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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 <,: -->
1046:{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 - 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 = iħX, = iħY, = iħZ, to distinguish between the two.
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445:. And please don't use for Minkowski-space an ℜ, 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.
1027:, 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.
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862:: 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.
561:. 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.
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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".
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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 25 May 2011, just to pick something out of arXiv at random (all of them having presentations of the Poincaré Lie algebra in them).
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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". —
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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"
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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
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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
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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
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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
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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".
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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
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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
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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
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The best way to think of this is to treat /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
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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
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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
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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
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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
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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.
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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.
1012:, 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
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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
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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
1098:, 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
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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!
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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
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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
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I have not yet found a math or science topic I couldn't make understandable to non-Jedi. For example, my explanation of tensors that my grandmother could understand (if the horrible woman wasn't in hell now).
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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.
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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
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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
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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.
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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.
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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.
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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.
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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
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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
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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...
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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
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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.
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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.
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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. -
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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
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actions discussed here are normally understood as hermitian matrices representing the generators on finite dimensional spaces (You may notice
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These groups and algebras are of principal interest in physics. Physicists untilize hermitean operators whose exponentials are unitary ones
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is the double cover). This section needs a cleanup, and more importantly we need to decide which group we call the Poincaré group:
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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. -
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clarified that preserving (up to sign) of proper time is an equivalent, but not the basic formulation (does anybody oppose?);
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Which "plane"? A Euclidian plane in a Euclidian space?? Does it hold when reflected through a sphere in a Spherical space?
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Yes, that sentence does not make sense to me either, but I have not worked with affine groups so I am not entirely sure.
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is correct, and is the more general term. In a group-theoretic article, the use of the general term is appropriate. —
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Coordinate transformations are not quite what is being referred to here; the concept is a coordinate-independent one.
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A Lorentz transformation is an example of an isometry in the precise sense that it preserves a distance function.
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The Lorentz group is the subgroup of the Poincaré group which does not move the origin of the Minkowski space.
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See e.g. Blagoje Oblak - BMS Particles in Three Dimensions, p. 80, who introduces the former as the
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Why isn't this explanation in the actual page? Much more useful to people just needing background.
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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The main problem with 2012 versions of the article is that it used such terms (and links) as “
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Can one of you wizards explain the poincare group in a way that Feynman would approve of?
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very thing which the physicist are talking about — the metric of Minkowsky space (or of
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The article does not properly distinguish between the three groups in section
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in the other (big conceptual change). Reference for this is Weinberg, vol I.
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the Lorentz group or its cover. There is a price to pay with either choice.
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at the WPM talk page. The goal is to feel out what we can adopt from
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at the bottom of section 1, and deleted my explanation that it is a
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may be chosen to make the limit of the representations prettier.
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ever ascend above its doomed starter status. The Galilean group is
645:"the full Poincaré group is the affine group of the Lorentz group"
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in the broader meaning) may also be in use. Perhaps the article
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section. I suspect that many readers will never have heard of
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789:” in a sense different than linked article suggest. My edit:
1829:{\displaystyle \mathbf {R} ^{1,3}\rtimes \mathrm {SL} (2,C)}
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world lines, not, say, only actual world lines of particles;
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Thanks for bringing this article back to a tolerable form.
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and what the objections are. Here's what occurred to me:
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2012 versions of basic explanation were unsatisfactory.
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needs to be updated, not the link to it avoided here. —
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are not able to describe fields with spin 1/2, i.e.
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440:useful in math/physics. ---2 July 2005 04:23 (UTC)
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2529:{\displaystyle \mathrm {Pin} (1,3)}
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700:) 19.00, 29 August 2012 (GMT+1)
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860:JRSpriggs’s “basic explanation”
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2361:Poincaré_group#Poincaré_group
2226:18:37, 12 November 2017 (UTC)
1313:17:56, 28 February 2015 (UTC)
1146:is a c-number, a function of
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995:17:50, 27 November 2013 (UTC)
974:16:13, 27 November 2013 (UTC)
943:16:17, 26 November 2013 (UTC)
903:12:49, 26 November 2013 (UTC)
332:{\displaystyle \partial _{t}}
293:Generators of the Lie algebra
244:and see a list of open tasks.
167:This article is supported by
132:Knowledge:WikiProject Physics
126:and see a list of open tasks.
2789:Start-Class physics articles
2058:13:41, 10 January 2017 (UTC)
635:11:41, 2 November 2011 (UTC)
612:15:51, 1 November 2011 (UTC)
461:18:28, 31 October 2012 (UTC)
135:Template:WikiProject Physics
1949:{\displaystyle (2\times 2)}
1185:section rather than in the
736:19:54, 29 August 2012 (UTC)
708:17:02, 29 August 2012 (UTC)
518:04:00, 31 July 2011 (UTC)
495:A simple explanation please
2835:
2759:Start-Class vital articles
2232:Inaccuracies with respect
1966:Projective representations
1535:Isometry (quadratic forms)
1390:pseudo-Riemannian manifold
1068:00:24, 26 March 2022 (UTC)
872:13:08, 17 April 2013 (UTC)
854:17:20, 15 April 2013 (UTC)
836:15:43, 15 April 2013 (UTC)
775:14:19, 15 April 2013 (UTC)
158:project's importance scale
1924:is the group of complex
1755:20:31, 13 July 2015 (UTC)
1740:20:13, 13 July 2015 (UTC)
1696:09:14, 2 March 2015 (UTC)
1650:08:41, 2 March 2015 (UTC)
1600:07:25, 2 March 2015 (UTC)
1558:16:31, 1 March 2015 (UTC)
1497:14:42, 1 March 2015 (UTC)
1402:11:42, 1 March 2015 (UTC)
1368:11:30, 1 March 2015 (UTC)
1354:07:12, 1 March 2015 (UTC)
1344:02:12, 1 March 2015 (UTC)
685:13:19, 3 April 2012 (UTC)
667:12:55, 3 April 2012 (UTC)
592:18:42, 31 July 2011 (UTC)
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1242:13:43, 22 May 2014 (UTC)
1203:05:11, 22 May 2014 (UTC)
1176:00:37, 22 May 2014 (UTC)
301:and merge material from
276:project's priority scale
1638:Metric_space#Definition
1221:absolute time and space
1082:added and aside on the
1004:Agreed, but...there is
489:04:47, 8 May 2006 (UTC)
479:03:48, 8 May 2006 (UTC)
233:WikiProject Mathematics
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1213:classical physics
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247:Mathematics
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2753:Categories
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1256:WP:UPFRONT
1010:convention
824:Rschwieb’s
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787:trajectory
555:trajectory
75:Relativity
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1688:JRSpriggs
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748:this note
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1884:. Here
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1531:isometry
1484:Isometry
1439:Isometry
1331:Isometry
1306:isometry
1056:unsigned
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846:Rschwieb
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