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! — Preceding unsigned comment added by 178.12.206.11 (talk) 12:49, 26 November 2013 (UTC)

Add reliable secondary sources to support your claim if you change the article. M∧Ŝc²ħεИ_τlk 16:17, 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)

the term 'hermitian' is not even defined in the category of real Lie algebras. I think one should not put common pysical practice over mathematical rigor... — Preceding unsigned comment added by 94.223.151.28 (talk) 17:50, 27 November 2013 (UTC)

Agreed, but...there is nothing nonrigorous about a convention, including this one. It has been chosen to agree with common Lie Algebraic practices and language in the whole of physics (Wigner's seminal 1939 classification of unitary reps essentially sets the tone: a sound one). A mathematician should not be that interested in this group... It is only discussed for its central importance in describing, indeed, governing, the real world. A working physicist cannot live without it. So, then, much of this article is a service introduction for mostly physics students. Many of these generators and their physical actions discussed here are normally understood as hermitian matrices representing the generators on finite dimensional spaces (You may notice J is a physical observable). Νοw, the unitary irreps of the associated compact su(2)×su(2) are finite dimensional, but they correspond to nonunitary irreps of the Lorentz group, 4d ones in that article--while the unitary irreps of the non-compact Lorentz are infinite-dimensional, a bit of a mathematical curiosity when it comes to physics. Familiar with this fact of life, the majority of physicists opt for hermitean operators and are normally mindful of the i's in all redefinitions that occur, or which there are many and treacherous. If you had useful explanatory phrases to protect readers unfamiliar with this dual thinking, they could be quite welcome. But, in either language, sooner or later, the i's are unavoidable. Cuzkatzimhut (talk) 19:36, 27 November 2013 (UTC)

The best way to think of this is to treat [A,B]/iħ as {A,B}, which - itself - is the Lie bracket of a Lie algebra; and also the Poisson bracket of a Poisson algebra (namely: that which arises from the double-dual of the Lie algebra, which is closely tied to the co-adjoint orbit framework). The latter {A,B} may be treated as the Lie bracket in the form that Mathematicians like to think of it in terms of, while the former [A,B] - when thought of as a commutator in the context of quantum-theoretic settings - provides an algebraic representation of {A,B} that is historically closer in line to how Physicists first encountered them. Strictly speaking, the mathematical form, which is tied to Poisson manifolds and symplectic geometry provides the correct notational language and framework for Lie algebras, since it is paradigm-neutral: it arises in both classical and quantum settings, as well as hybrids thereof (e.g. quantum theories with superselection), while the Physicists' language only applies within the quantum setting, obscures the connection to the classical limit and to classical physics and is already derivable from the Poisson manifold / symplectic formulation as an algebraic representation. I tend to use the curly brackets for the version without the i's, e.g. {Y,Z} = X, {Z,X} = Y, {X,Y} = Z for SO(3) in the context of symplectic geometry, co-adjoint orbit methodology, Poisson manifolds, while using the square brackets for applications in quantum theory [Y,Z] = iħX, [Z,X] = iħY, [X,Y] = iħZ, to distinguish between the two.

Both conventions are - and have long been - in widespread use, even in the Physics community; e.g. Currie, Jordan & Sudarshan, "Relativistic Invariance and Hamiltonian Theories of Interacting Particles", in Reviews of Modern Physics 35(2) April 1963; or Dirac "Forms of Relativistic Dynamics", in Reviews of Modern Physics 21(3) July 1949, both use the Mathematician's form of the Lie bracket, without the i's; as do more recent papers, like "Newtonian Gravity and the Bargmann Algebra", Andringa et al. arXiv:101145v3 [hep-th] 25 May 2011, just to pick something out of arXiv at random (all of them having presentations of the Poincaré Lie algebra in them). — Preceding unsigned comment added by 2603:6000:AA4D:C5B8:0:3361:EAF8:97B7 (talk) 00:24, 26 March 2022 (UTC)

Group contraction requires explanation????

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)

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)

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)

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)

Lorentz invariance vs. isometry

Relating to this edit comment (isometry means invariance of distance; in this case the precise term is Lorentz invariance):

• Coordinate transformations are not quite what is being referred to here; the concept is a coordinate-independent one.
• A Lorentz transformation is an example of an isometry in the precise sense that it preserves a distance function.

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)

Lorentz invariant is correct since spacetime is not a metric space. If you read Isometry you will see that it doesn't make sense to use this term without the underlying structure. In physics literature, phrases like "Minkowski metric" or "metric tensor" are used frequently, but more mathematically aware authors prefer "Minkowski squared interval" and pseudo-Riemannian geometry to show they understand something is at stake in the terminology. Though "usage" is important for terminology, as a general reference work spanning all specialties, accuracy has been a standard too.Rgdboer (talk) 02:12, 1 March 2015 (UTC)

It does have the underlying structure that defines a distance measure: the (indefinite) quadratic form, which is the Minkowski squared interval to which you refer. From a group theory perspective, this is on exactly equal footing with the Euclidean metric of the Euclidean group. The term "isometry" relates to a distance function; the fact that it is not a metric in the normal sense of that word does not change the definition of isometry. I agree that suitable terminology is important, and the terms you mention are good (we should use them where the weaker terms are used), but I do not see that this changes the validity of the term "isometry". —Quondum 07:12, 1 March 2015 (UTC)

It is not a metric, and I would not call it a distance either. It is confusing to use the same term for something with essential differences. - Patrick (talk) 11:30, 1 March 2015 (UTC)

These articles which Patrick has been changing are primarily for physicists. Physicists use the terms "metric" and "isometry". So Patrick is making the articles less readable and inconsistent with the sources.

Also the physicists are correct and the mathematicians are wrong here. The original source of the concept of a metric is the physical measurements in Euclidean geometry (a spatial slice of Minkowsky space) using a compass and straight edge or ruler. Thus what was originally being measured was the very thing which the physicist are talking about — the metric of Minkowsky space (or of pseudo-Riemannian manifold). JRSpriggs (talk) 11:42, 1 March 2015 (UTC)

The article said "The Poincaré group is the group of Minkowski spacetime isometries", with this link to the article Isometry. What is meant is not an isometry as defined there, and not even a pseudometric (events can have a distance 0 while with respect to causality they are quite different) or semimetric (no triangle inequality) as defined at the end of the article, but a pseudo-semimetric. Thus the quoted sentence does not explain at all what the Poincaré group is.

Linking to Metric tensor and/or Metric tensor (general relativity) instead of Isometry may already be a big improvement. Looking at these articles it seems that "metric" is common, and when done very carefully, with the right links and necessary explanation, can be used without problems. "Isometry" could then be defined as a map preserving this metric, to provide a shorthand for "coordinate transformation with Lorentz invariance". - Patrick (talk) 14:42, 1 March 2015 (UTC)

At least the terms "orthogonality" and "orthogonal group" seem to have expanded to include indefinite bilinear forms, but "metric" has not. Please consider that expanded definitions of isometry (see Isometry (quadratic forms), Isometry (Riemannian geometry); also Symmetry (physics)#Conservation laws and symmetry seems to refer to isometry in the broader meaning) may also be in use. Perhaps the article Isometry needs to be updated, not the link to it avoided here. —Quondum 16:31, 1 March 2015 (UTC)

The article Isometry does not appear to say that the distance (or squared distance) being preserved is positive (or even non-negative). If you see that in it, please indicate exactly where it says that. JRSpriggs (talk) 07:25, 2 March 2015 (UTC)

It refers to Metric space, for that see Metric_space#Definition. - Patrick (talk) 08:41, 2 March 2015 (UTC)

Thanks for bringing this article back to a tolerable form. JRSpriggs (talk) 09:14, 2 March 2015 (UTC)

Which "plane"?

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 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)

Citation needed for new addition

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)

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)

Inaccuracies with respect $\mathrm {O} (1,3)$, $\mathrm {SO} (1,3)$, and $\mathrm {SO} _{0}(1,3)$

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)