Talk:Poincaré group

Generators of the Lie algebra

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)

Poincaré algebra

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)

Yes it is. I've made this explicit. -- Fropuff 04:47, 8 May 2006 (UTC)

A simple explanation please

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)

The main thing you need to understand is what an isometry is. It is a way in which the contents of spacetime could be shifted that would not affect the proper time along a trajectory between events. For example, if everything was postponed by two hours including two events and the path you took to go from one to the other, then the time interval between the events recorded by a stop-watch you carried with you would be the same. Or if everything was shifted five miles to the west, you would also see no change in the interval. It turns out that the length of a rod is also unaffected by such a shift.

If you ignore the effects of gravity, then there are ten basic ways of doing such shifts: translation through time, translation through any of the three dimensions of space, rotation (by a fixed angle) around any of the three spatial axes, or a boost in any of the three spatial directions. 10=1+3+3+3. If you combine such isometries together (do one and then the other), the result is also such an isometry (although not generally one of the ten basic ones). These isometries form a group. That is, there is an identity (no shift, everything stays where it was), and inverses (move everything back to where it was), and it obeys the associative law. The name of this particular group is the "Poincaré group". I hope that clears things up. JRSpriggs (talk) 18:42, 31 July 2011 (UTC)

Why isn't this explanation in the actual page? Much more useful to people just needing background. — Preceding unsigned comment added by 24.218.104.144 (talk) 15:51, 1 November 2011 (UTC)

Thanks for the suggestion. I just copied it into the article as the new section "Simple explanation" ahead of the previous material which I renamed the "Technical explanation" section. JRSpriggs (talk) 11:41, 2 November 2011 (UTC)

"the full Poincaré group is the affine group of the Lorentz group"

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)

Yes, that sentence does not make sense to me either, but I have not worked with affine groups so I am not entirely sure.

The Lorentz group is the subgroup of the Poincaré group which does not move the origin of the Minkowski space. JRSpriggs (talk) 13:19, 3 April 2012 (UTC)

That sentence seems wrong to me too. As I understand, affine transformations are a more general set of transformation which do not conserve distances. As such, the Poincare group can't be the group of affine transformations. Of course, I am not an expert on the subject, so I can certainly be wrong on this. Npoles (talk) 19.00, 29 August 2012 (GMT+1) —Preceding undated comment added 17:02, 29 August 2012 (UTC)

I  changed the wording. Incnis Mrsi (talk) 19:54, 29 August 2012 (UTC)

Discussion of early April 2013 edits

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:

• Firstly, this being a mathematical topic used by physicists, that intro should definitely be able to incorporate both disciplines. The intro right now is really short: we can probably have both and remain clear.

• Secondly, I think I see the problem with using "isometries" but I have to ask this question to make sure. While the WP isometry article says such a transformation "preserves distances", I'm also aware of a extended use of "isometry" relative to a bilinear form meaning "preserves the bilinear form". In the latter sense, it doesn't seem to be an abuse of terminology at all, but if one is believing the wlinks, then yes, it is. I think using "preserves the interval" and explaining that the interval is "like distance" is a good introductory explanation, but I like the "preserves the metric" explanation in the technical part. In any case, we should use a little care when writing "isometry". Rschwieb (talk) 14:19, 15 April 2013 (UTC)

The main problem with 2012 versions of the article is that it used such terms (and links) as “isometry” and “trajectory” in a sense different than linked article suggest. My edit:

• named explicitly the thing to be conserved: the interval between events (or, the same, the pseudo-Euclidean magnitude of a vector);
• clarified that vectors of all three types are preserved, and explained what it means in details;
• clarified that preserving (up to sign) of proper time is an equivalent, but not the basic formulation (does anybody oppose?);
• clarified that proper time can be calculated along a world line only, and that we have to consider all possible world lines, not, say, only actual world lines of particles;
• clarified that the time can be reversed (which all physical reasoning about stop-watches apparently contradict to);

Was this an improvement or a degradation? One can use whatever terms, even “isometry”, but one has to define the quantity it should conserve: vector magnitude? dot product? metric tensor in the sense of a Riemannian manifold? The same about “trajectory”: Rschwieb’s 2012 versions of basic explanation were unsatisfactory. Incnis Mrsi (talk) 15:43, 15 April 2013 (UTC)

Since I did not attempt any basic explanation, your reading of the previous comment was unsatisfactory. The point was that preserving the semi-Riemannian metric does not seem like a basic explanation. Appealing to distance or pseudo-distance seems better. Let me know if you have any more questions about what I meant. Rschwieb (talk) 17:20, 15 April 2013 (UTC)

I am completely dissatisfied with JRSpriggs’s “basic explanation”: it is a mixture of non-elementary facts (such as preserving only proper times is sufficient for preserving other structure), unclear statements (see above), misleading internal links (see above), and bizarre wording and metaphors (such as “the contents of spacetime could be shifted”). If two editors defend this version against my (radical) proposal, then specific issues I addressed should be considered. Incnis Mrsi (talk) 13:08, 17 April 2013 (UTC)

Whats with this imaginary unit?

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!