Talk:Poincaré group

message: 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
:  \partial_t 
:  x \partial_t + t \partial_x 
:  -y \partial_x + x \partial_y 
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)
thread_title: Generators of the Lie algebra

message: 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)
replies: text: Yes it is. I've made this explicit. -- Fropuff 04:47, 8 May 2006 (UTC)
thread_title: Poincaré algebra

message: 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)
replies: text: Perhaps it would be more appropriate in the Poincaré group#Technical explanation section rather than in the Poincaré group#Basic explanation section. I suspect that many readers will never have heard of group contraction (by that name) before; certainly I had not. JRSpriggs (talk) 05:11, 22 May 2014 (UTC)
replies: text: I also find the term strange (but that means nothing: I'm a layman) and would have preferred something like "limiting group".  However, in the sentence "In classical physics the Galilean group is a comparable 10-parameter group that acts on absolute time and space", the phrase with link "related to the Poincaré group by group contraction" added by Cuzkatzimhut did not seem out of place. This is not technical, nor the fact that there is a name given to this relationship. I disagree with this revert. —Quondum 13:43, 22 May 2014 (UTC)
replies: text: According to WP:UPFRONT the division of this article into Basic explanation and Technical explanation is appropriate. Since the Galilean group is an intuitive concept, unstated for centuries until the electromagnetic revolution, mentioning it in the Basic section is also appropriate. The contraction group concept is not elementary, requiring introduction of limits, so mentioning it is appropriate to the Technical section.Rgdboer (talk) 22:02, 24 May 2014 (UTC)
thread_title: Group contraction requires explanation????

message: 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)
replies: text: The sentence starts with "In Minkowski space [...]".  in this context, the term plane is unambiguous (intuitively the same as a plane in a Euclidean space, but the concept of length changes, so not a Euclidean space).  I don't particularly see how this could be made clearer, since the explanation of these concepts belongs in the respective articles, not here.  —Quondum 20:31, 13 July 2015 (UTC)
thread_title:  Which "plane"? 

message: The new addition
:In quantum field theory the universal cover of the Poincaré group
::\mathbf{R}^{1,3} \rtimes \mathrm{SL}(2,C) 
:is more important, because representations of  \mathrm{SO}(1,3) are not able to describe fields with spin 1/2, i.e. fermions.  Here  \mathrm{SL}(2,C) is the group of complex  (2 \times 2)  matrices with unit determinant
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 \mathrm{SO}(3, 1) in favor of \mathrm{SL}(2, C) in the other (big conceptual change). Reference for this is Weinberg, vol I. YohanN7 (talk) 13:41, 10 January 2017 (UTC)
replies: text: I agree that one could formulate it more elegantly, but what is meant (according to my knowledge) is that for fermions you need either the projective representations or the representations of the Spin group (here  \mathrm{SL}(2,C) = \mathrm{Spin}(1,3) ). As the projective representations of \mathrm{SO}(3, 1) can be lifted to "normal" representations of  \mathrm{SL}(2,C), physicists usually talk about the lift and not about the projective representation.EduardoW (talk) 18:37, 12 November 2017 (UTC)
thread_title:  Citation needed for new addition 

message: The article does not properly distinguish between the three groups in section Poincaré_group#Poincaré_group. For example, \mathrm{Spin}(1,3) is a double cover of the zero-connected component of \mathrm{SO}_0(1,3), not of \mathrm{O}(1,3) (where \mathrm{Pin}(1,3) is the double cover). This section needs a cleanup, and more importantly we need to decide which group we call the Poincaré group:
:\mathrm{IO}(1,3) := R^{1,3} \rtimes \mathrm{O}(1,3)  or  \mathrm{ISO}_0(1,3) := R^{1,3} \rtimes \mathrm{SO}_0(1,3)?
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)
thread_title:  Inaccuracies with respect <math>\mathrm{O}(1,3)</math>, <math>\mathrm{SO}(1,3)</math>, and <math>\mathrm{SO}_0(1,3)</math>