This page is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
We read: Lorentz transformations which ... preserve orientation are called proper, and as linear transformations they have determinant +1.. It is unclear whether the "proper" transformations are those of the restricted group, whch preserve orientation, or those of the restricted group plus those which reverse orientation and time, which have determinant +1. 87.240.241.192 ( talk) 13:45, 18 October 2012 (UTC)
To do: add section on Lie algebra, and one on representations of the Lorentz group. Discuss relationship of the Lorentz group to special and general relativity. -- Fropuff 16:45, 11 Feb 2004 (UTC)
It is a 6-dimensional noncompact Lie group which is neither connected, nor simply connected.
Can something be simply connected without being connected? Josh Cherry 04:44, 18 Nov 2004 (UTC)
Technically speaking, no. What is meant is that the connected components of the Lorentz group are themselves not simply connected. We could say this instead but it's a little more wordy. -- Fropuff 06:21, 2004 Nov 18 (UTC)
Hi all, I added a bunch of improvements to this article yesterday, but unfortunately the server lost all my work! I am trying again today.
I plan to
Because of the close connection with Möbius transformations, I plan to also improve the article on Möbius transformations, putting the more mathematical material in the latter article, and putting more of the physical interpreation in this one.
At present, Wikipedia lacks a suitable discussion of
Eventually, I hope to repair this gap by writing new articles and revising more old ones as appropriate.--- CH (talk) 2 July 2005 22:09 (UTC)
OK, I've pretty much carried out the above plans. I still need to add a figure showing the lattice of subalgebras (up to conjugacy).
I also need to create figures showing the flow lines of parabolic, elliptic, hyperbolic, and loxodromic flow lines. I think I see how to create .png images, but does anyone know how to create an animated picture which can be used on Wiki, say using Maple? Are anitmated .gif images acceptable?-- CH (talk) 3 July 2005 04:05 (UTC)
Article states:
Perchance the diff eq you are looking for is the Picard-Fuchs equation? or are you looking for the hypergeometric differential equation? By point symmetry group, do you mean the monodromy group? The Mobius transforms are the monodromy group of this differential equation.
I think this is incorrect, I think its not the group of holomorphisms but the monodromy action which are the Mobius transforms. I started to try to write this up in Riemann's differential equation, (scroll to the bottom, fractional linear transformations), and the cluster of related articles (e.g.the bottom of the article hypergeometric differential equation), but got distracted. I mean to come back later and finish this. -- linas 3 July 2005 18:11 (UTC)
Hmm... I seem to already be over the recommended size limit, but haven't even mentioned the irreps or invariants. (These are important in physics as well as math, so there is a case that they should be mentioned in this article.) I think there should be at least short sections briefly indicating at least some of the most basic results. How serious is it that the article seems to be over 30 KB? I'll stop adding more until someone let's me know about this.-- CH (talk) 3 July 2005 19:29 (UTC)
Whoever changed complex closed curve to complex projective line, I don't think you are helping, since existing articles will only confuse readers if they go there from this article. My idea was to leave red links until improved articles can be written. The red links might encourage some of you to write the missing articles! If you disagree, maybe we should discuss this here before you make more apparently minor changes of wording? If you guess wrong about my intentions, you could change a correct statement into an incorrect one-- CH (talk) 3 July 2005 19:45 (UTC)
P.S. To mention some specific red links I'd like to leave red for the present: I intend to write articles on Bianchi group (this should even be a category with separate articles for each group) and Kleinian geometry. Someone else can write one on deformation retracts. Can the well intentioned but misleading redirect for complex closed curve be removed? Possibly someone can write a proper article on twistor theory and then this "controversy" can be cleared up by having this article refer to that one.-- CH (talk) 3 July 2005 19:51 (UTC)
FYI, be aware that here at UT Austin, the math dept calls SL(2,O_K) the Bianchi group. See [1] and [2] and [3] Perhaps you should call it Bianchi classification? For upper triangular matrices, we have the article Borel subgroup. linas 17:49, 10 July 2005 (UTC)
This article is getting a little long, even though more needs to be said. Perhaps we should split off the Lie algebra stuff to its own article (e.g Lie algebra of the Lorentz group). We should probably also start an article on representations of the Lorentz group. -- Fropuff 8 July 2005 23:50 (UTC)
Okay, just a suggestion. I was actually thinking more in terms of readablity then length. The length doesn't really bother me. Certainly, a little more needs to be said about the Lie algebra. Although you object to Linas's notation for the Pauli matrices (see below), I think it is wrong not to include mention of the matrices themselves in this article. The fact that generate the Lorentz algebra is both important and useful. For the math student who hasn't seen these before, it takes only 10 seconds to look at the definition and be satisfied.
Regarding the introductory pedantic comment, I have no objections to removing it. I think it adds very little. -- Fropuff 03:20, 11 July 2005 (UTC)
I am somewhat surprised on skimming the new article that no mention is made of the spinor component notation. These are the Pauli matrices
with the vector index μ running from 0..3 and the spinor indeces a and running from 1,2. Thus a four-vector could be represented by a pair of spinors by making the contraction
and likewise a pair of spinors could be made into a vector. I really liked this notation, as it made very clear that there are two distinct reps, a complex 2-D and a real 4-D rep, and that it was the isomorphism of lorentz to SL(2,C) is actually given by . It also makes clear how SU(2) is a double covering of SO(3). It also provides a bridge to the reps of SU(3) via which is important to quark physicists. Back when I was learning supersymmetry, the vector-spinor-index notation was de-rigeur. So I am somewhat surprised to not see this in this article. All in all, coming from an old-fashioned physics background, this article, as currently structured, comes off as a very non-standard treatment of the Lorentz group that would leave many practicing physicists scratching thier heads.
Let me put it another way: the intro mentions Maxwell eqns, special rel, and Dirac eqn. None of these require any knowledge of the parabolic/elliptic/etc. distinctions. Nor do these need the the Bianchi subgroup bits or talk about stabilizers. The third requires the spinor algebra notation, but little else. So the article promises to talk about physics, but then does anything but. The night-sky null-vector bit is interesting, but would normally be considered a curiosity ... yet it gets prominent billing in the article. This is confusing. linas 01:53, 11 July 2005 (UTC)
Let me put it a third way: physicists are usually insterested only in the spin-1/2, the spin-1, the spin 3/2 (the supersymetric -ino/gauge ghost things) and spin-2 (the GR tensor). The current article fails to distinguish spin 1/2 from spin-1 and makes no mention of the other two. linas 02:27, 11 July 2005 (UTC)
Hey, Sorry, I was not trying to question your experience or judgement. This is not its not a bad article; seems you did a good job. But you deeply misunderstand what I'm interested in and where I'm coming from; I have absolutely no desire whatsoever in writing about the Lorentz group; my interests lie elsewhere; this conversation is a pleasent distraction with a newcomer to wikipedia. I applaud your work; I'm glad you're here; I was merely trying make helpful comments.
However, I am under the impression that most math students couldn't care less about the Lorentz group; they study other things. The people who really care about the Lorentz group are physicists. There are two classes of physicists: those who need a text-book style treatment so that they can design particle accelerators or teach Maxwell and Dirac equations. For them, a clear statement of representation theory matters, and this article is currently lacking clear talk about representations. The other class of physicists are the string theorists, who are interested in Riemann surfaces and Fuchsian groups and the like; but thier level of needs and understanding are a few light-years beyond this article. Its not clear that either group is very well served by the current article. Don't get me wrong, if I was back in school hitting this for the first time, I would find it absolutely fascinating. Its a good article. But if I was in school, I'd also be concerned at how little it overlapped my textbooks. OK, now for the hard knocks: the current treatment is very different from e.g. the Moshe Carmeli treatment or chapters 10,11 of Fulton & Harris. There are standard names for the generators of the algebra, these aren't even named in this article.
I would have been happier if the effort put into conjugacy classes had been put into the article on the Mobius transforms instead, which duplicates a lot of this material. The treatment of the Lie algebra is also unusual; a more standard treatment talks about the structure constants and the generators. The bit about subgroups, covering groups and topology could indeed be moved to another article. To conclude, let me be clear: this is a good article. But as to culture and subjectiveness, WP readers are seeped in culture; I won't be the first or last to think or say what I just said. Although I've already said far far too much.
OK, in fact, I write really bad articles, but that's me :) I hold everyone else up to a higher standard. So yes, I live in the proverbial glass house, and yet I'm chunking rocks. Really, glad to have you here. Please take this kindly. linas 04:53, 11 July 2005 (UTC)
I've only been participating in Wikipedia for month or so. It's horrible that I am already involved in some kind of edit war.
A few weeks ago I completely reorganized and added much, much new material to this article. I thought hard about what to say, how to say it and in what order, what notation to use, etc. I worked really hard on this article. My goal was to collect and clearly present the most generally useful facts (for a broad audience) about the Lorentz group in the most elementary way possible. For example, I conciously attempted to remain as far as possible in the world of matrix Lie groups, because most undergraduates are far more likely to grasp matrices and systems of ordinary differential equations than more abstract concepts from graduate level Lie theory courses or string theory courses which I feel belongs in companion special topics articles written to address the needs of more sophisticated audiences.
A user called Linas seems to have a very very different ideas about the goals of an article on the Lorentz group. For example he complained above that in this article I've ignored the needs of students of string theory. Likewise, he has every different ideas about what topics should be emphasized. In particular, he wants to see in this article much, much more about representations, including infinite dimensional representations(!), than I do. As should be clear from the above, I too would in fact like to see at least one companion article on a special topic which is very important but too complicated to discuss in this article without unbalancing it. Accordingly I proposed what I feel is very reasonable procedure which I would hope could make everyone happy:
I even said I was going to add a few paragraphs on representations to this article, giving my own very brief summary of the basic facts, which I am confident I can do in a way which fits in gracefully with the rest of this article.
Unfortunately, it seems that Linas wants to delete material which I worked hard to include and to explain in an elementary way, in favor of much more advanced material which he wants to add. However, adding this material in my view would
Is it really too much to ask that Linas adhere to my request above, given that I've worked very hard on this and given that he and I seem to have completely different visions about what to say and how to say it? Is there someway to appeal to the Wiki maintainers to mediate this dispute? Unfortunately, by now I am so disgusted by my interactions with Linas that I want nothing to do with him ever again, but perhaps someone who "knows" us both (on Wiki) can relay my request as above.
Again, I stress that I myself believe that there is a great deal one could legitimately say about the Lorentz group, and I conciously chose not to attempt to say very much in this article about some important topics, particuarly representations. After all, the Lorentz group and its closest relatives, particularly SL(2,C) has been the subject of entire books from various points of view, books which string theorists must perhaps master. So I think that breaking up articles into an elementary one (this article) and the more elaborate one focusing on the needs of string theory students, which Linas outlined above, is a very reasonable solution.
Does anyone other than Linas really think that my proposal is unreasonable?
Is there some formal appeals process for mediating this dispute?
Thanks!--- CH (talk) 04:24, 17 July 2005 (UTC)
Charles, please, this is really getting frustrating. I don't know how I could possibly have made it clearer that I feel that the basic facts about representations of Lorentz group should be briefly described in an elementary way in this article. I even said that I intend to do this. The dispute with Linas is what I said it is above: whether to remove the material I added (see the history page) and replace it with high level material on representations, which I think inappropriate, or to have Linas write a companion article.
Give me a chance, OK--- I have other pans in the fire, and I didn't get to it 20 minutes after writing the above. Just wait and see what I do in the next few days, OK?
As for NPOV tag, well, one of the pans I just mentioned was searching for information about dispute tags. The proposed "inclusion" template would have been perfect, but apparently that proposal was rejected long ago.
Thanks for your comments, but you failed to say whether you agree with my proposal? If so, what don't you like about it? (Maybe you should reread the entire page above to make sure you have a reasonable idea of what issues I have in mind, which have already been discussed, such as length.) And can you help me with my question about how to resolve this dispute? I am trying to avoid an edit war with Linas.--- CH (talk) 05:45, 17 July 2005 (UTC)
Hi, Charles, thanks. I don't really disagree with anything you say, but given the concerns on my talk page I have concluded that I am not after all cut out to contribute to Wikipedia. (I've tried to explain this at some length on my user page.)
This page is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
We read: Lorentz transformations which ... preserve orientation are called proper, and as linear transformations they have determinant +1.. It is unclear whether the "proper" transformations are those of the restricted group, whch preserve orientation, or those of the restricted group plus those which reverse orientation and time, which have determinant +1. 87.240.241.192 ( talk) 13:45, 18 October 2012 (UTC)
To do: add section on Lie algebra, and one on representations of the Lorentz group. Discuss relationship of the Lorentz group to special and general relativity. -- Fropuff 16:45, 11 Feb 2004 (UTC)
It is a 6-dimensional noncompact Lie group which is neither connected, nor simply connected.
Can something be simply connected without being connected? Josh Cherry 04:44, 18 Nov 2004 (UTC)
Technically speaking, no. What is meant is that the connected components of the Lorentz group are themselves not simply connected. We could say this instead but it's a little more wordy. -- Fropuff 06:21, 2004 Nov 18 (UTC)
Hi all, I added a bunch of improvements to this article yesterday, but unfortunately the server lost all my work! I am trying again today.
I plan to
Because of the close connection with Möbius transformations, I plan to also improve the article on Möbius transformations, putting the more mathematical material in the latter article, and putting more of the physical interpreation in this one.
At present, Wikipedia lacks a suitable discussion of
Eventually, I hope to repair this gap by writing new articles and revising more old ones as appropriate.--- CH (talk) 2 July 2005 22:09 (UTC)
OK, I've pretty much carried out the above plans. I still need to add a figure showing the lattice of subalgebras (up to conjugacy).
I also need to create figures showing the flow lines of parabolic, elliptic, hyperbolic, and loxodromic flow lines. I think I see how to create .png images, but does anyone know how to create an animated picture which can be used on Wiki, say using Maple? Are anitmated .gif images acceptable?-- CH (talk) 3 July 2005 04:05 (UTC)
Article states:
Perchance the diff eq you are looking for is the Picard-Fuchs equation? or are you looking for the hypergeometric differential equation? By point symmetry group, do you mean the monodromy group? The Mobius transforms are the monodromy group of this differential equation.
I think this is incorrect, I think its not the group of holomorphisms but the monodromy action which are the Mobius transforms. I started to try to write this up in Riemann's differential equation, (scroll to the bottom, fractional linear transformations), and the cluster of related articles (e.g.the bottom of the article hypergeometric differential equation), but got distracted. I mean to come back later and finish this. -- linas 3 July 2005 18:11 (UTC)
Hmm... I seem to already be over the recommended size limit, but haven't even mentioned the irreps or invariants. (These are important in physics as well as math, so there is a case that they should be mentioned in this article.) I think there should be at least short sections briefly indicating at least some of the most basic results. How serious is it that the article seems to be over 30 KB? I'll stop adding more until someone let's me know about this.-- CH (talk) 3 July 2005 19:29 (UTC)
Whoever changed complex closed curve to complex projective line, I don't think you are helping, since existing articles will only confuse readers if they go there from this article. My idea was to leave red links until improved articles can be written. The red links might encourage some of you to write the missing articles! If you disagree, maybe we should discuss this here before you make more apparently minor changes of wording? If you guess wrong about my intentions, you could change a correct statement into an incorrect one-- CH (talk) 3 July 2005 19:45 (UTC)
P.S. To mention some specific red links I'd like to leave red for the present: I intend to write articles on Bianchi group (this should even be a category with separate articles for each group) and Kleinian geometry. Someone else can write one on deformation retracts. Can the well intentioned but misleading redirect for complex closed curve be removed? Possibly someone can write a proper article on twistor theory and then this "controversy" can be cleared up by having this article refer to that one.-- CH (talk) 3 July 2005 19:51 (UTC)
FYI, be aware that here at UT Austin, the math dept calls SL(2,O_K) the Bianchi group. See [1] and [2] and [3] Perhaps you should call it Bianchi classification? For upper triangular matrices, we have the article Borel subgroup. linas 17:49, 10 July 2005 (UTC)
This article is getting a little long, even though more needs to be said. Perhaps we should split off the Lie algebra stuff to its own article (e.g Lie algebra of the Lorentz group). We should probably also start an article on representations of the Lorentz group. -- Fropuff 8 July 2005 23:50 (UTC)
Okay, just a suggestion. I was actually thinking more in terms of readablity then length. The length doesn't really bother me. Certainly, a little more needs to be said about the Lie algebra. Although you object to Linas's notation for the Pauli matrices (see below), I think it is wrong not to include mention of the matrices themselves in this article. The fact that generate the Lorentz algebra is both important and useful. For the math student who hasn't seen these before, it takes only 10 seconds to look at the definition and be satisfied.
Regarding the introductory pedantic comment, I have no objections to removing it. I think it adds very little. -- Fropuff 03:20, 11 July 2005 (UTC)
I am somewhat surprised on skimming the new article that no mention is made of the spinor component notation. These are the Pauli matrices
with the vector index μ running from 0..3 and the spinor indeces a and running from 1,2. Thus a four-vector could be represented by a pair of spinors by making the contraction
and likewise a pair of spinors could be made into a vector. I really liked this notation, as it made very clear that there are two distinct reps, a complex 2-D and a real 4-D rep, and that it was the isomorphism of lorentz to SL(2,C) is actually given by . It also makes clear how SU(2) is a double covering of SO(3). It also provides a bridge to the reps of SU(3) via which is important to quark physicists. Back when I was learning supersymmetry, the vector-spinor-index notation was de-rigeur. So I am somewhat surprised to not see this in this article. All in all, coming from an old-fashioned physics background, this article, as currently structured, comes off as a very non-standard treatment of the Lorentz group that would leave many practicing physicists scratching thier heads.
Let me put it another way: the intro mentions Maxwell eqns, special rel, and Dirac eqn. None of these require any knowledge of the parabolic/elliptic/etc. distinctions. Nor do these need the the Bianchi subgroup bits or talk about stabilizers. The third requires the spinor algebra notation, but little else. So the article promises to talk about physics, but then does anything but. The night-sky null-vector bit is interesting, but would normally be considered a curiosity ... yet it gets prominent billing in the article. This is confusing. linas 01:53, 11 July 2005 (UTC)
Let me put it a third way: physicists are usually insterested only in the spin-1/2, the spin-1, the spin 3/2 (the supersymetric -ino/gauge ghost things) and spin-2 (the GR tensor). The current article fails to distinguish spin 1/2 from spin-1 and makes no mention of the other two. linas 02:27, 11 July 2005 (UTC)
Hey, Sorry, I was not trying to question your experience or judgement. This is not its not a bad article; seems you did a good job. But you deeply misunderstand what I'm interested in and where I'm coming from; I have absolutely no desire whatsoever in writing about the Lorentz group; my interests lie elsewhere; this conversation is a pleasent distraction with a newcomer to wikipedia. I applaud your work; I'm glad you're here; I was merely trying make helpful comments.
However, I am under the impression that most math students couldn't care less about the Lorentz group; they study other things. The people who really care about the Lorentz group are physicists. There are two classes of physicists: those who need a text-book style treatment so that they can design particle accelerators or teach Maxwell and Dirac equations. For them, a clear statement of representation theory matters, and this article is currently lacking clear talk about representations. The other class of physicists are the string theorists, who are interested in Riemann surfaces and Fuchsian groups and the like; but thier level of needs and understanding are a few light-years beyond this article. Its not clear that either group is very well served by the current article. Don't get me wrong, if I was back in school hitting this for the first time, I would find it absolutely fascinating. Its a good article. But if I was in school, I'd also be concerned at how little it overlapped my textbooks. OK, now for the hard knocks: the current treatment is very different from e.g. the Moshe Carmeli treatment or chapters 10,11 of Fulton & Harris. There are standard names for the generators of the algebra, these aren't even named in this article.
I would have been happier if the effort put into conjugacy classes had been put into the article on the Mobius transforms instead, which duplicates a lot of this material. The treatment of the Lie algebra is also unusual; a more standard treatment talks about the structure constants and the generators. The bit about subgroups, covering groups and topology could indeed be moved to another article. To conclude, let me be clear: this is a good article. But as to culture and subjectiveness, WP readers are seeped in culture; I won't be the first or last to think or say what I just said. Although I've already said far far too much.
OK, in fact, I write really bad articles, but that's me :) I hold everyone else up to a higher standard. So yes, I live in the proverbial glass house, and yet I'm chunking rocks. Really, glad to have you here. Please take this kindly. linas 04:53, 11 July 2005 (UTC)
I've only been participating in Wikipedia for month or so. It's horrible that I am already involved in some kind of edit war.
A few weeks ago I completely reorganized and added much, much new material to this article. I thought hard about what to say, how to say it and in what order, what notation to use, etc. I worked really hard on this article. My goal was to collect and clearly present the most generally useful facts (for a broad audience) about the Lorentz group in the most elementary way possible. For example, I conciously attempted to remain as far as possible in the world of matrix Lie groups, because most undergraduates are far more likely to grasp matrices and systems of ordinary differential equations than more abstract concepts from graduate level Lie theory courses or string theory courses which I feel belongs in companion special topics articles written to address the needs of more sophisticated audiences.
A user called Linas seems to have a very very different ideas about the goals of an article on the Lorentz group. For example he complained above that in this article I've ignored the needs of students of string theory. Likewise, he has every different ideas about what topics should be emphasized. In particular, he wants to see in this article much, much more about representations, including infinite dimensional representations(!), than I do. As should be clear from the above, I too would in fact like to see at least one companion article on a special topic which is very important but too complicated to discuss in this article without unbalancing it. Accordingly I proposed what I feel is very reasonable procedure which I would hope could make everyone happy:
I even said I was going to add a few paragraphs on representations to this article, giving my own very brief summary of the basic facts, which I am confident I can do in a way which fits in gracefully with the rest of this article.
Unfortunately, it seems that Linas wants to delete material which I worked hard to include and to explain in an elementary way, in favor of much more advanced material which he wants to add. However, adding this material in my view would
Is it really too much to ask that Linas adhere to my request above, given that I've worked very hard on this and given that he and I seem to have completely different visions about what to say and how to say it? Is there someway to appeal to the Wiki maintainers to mediate this dispute? Unfortunately, by now I am so disgusted by my interactions with Linas that I want nothing to do with him ever again, but perhaps someone who "knows" us both (on Wiki) can relay my request as above.
Again, I stress that I myself believe that there is a great deal one could legitimately say about the Lorentz group, and I conciously chose not to attempt to say very much in this article about some important topics, particuarly representations. After all, the Lorentz group and its closest relatives, particularly SL(2,C) has been the subject of entire books from various points of view, books which string theorists must perhaps master. So I think that breaking up articles into an elementary one (this article) and the more elaborate one focusing on the needs of string theory students, which Linas outlined above, is a very reasonable solution.
Does anyone other than Linas really think that my proposal is unreasonable?
Is there some formal appeals process for mediating this dispute?
Thanks!--- CH (talk) 04:24, 17 July 2005 (UTC)
Charles, please, this is really getting frustrating. I don't know how I could possibly have made it clearer that I feel that the basic facts about representations of Lorentz group should be briefly described in an elementary way in this article. I even said that I intend to do this. The dispute with Linas is what I said it is above: whether to remove the material I added (see the history page) and replace it with high level material on representations, which I think inappropriate, or to have Linas write a companion article.
Give me a chance, OK--- I have other pans in the fire, and I didn't get to it 20 minutes after writing the above. Just wait and see what I do in the next few days, OK?
As for NPOV tag, well, one of the pans I just mentioned was searching for information about dispute tags. The proposed "inclusion" template would have been perfect, but apparently that proposal was rejected long ago.
Thanks for your comments, but you failed to say whether you agree with my proposal? If so, what don't you like about it? (Maybe you should reread the entire page above to make sure you have a reasonable idea of what issues I have in mind, which have already been discussed, such as length.) And can you help me with my question about how to resolve this dispute? I am trying to avoid an edit war with Linas.--- CH (talk) 05:45, 17 July 2005 (UTC)
Hi, Charles, thanks. I don't really disagree with anything you say, but given the concerns on my talk page I have concluded that I am not after all cut out to contribute to Wikipedia. (I've tried to explain this at some length on my user page.)