Talk:Lorentz group

Old Discussion Archived

This article was changed beyond recognition when I greatly expanded it in July 2005 (and I still have more to do). Unfortunately, in the middle of this process, a serious misunderstanding arose which led to a bit of a kerfluffle. Fortunately, this has been mediated, and I don't think the "flap" is very edifying, so I have archived it at Talk:Lorentz group/Archive.

Please add any comments/suggestins on present article below. I've put in a lot of work on this, so I hope all good Wikipedias will ask me for a response before making major changes. TIA---CH (talk) 04:47, 6 August 2005 (UTC)

Left/Right groups?

What is meant exactly by the left and right groups in the double covering SU(2) → SO(3)? -- Fropuff 06:02, 14 October 2005 (UTC)

You still haven't answered my question CH. This needs to be clarified. At any rate the section on topology needs to be corrected: SO⁺(1,3) is a trivial bundle over H³; there is no "twist" (any fiber bundle over a contractible base is trivial). -- Fropuff 18:44, 23 January 2006 (UTC)

Sorry, I guess I didn't see your comment before. Yeah, this could be clearer! I meant left and right side of the arrow, not left and right multiplication. Feel free to untwist! A reference specifically discussing the topology of the restricted Lorentz group would be useful. I don't have Hall's book in front of me but there might be something useful in there.---CH 16:20, 24 January 2006 (UTC)

Ahh! Thank you. I couldn't figure out what in the world you meant by that. -- Fropuff 16:30, 24 January 2006 (UTC)

Topology

The article does not state which topology is used for the Lorentz group but mentions connection properties which depend on the topology. I assume it is topology induced by the operator norm which is the same topology that results when the Group is viewed as a finite dimensional vector space? The Infidel 10:41, 21 January 2006 (UTC)

The topology of O(3,1) is the subspace topology inherited from GL(4,R), which itself can be viewed as an open subset in R¹⁶ (with the Euclidean topology). The Lorentz group is not a vector space (but it sits inside one), so I'm not sure what you mean by that last statement. -- Fropuff 16:38, 21 January 2006 (UTC)

Thank you. The last statement should be "as (topological) subspace of a finite dimensional vector space". I think we should mention the topology briefly in the article. Any objections? The Infidel 20:17, 22 January 2006 (UTC)

Hi, Infidel, I don't understand what you found confusing about the existing description. Why did you think I might be refering to functional analysis in the context of a finite dimensional real Lie group? Have you seen the book by Hatcher which I mention in the references?

If I understand what you found confusing, I can try to improve this bit myself. I'd like to try to keep the style and emphasis as internally consistent as possible, which probably means that wherever possible I should make any neccessary changes myself.---CH 01:07, 23 January 2006 (UTC)

I have no special background of Lie groups, so I just wondered on which topology the unconnectedness is based. And shortly my mind went astray considering open-compact topology and the like ... The Infidel 18:32, 23 January 2006 (UTC)

Students beware

I completely rewrote the August 2005 version of this article and had been monitoring it for bad edits, but I am leaving the WP and am now abandoning this article to its fate.

Just wanted to provide notice that I am only responsible (in part) for the last version I edited; see User:Hillman/Archive. I emphatically do not vouch for anything you might see in more recent versions. I hope for the best, but unfortunately relativity theory attracts many cranks, and at least some future versions of this article are likely to have been vandalized or to contain slanted information, misinformation, or disinformation.

Good luck to all students in your search for information, regardless!---CH 01:50, 1 July 2006 (UTC)

Lie algebra

"its Lie algebra is reducible and can be decomposed to two copies of the Lie algebra of SL(2,R)"

Really? sl(2,R) has rank 1 (because it contains so(2)), so sl(2,R)^2 has rank 2, but so(3,1) has rank 1 (since sl(2,C) has rank 1). What's going on here? Adam1729 07:18, 2 October 2007 (UTC)

No further comment so far? Strictly speaking, the statement is false. As the author has shown before, so(3,1) = sl(2,C) and sl(2,C) = sl(2,R)⊕sl(2,R) as vector spaces but not as Lie algebras. I think however that the author actually refers to the complexified Lie algebra, which is so(4,C) and hence coincides with the complexification of so(4), so the algebra decomposition so(4) = su(2)⊕su(2) alluded to in the article induces so(4,C) = sl(2,C)⊕sl(2,C).

This was fixed, somewhen long ago. The confusion is presumably due to Dirac spinors aka bispinors which is a valid related topic. 67.198.37.16 (talk) 02:48, 4 December 2020 (UTC)

Conjugacy classes

There are infinitely many conjugacy classes. Every rotation of order m is in a different conjugacy class from every rotation of order n. I suspect that when the article says "conjugacy class" it really means "topological closure of conjugacy class". Is this right? Adam1729 07:24, 2 October 2007 (UTC)

Is this the right matrix?

In the text I see this:

$\left[{\begin{matrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\end{matrix}}\right]$

Should it be this:

$\left[{\begin{matrix}1&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&1\end{matrix}}\right]$

I admit to being way over my head to even try to read this article, for the former matrix looks quite odd to me, while the latter looks commonplace.  Randall Bart   Talk  04:39, 17 December 2009 (UTC)

The first matrix is the rotation generator (infinitesimal excursion from the identity), iJ_z, which I now labelled. The "commonplace" one you are adducing is a full finite rotation by π/2, namely exp(iπ J_z/2). Try power series expansion of the exponential. This highlights the article's deficiency in not identifying the geometrical entities with the conventional operators employed in quantum mechanics and particle physics, at the very least. Cuzkatzimhut (talk) 19:49, 29 November 2013 (UTC)

I agree. I think it gives far too much emphasis on the loxodromic, etc subgroups which is interesting and all, but really first and foremost should be the physics applications, notation, conventions, including spin. 67.198.37.16 (talk) 02:45, 4 December 2020 (UTC)

How about some basic facts?

Is the Lorentz group a). an abelian group, b). non-abelian group, or c). Sometimes one and sometimes the other? The only thing that I say even connected to these clasifications is a mention of an abelian subgroup here and there. 98.81.17.64 (talk) 21:46, 4 August 2010 (UTC)

I've tweaked the lead to reflect this classification (non-abelian), which is relevant. The article structure needs work. — Quondum^☏ 13:30, 15 August 2012 (UTC)

Notation of connected components and covering groups

In Lorentz group#Covering groups it is said that Spin(1,3) is the double cover of SO(1,3), and the double cover of $SO^{+}(1,3)$, i. e. $SL(2,\mathbb {C} )$, is called $Spin^{+}(1,3)$. In Spin group#Indefinite signature, the double cover of $SO^{+}(1,3)=SO^{+}(3,1)$ (which is denoted by $SO_{0}(3,1)$ there) is called $Spin(3,1)$. The notations should be unified in Wikipedia: it should b decided whether $SO^{+}(p,q)$ or $SO_{0}(p,q)$ should be used for the connected component of identity of $SO(p,q)$ and whether $Spin(p,q)$ or $Spin^{+}(p,q)$ should be used to denote the double cover of the identity component. Jaan Vajakas (talk) 12:14, 15 August 2012 (UTC)

This appears to be fixed (!?) 67.198.37.16 (talk) 02:41, 4 December 2020 (UTC)

The Lie algebra of the Lorentz group

The Lie algebra of SO⁺(1;3), so(1;3), cannot be decomposed into two copies of sl(2;R). It can't be decomposed at all, because it is simple. What is true is that its complexification, so(1;3)_C can be decomposed into sl(2;C) ⊕ sl(2;C). YohanN7 (talk) 20:33, 3 October 2012 (UTC)

Removed this: "Since the Lorentz group is SO⁺(1,3), its Lie algebra is reducible and can be decomposed to two copies of the Lie algebra of SL(2,R), as will be shown explicitly below (this is the Minkowski space analog of the SO(4) → SU(2) × SU(2) decomposition in a Euclidean space). In particle physics, a state that is invariant under one of these copies of SL(2,R) is said to have chirality, and is either left-handed or right-handed, according to which copy of SL(2,R) it is invariant under."

It wasn't used (and much less proved because it's false) anyway. YohanN7 (talk) 19:01, 8 March 2013 (UTC)

Style problems

The article reads too much like bad lecture notes. It's full of writing in the second person and leaving the reader in suspense ("We will later find such and such..."), working through calculations without telling us where they are going. ("Consider the following..." followed by piles of unmotivated calculations. "But then consider the following!" (jarring and upsetting twist which makes the subject look difficult)) Hopefully I'll be able to make some improvements so it's not necessary to plough through the article line-by-line to understand what it is talking about. Count Truthstein (talk) 17:34, 8 March 2013 (UTC)

I fixed a handful. There are still 13 "we" left int it.67.198.37.16 (talk) 02:27, 4 December 2020 (UTC)

map from SL(2,C) to SO+(1,3)

In the section on "Relation to the Möbius group" there is the sentence "SL(2,C) ... is isomorphic to a subset of the Lorentz group". This statement indicates that the map from SL(2,C) to SO+(1,3) is injective. But if I have got this right, this isn't true. The map is surjective, as the next paragraph explains, and the kernel of the map is ±I. - Subh83 (talk | contribs) 19:14, 31 May 2013 (UTC)

It looks like it was fixed. 67.198.37.16 (talk) 02:25, 4 December 2020 (UTC)

"Lorentz transformations are examples of linear transformations"

The article claims that LTs are examples of linear transformations, but isn't half the Lorentz group antilinear, i.e. precisely *not* linear transformations? Matthew.daniels (talk) 15:02, 11 September 2013 (UTC)

Time reversal is a linear operation in the context of the Lorentz group. When one considers the action of time reversal in quantum mechanics, it does become an anti-linear operator on the corresponding Hilbert space, but that is particular to quantum mechanics. --Mark viking (talk) 18:08, 11 September 2013 (UTC)

I have updated Representation theory of the Lorentz group with a new section, Time-reversal, and tried to explain the reasons for the antilinearity. YohanN7 (talk) 09:39, 18 December 2013 (UTC)

Lorentz_group#Parabolic

The formulas are correct only if the matrix is from SL(2,R), i.e. if alpha is real. Otherwise they don't give the right results.--Café Bene (talk) 04:20, 21 November 2013 (UTC) Corrected it.--Café Bene (talk) 04:39, 21 November 2013 (UTC)

It looks still wrong to me Mario Geiger. The generator of Lorentz has to satisfy

$\eta E+E^{T}\eta =0$

where $\eta$ is the metric and $E$ is a generator. But $Im(\alpha )$ violates that. — Preceding unsigned comment added by Geiger.mario (talk • contribs) 15:24, 29 March 2020 (UTC)

You are correct, and the article is still broken. Some minus sign got dropped somewhere. I'm too lazy to figure out all the required corrections. It should not be hard, just tedious. 67.198.37.16 (talk) 02:23, 4 December 2020 (UTC)

Fix this statement

In optics, this construction is known as the Poincaré sphere.

I removed this statement because, as written, it's a bit out of context. In particular, I could not see what was meant by "this construction" other than the fact that the groups SU(2) and/or PSL_2(C) are used to do the Poincare sphere of optics. If the statement can be sharpened to make the connection clearer, and put back in, that would be great.

198.129.64.80 (talk) 17:22, 4 May 2016 (UTC)

Preserved event types

The following statement is found in the section on the parabolic conjugacy class:

Parabolic Lorentz transformations are often called null rotations, since they preserve null vectors, just as rotations preserve timelike vectors and boosts preserve spacelike vectors.

In fact, all Lorentz transformations respect the three event types, null, timelike, and spatial. The statement suggests otherwise and should be corrected. — Rgdboer (talk) 03:12, 21 December 2017 (UTC)

Watch Out!

Many aspects of this article are semi-correct, misleading and irritating. Do not trust the statements made in here and assume that common standards in formulae are wrong.

Example: Boosts do NOT preserve spacelike vectors, they preserve the FACT that a vector is spacelike, not the vector itself. Moreover, timelikeness and spacelikeness is preserved by ALL Lorentz transformations, this is NOT characteristic of boosts or rotations, as suggested in the article. While this has been corrected in parts, there still is a problem pedagogically: It is by the very definition of the Lorentz transformation that timelikness and spacelikeness is preserved. The way it is written it still can be read as characterization of boosts resp. rotations.

You are replicating the comment right above. I wouldn't disagree with you the pedagogy is suboptimal. Can you suggest or perform unexceptional improvements, instead of shaking warnings at the public? Cuzkatzimhut (talk) 17:02, 18 May 2018 (UTC)

Removed oracular remark on name. The origin of the name should be apparent in the night sky section. Cuzkatzimhut (talk) 19:36, 18 May 2018 (UTC)

Null rotations ?

This article has a list of references but lacks inline references, particularly for the dubious "null rotation". These types are said to be parabolic transformations, such as the translations found in the Mobius group. But parabolic transformations have only one fixed point (at infinity), while rotations and boosts have two (their center and infinity). Without a reference this assertion of null rotations can be questioned. Furthermore, given that Lorentz and Mobius groups differ on having parabolic subgroups perhaps the isomorphism is also suspect (though in that case we have "reliable sources" claiming the isomorphism ! ) — Rgdboer (talk) 01:19, 3 May 2019 (UTC)

In fact, Galilean transformations, which are geometrically shear transformations, are the parabolic transformations of space and time. The naive physics of absolute space and time uses these motions to relate observers. These transformations are not in the Lorentz group. — Rgdboer (talk) 22:19, 3 May 2019 (UTC)

A search for "null rotation" with mathscinet turned up two articles : MR3534994, MR1926987 The first concerns surfaces generated by rotation in a null line, the second says a null rotation is a boost + rotation. Neither reference supports the Lorentz group element under discussion here.

Ronald Shaw uses the term for the identity map in MR0253695. The term appears to be used in the context of Newman-Penrose formalism ( MR970004 ) but there is no mention of null rotations in that article. — Rgdboer (talk) 01:08, 5 May 2019 (UTC)

Searching the Encyclopedia turns up "null rotation" as the identity map in several articles. In the § Covering group one finds the following unreferenced statement:

(In applications to quantum mechanics, the special linear group SL(2, C) is sometimes called the Lorentz group.)

As it stands, the article uses the Mobius-Lorentz group isomorphism to assert the null rotations. Reliable source needed. Rgdboer (talk) 21:45, 5 May 2019 (UTC)

Among the distinctions between groups, the Möbius group is not a Lie group (though it has six degrees of freedom as a projective linear group in M(2,C) ); both groups acts on manifolds but they are different manifolds. A question has been posted at Wikipedia:Reference_desk/Mathematics#Möbius—Lorentz_group_isomorphism_?. — Rgdboer (talk) 22:46, 8 May 2019 (UTC)

A detailed and topological approach to the isomorphism, with no mention of Null rotation:

• Waldyr Muniz Olivia (2002) Geometric Mechanics, appendix B: Mobius transformations and the Lorentz group, pages 195 to 221, Springer ISBN 3-540-44242-1 MR1990795

Connection within the groups is identified as an issue. — Rgdboer (talk) 22:57, 9 May 2019 (UTC)

The two structures are topological groups. Removing {1} from the Lorentz group separates it into two components, rotations and boosts. The Mobius group does not come apart from removing a point. — Rgdboer (talk) 23:21, 9 May 2019 (UTC)

Null rotations are a standard topic in general relativity, especially if Roger Penrose is involved. They are absolutely key, because they tell you how to move around on the light cone, which is something you need to know while falling into a black hole. Check out flag manifold and generalized flag variety and twistor theory and surrounding topics. 67.198.37.16 (talk) 01:57, 4 December 2020 (UTC)

Overall structure and content of this article

I'm rather deeply unhappy about the overall structure and content of this article. The lede opens promisingly:

For example, the following laws, equations, and theories respect Lorentz symmetry:

• The kinematical laws of special relativity
• Maxwell's field equations in the theory of electromagnetism
• The Dirac equation in the theory of the electron
• The Standard model of particle physics

The Lorentz group expresses the fundamental symmetry of space and time of all known fundamental laws of nature. In general relativity physics, in cases involving small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as that of special relativity physics.

Yayy!! And then after that, physics is never-ever mentioned again. Instead we get treated to a vast quantity of nearly incomprehensible stuff about loxodromic, etc. subgroups. Now, I actually spent more than a few months on the mobius transform, and so I actually know the loxodromic lingo (I even wrote chunks of the articles that dig into it, if I recall.) But, from the physics standpoint, this type of material never-ever appears in any of the standard textbooks in E&M, reviews of special relativity, intro-to-quantum, etc. because it is simply not needed to understand kinematics, maxwells eqns, spinors etc. The current contents of this article has very little or almost nothing to do with what any basic undergraduate would ever encounter or need to know about the Lorentz group. There needs to be at least:

• A review of the standard terminology and notation
• Much greater articulation of the relationship to spinors and all things spin-related
  • e.g. the relationship to bispinor
  • e.g. that SL(2,C) is isomorphic to the symplectic Sp(2,C) which is where the spinors come from! (!!!)
• A review of at least the basic representation theory. Sure, there is some other article, devoted to the representation theory, but this article should cover at least the basics - at least 6 or 8 paragraphs, at least 6 or 8 formulas.
• A basic demonstration of Lorentz invariance/covariance of some aspect of E&M, maybe maxwells eqns
• A basic demonstration of Lorentz invariance of some aspect of spin, e.g. either weyl or dirac eqn.
• More about Casimir operators, e.g. that mass is invariant.
• Greater usage of terminology like "spacelike", "timelike" in reference to the one/two sheeted hyperbolas.
• Much much more about the "Weyl" representation (the map with the Pauli matrices). Right now, its written as if it was "one weird trick" clickbait that fell from the sky, pronounced by some oracle at Delphi, when in fact its fundamental.
• Use the Lie-algebra stuff as a teaching moment for Lie groups. The Lorentz group, specifically, SL(2,C) is an excellent foot-in-the-door for Lie algebras. This article could be written to explain Lie algebras to the reader, instead of expecting the reader to already know them, before starting... (sl(2,C) is the defacto #1 most commonly used ingredient when constructing the semi-simple Lie algebras (in the textbooks that I can recall)).
• At least a few breaths of air about infinite-dimensional reps, at the very end.

Other dings:

• The use of eta for some real parameter of some subgroup is unfortunate, because eta is the conventional symbol for the flat spacetime metric.
• The night sky stuff is intellectually interesting, but irrelevant from any practical standpoint, it needs to be moved elsewhere.
• Stop it with the first person plural "We can easily..." style. It quite literally makes things harder to understand.

I'd like to suggest that maybe 80% of the content of this article (most of the content, starting with "Mobius group", and most of everything after that) be moved to a new article, Subgroups of the Lorentz group, and we should start all over again, with a clean slate, explaining the Lorentz group following bog-standard textbook presentations. Starting at a level that a college undergraduate might get in some class on special relativity. Yes, getting this done would be a huge amount of work, but really... Simply by nuking much of the existing content might allow new, better content to grow and fill the space. Sigh. I hate being a bombastic jerk, but this article is just appallingly bad. 67.198.37.16 (talk) 03:49, 4 December 2020 (UTC)

Error in "Commutation relations of the Lorentz algebra"

I believe the definition given for $M^{\mu\nu}$ is inconsistent with the commutation relations given. The section defines what it calls $(M^{\mu\nu})_{\rho\sigma}$ purely in terms of delta functions: basically, for $\mu < $\nu$, as a basis of anti-symmetric matrices, each with +1 in an UR and -1 in the corresponding LL entry. Notice $\mu$ and $\nu$ are already raised and the metric $\eta$ is not involved in the definition. If matrix multiplication is defined in the usual way (that also does not involve the metric), i.e. $(AB)_{\alpha\beta} = A_{\alpha\kappa} B_{\kappa\beta}$, then there is no way that commutator between $M^{\mu\nu}$ and $M^{\rho\sigma}$ can depend on the metric tensor $\eta$, as the next line asserts that it does. For example, with the definition given, $[M^{01},M^{02}] = -M^{12}$, not $-\eta^{00} M^{12}$. My only guess as to what the author means here is that he wants to define $(AB)_{\alpha\beta} = A_{\alpha\kappa} B_{\lambda\beta} \eta^{\kappa\lambda}$. But matrix multiplication is matrix multiplication -- once you define the entries, you have to multiply in the usual way. Ichbin4 (talk) 07:40, 21 July 2023 (UTC)

Some beautiful figures that might be nice to reproduce

At this twitter thread (archive) some beautiful illustrations from Penrose & Rindler's "Spinors and Spacetime", pages 27 and 29 I believe. They show the motion of null directions (as a celestial sphere), for boosts, rotations, boost and rotation ("four-screw"), and null rotations. I think they could work well as animations. --Nanite (talk) 18:16, 26 February 2024 (UTC)