approve of? Helvitica Bold 04:00, 31 July 2011 (UTC) — Preceding unsigned comment added by Helvitica Bold ( talk • contribs)
This wording does not match to the definition in the affine group article. Either this "affine group of …" is a deeply substandard term, or we miss a dab hatnote. Incnis Mrsi ( talk) 12:55, 3 April 2012 (UTC)
This post is meant to pick up from this note at the WPM talk page. The goal is to feel out what we can adopt from these edits and what the objections are. Here's what occurred to me:
The Poincare algebra is a REAL Lie algebra, why is everybody nowadays writing the commutators with the imaginary unit? This drives me crazy! There is no $i$ in the category of real Lie algebras. So please get rid of this! — Preceding unsigned comment added by 178.12.206.11 ( talk) 12:49, 26 November 2013 (UTC)
These groups and algebras are of principal interest in physics. Physicists untilize hermitean operators whose exponentials are unitary ones with the inclusion of an i. The commutator of two hermitean operators is antihermitean, hence the i, to redress the imbalance in the Lie algebra expressions. Absorbing the i in the generators to make them antihermitean can only unleash untold grief, which has done its unwholesome damage in the past, and has been deprecated by decades-old consensus. Cuzkatzimhut ( talk) 16:13, 27 November 2013 (UTC)
User:Rgdboer added and aside on the Galilean group at the bottom of section 1, and deleted my explanation that it is a group contraction of the Poincaré as c→∞, leaving it's "comparability" to it as the only excuse of it being discussed at such a crucial section. Since I have no clue what "explanation" is required, beyond the self-evident explanation in group contraction, I terminate my involvement in this business, leaving it to somebody else to satisfy the exigeant. I have no intention of entering in an edit war, and am herewith deleting this article off my watch list, and wondering how it could ever ascend above its doomed starter status. The Galilean group is always (non-negotiably!) introduced as a group contraction of the Poincare. That is, in the language of section 2 here, mapped to section 4 of Galilean group, M ↦ L ; P ↦P ; P₀ ↦ H/a ; K ↦ a C where a is a c-number, a function of c diverging as c→∞ : Given this divergence, the commutation relations of the Poincare contract to those of the Galilean group. The contraction parameter a may be chosen to make the limit of the representations prettier.
I believe it is the article on the Galilean group that could take this "explanation", and not this one--it's got enough asides and loopy non-sequiturs to make it far less useful to the novice than it could be. I suspect this article could benefit from a minimal statement on the contractive origin of the Galilean group, if it has to be brought up at all, and that should be enough. But I'm through. Cuzkatzimhut ( talk) 00:37, 22 May 2014 (UTC)
Relating to this edit comment (isometry means invariance of distance; in this case the precise term is Lorentz invariance):
Thus, the term isometry is correct, and is the more general term. In a group-theoretic article, the use of the general term is appropriate. — Quondum 17:56, 28 February 2015 (UTC)
reflection through a plane (three degrees, the freedom in orientation of this plane);
Which "plane"? A Euclidian plane in a Euclidian space?? Does it hold when reflected through a sphere in a Spherical space?
-- 216.52.207.72 ( talk) 20:13, 13 July 2015 (UTC)
The new addition
needs a citation. My understanding is that you can take the symmetry group of spacetime to be either the Lorentz group or its cover. There is a price to pay with either choice. Projective representations in one case and the tossing of in favor of in the other (big conceptual change). Reference for this is Weinberg, vol I. YohanN7 ( talk) 13:41, 10 January 2017 (UTC)
The article does not properly distinguish between the three groups in section Poincaré_group#Poincaré_group. For example, is a double cover of the zero-connected component of , not of (where is the double cover). This section needs a cleanup, and more importantly we need to decide which group we call the Poincaré group:
See e.g. Blagoje Oblak - BMS Particles in Three Dimensions, p. 80, who introduces the former as the Poncaré group and the latter as the connected Poincaré group, but then uses "Poincaré group" for the latter as the former is not relevant for the rest of the book. EduardoW ( talk) 18:51, 12 November 2017 (UTC)